Category: CBSE Notes

By going through these CBSE Class 11 Chemistry Notes Chapter 2 Structure of Atoms, students can recall all the concepts quickly.

Structure of Atoms Notes Class 11 Chemistry Chapter 2

Sub-Atomic Particles: Dalton’s Atomic Theory regarded the atom as the ultimate particle of matter. It explained satisfactorily various laws of chemical combination like the law of conservation of mass, the law of constant composition, and the law of multiple proportions. However, it failed to explain the existence of sub-atomic particles which were later discovered like electrons and proton.

Discovery of Electron: Michael Faraday suggested the particular nature of electricity. When he passed electricity through a solution of an electrolyte, chemical reactions occurred at the electrodes with liberation and deposition of matter at the electrodes.

Michael Faraday discovered the sub-atomic particles like electrons from his well-known experiments in partially evacuated glass tubes called cathode ray discharge tubes. The cathode ray tube is made of glass containing two thin pieces of metal, called electrodes, sealed in it.

At very low pressure and at high voltage, current starts flowing through a stream of particles from cathode to anode. These were called cathode rays or cathode ray particles

Characteristics of Cathode rays:

1. The cathode rays start from the cathode and move towards the anode.
2. These rays travel in straight lines.
3. Cathode rays are made up of material particles.
4. On applying an electric field, these rays are deflected towards the positive plate. This shows that cathode rays carry a negative charge. These negatively charged particles are Electrons.
5. Cathode rays produce a heating effect.
6. They produce X-rays when they strike against the surface of hard metals like tungsten, molybdenum, etc.
7. They produce green fluorescence when they stride zinc sulfide.
8. They affect the photographic plates.
9. They possess a penetrating effect.
10. They possess the same charge/mass ratio.

\(\frac{\text { Charge }}{\text { Mass }}=\frac{e^{-}}{m}\) = 1.76 × 10⁸ coulombs/g Mass m
Charge = e^– = 1.60 × 10^-19 coulombs or 4.8 × 10^-10 esu

Thus the mass of electron m
= \(\frac{e}{e / m}=\frac{1.60 \times 10^{-19}}{1.76 \times 10^{8}}\)
= 9.11 × 10^-28 g
= 9.11 × 10^-31 kg
Thus, it is concluded that electrons are the basic constituent of all atoms.

The amount of deviation of the particles from their path in the presence of an electrical or magnetic field depends upon

• The magnitude of the negative charge on the particle, the greater the magnitude of the charge on the particle, the greater is the deflection.
• The mass of the particle-lighter the particle, the greater is the deflection.
• The strength of the electrical or magnetic field.

The deflection of electrons from their original path increases with the increase in the voltage or strength of the magnetic field.

Thus electron can be defined as the fundamental particle which carries one unit negative charge and has a mass nearly equal to \(\frac{1}{1837}\)th of that of the hydrogen atom.

Discovery of Protons and Newtons: Anode rays or Canal rays: If a perforated cathode is used in the discharge tube experiment, it is found that certain type of radiations also travels from anode to cathode.

Production of Anode rays or Positive rays

Thus anode rays are not emitted from the anode but are produced in the space between the anode and cathode.

Properties of Positive rays/canal rays:

1. The anode-rays originate in the region between two electrodes in the discharge tube.
2. These rays are made of material particles.
3. These rays are positively charged.
4. These rays produce heat when striking against a surface.
5. The magnitude of the charge on anode-rays varies from particle to particle depending on the number of electrons lost by an atom or molecule.
6. The mass of positive particles which constitute these rays depend upon the nature of the gas in the tube.
7. The charge/mass (e/m) ratio of anode-rays is not constant but depends upon the nature of gas in the tube. The value of e/m is greatest for the lightest gas, hydrogen.

The electric charge on the lightest positively charged particle from hydrogen gas was found to be exactly equal in magnitude but opposite in sign to that of the electron. This lightest positively charged particle from hydrogen gas was named a proton. The mass of a proton is almost 1836 times that of the electron.

When hydrogen gas is taken inside the tube
Charge on these particles = 1.6 × 10^-19 coulomb
\(\frac{\text { Charge }}{\text { mass }}\) = 958 × 10⁴ coulombs/g for each particle
∴ mass on each particle = \(\frac{1.6 \times 10^{-19}}{9.58 \times 10^{4}}\) = 1.67 × 10^-24 g

This mass is nearly the same as that of hydrogen atom.

Therefore, a Proton may be defined as the fundamental particle which carries one unit positive charge and has a mass nearly equal to that of the hydrogen atom.

Chadwick in 1932 discovered the 3rd sub-atomic neutral particle and named it Neutron. He bombarded a thin sheet of beryllium by a-particles to discover neutrons. Neutron is a neutral particle carrying no charge and has a mass slightly greater than that of proton/

Thomson Model of Atom:
J.J. Thomson proposed that an atom is a sphere (radius approximately 10-10 m) of positive electricity and electrons are embedded into like the seeds of watermelon. An important figure of this model is that the mass of the atom is uniformly distributed over the atom.

Thomson model of the atom

Later on, this model was rejected as Sphere electrons are mobile. Thomson model of the atom

Rutherford’s Nuclear Model of Atom: Rutherford bombarded very thin gold foil with a-particles.

The observations of this a-particle scattering experiment were:

1. Most of the a-particles passed through the gold foil undeflected.
2. A small fraction of the a-particles was deflected through small angles.
3. A very few a-particles (~ 1 in 20,000) bounced back, that is, were deflected by nearly 180°.

Conclusions:

1. Atom is hollow from within. There is empty space within the atom as most of the a-particles passed undeflected.
2. A few positively charged a-particles were deflected. These must have been deflected by some positively charged body present within the atom. This positively charged body is very small as compared to the size of the atom.
3. Calculations by Rutherford showed that the radius of this positive center called Nucleus is only 10^-15 m as compared to the radius of the atom which is about 10^-10 m.

Rutherford’s scattering experiments

On the basis of the above observations and conclusion, Rutherford proposed the nuclear model of the atom.
1. The positive charge and most of the mass of the atom were densely concentrated in an extremely small region. This very small portion of the atom was called the nucleus by Rutherford.

Scattering of a-particles by
(a) a single atom (b) a group of atoms

2. The nucleus is surrounded by electrons that move around the nucleus at a very high speed in circular paths called orbits. Thus, Rutherford’s model of the atom is similar to the solar system in which the nucleus is like the sun and moving electrons are like revolving planets.

3. Electrons and the nucleus are held together by electrostatic forces of attraction.

Atomic Number and Mass Number: Atomic Number (Z) is the number of protons present in the nucleus.
As an atom is electrically neutral, the no. of protons in the nucleus is equal to the number of electrons moving outside it.
No. of protons in hydrogen (Z) = 1
= no. of electrons in a neutral atom

No. of protons in sodium (Z) = 11
= no. of electrons in a neutral atom

While the positive charge of the nucleus is due to protons, the mass of the nucleus, due to both protons and neutrons. As both protons and neutrons are present in the nucleus, they are collectively called Nucleons.

The total no. of nucleons is termed as the Mass number (A) = No. of protons (Z) + No. of neutrons (n)
∴ No. of neutrons n = A – Z

→ Isobars and Isotopes: The composition of any atom of symbol, X can be represented by _Z^AX.

→ Isobars are defined as the atoms of different elements with the same mass number but a different atomic number, e.g., ₆¹⁴C and ₇¹⁴N

→ Isotopes are the atoms of the same element with the same atomic number but different mass numbers. Protium (₁¹H), deuterium (₁²D), and tritium (₁³T) are the isotopes of hydrogen.

Similarly, ₁₇³⁵Cl and ₁₇³⁷Cl are the isotopes of chlorine.

The chemical properties of atoms are controlled by the number of electrons which are determined by the no. of protons in the nucleus. No. of neutrons present in the nucleus have very little effect on the chemical properties of an element. Thus all the isotopes of a given element show the same chemical behavior.

→ Drawbacks of Rutherford Model of Atom:
1. According to Maxwell, charged particles when moving, dissipates energy in the form of electromagnetic radiations.
Slowly, the distance between the moving electron from the nucleus decreases. Calculations show that it should take an electron only 10^-8 to spiral into the nucleus.

But this does not happen. Thus Rutherford’s model cannot explain the stability of an atom.

2. Another serious drawback of the Rutherford model is that it says nothing about the electronic structure of atoms, i.e., how the electrons are distributed around the nucleus and what are the energies of these electrons.

→ Developments leading to Bohr’s Model of Atom: Neils Bohr improved upon the model of the atom as proposed by Rutherford.

Two developments played a major role in the formulation of Bohr’s model of the atom.

1. Electromagnetic radiations possess both wave-like and particle-like properties.
2. Quantization of electronic energy levels in atoms.

→ Wave Nature of Electromagnetic Radiation: Maxwell was the first to suggest that charged bodies moving under acceleration, produce alternating electrical and magnetic fields. These fields are transmitted in the form of waves called electromagnetic waves or electromagnetic radiations.

Properties associated with electromagnetic wave motion:

1. Electric and magnetic fields are perpendicular to each other and both are perpendicular to the direction of propagation of the wave.
2. These waves do not require medium and can move in a vacuum.
3. There are many types of electromagnetic radiations that differ from one another in wavelength (or frequency). They constitute electromagnetic spectrum (shown below). A visible part of the spectrum (around 10¹⁵ Hz) is only a small part of it.
4. Different kinds of units are used to represent electromagnetic radiation.
   SI unit for frequency (v -nu) is hertz (Hz, s^-1).

It is defined as the number of waves that pass through a given point in space in one second.
SI unit for wavelength (λ) should be a meter.
(a) Wavelength (λ): It is the distance between two consecutive points which are in the same phase along the direction of propagation. Depending upon the magnitude, the wavelength is expressed either in cm, micron, millimicron, Angstrom unit, or in nanometer.
1 Angstrom (Å) = 10^-8 cm = 10^-10 m
1 micron (g) = 10^-4 cm = 10^-6 m.
1 nanometer = 10^-9 m

(b) Frequency (v): It is defined as the number of wavelengths travelled in one second. Therefore,
v = \(\frac{\text { Velocity of the radiation }}{\text { Wavelength of the radiation }}=\frac{c}{\lambda}\)

The frequency is expressed in cycles per second or in Hertz (Hz) units.
1 Hz = 1 cycle/s

(c) Wave number (\(\bar{v}\)): It is defined as the number of wavelengths which can be accommodated in one cm length along the direction of propagation. Therefore,
Wave number (\(\bar{v}\)) = \(\frac{\text { Frequency of the radiation }(v)}{\text { Velocity of the radiation }(c)}=\frac{v}{c}\)

The wave number is generally expressed in the units of cm^-1, although it is not the SI unit.
Frequency = Velocity × wave number
or
v = c\(\bar{v}\)

Relationship between velocity, wavelength, and frequency of wave:
c = V × λ
where c = velocity, v = frequency, λ = wavelength

Electromagnetic spectrum: The different types of electromagnetic radiations differ only in their wavelengths and have frequencies.

The wavelengths increase in the following order.
Cosmic rays < γ-rays < X-rays < Ultra-violet rays < Visible rays < Infrared < Microwaves < Radio waves

(a) The spectrum of electromagnetic radiation

(b) Visible spectrum: The visible spectrum is onlv a small part of the entire spectrum

In a vacuum, all types of electromagnetic radiations, regardless of wavelength, travel at the same speed, i.e., 3.0 × 10⁸ ms^-1
This is called the speed of light and is given the symbol ‘c’.

Particle Nature of Electromagnetic Radiation: Planck’s Quantum Theory
The wave nature of electromagnetic radiation could explain experimental phenomena such as diffraction and interference. However, the experimental observations such as the emission of, radiation from a hot body, and the photoelectric effect could not be explained in terms of the wave nature of light.

→ Black body radiation: All hot bodies emit electromagnetic radiation. At high temperatures, a part of these radiations lies in the visible region of the spectrum With a further increase in the temperature of the body, the proportion of the higher frequency radiation increases. An ideal body that emits and absorbs radiations of all frequencies is called a black body.

Max Planck found that the characteristics of black body radiation could be accounted for by proposing that each electromagnetic oscillator, viz., an atom or a molecule, can emit or absorb only a certain discrete quantity of energy. This limitation of the energy of an object to discrete values is called the quantization of energy. According to Planck, the energy of an oscillator of frequency v is restricted to an integral multiple of the quantity, hv, where h is called the Planck’s constant.

→ Plank gave the name quantum to the smallest quantity of energy that can be emitted or absorbed in the form of electromagnetic radiation. The energy (E) of a quantum of radiation is proportional to its frequency (v) and is expressed as
E = h v

h is called Planck’s constant and has the value 6.626 × 10^-34 Js.
Thus, according to Planck
E = n h v
where n = 0,1, 2 , h = 6.626 × 10^-34 Js

The smallest amount of energy (n = 1) is then given by
E = h v

The energy to h v is called a quantum of energy that can be emitted or absorbed in the form of electromagnetic radiation.

→ The energy of Electromagnetic Radiation: All electromagnetic radiations are associated with a certain amount of energy.

According to Einstein

1.The radiation energy is emitted or absorbed in the form of small packets of energy. Each such packet of energy is called a quantum or photon. Each quantum has a certain discrete amount of energy associated with it.

2. Energy associated with a quantum or photon (e) is proportional to the frequency (v) of the radiation
Then E ∝ v
or
E = hv

where h is a constant called Planck’s constant. This constant (h) has value of 6.626 × 10^-34 Joule second (Js) or 3.99 × 10^-13 kJs mol^-1
The above relation may be written as ε = \(\frac{h c}{\lambda}\)

3. The energy associated with Avogadro’s number (N) of quanta is called an Einstein of energy (E). Thus, the Einstein of energy associated with the radiation of frequency v is. E = N_A hv

Photoelectric Effect: When a beam of light of suitable wavelength falls on a clean metal plate (such as cesium) in a vacuum, electrons are emitted from the surface of the metal plate. This phenomenon involving the emission of electrons from the surface of a metal by the action of light is known as the photoelectric effect. The electrons so emitted are called photoelectrons.

Photoelectric effect

The three important facts about the photo-electric effect observed are
1. The electrons are ejected only if the radiation striking the surface of the metal has at least a certain minimum frequency called threshold frequency (v_o). If the frequency is less than v_o, no electrons are ejected.
This value (v_o) is called Threshold Frequency. The minimum energy required to eject the electron (hv_o) is called the work function.

2. The velocity (and hence the kinetic energy) of the electron ejected depends upon the frequency of the incident radiation and is independent of its intensity.

3. The number of photoelectrons ejected is proportional to the intensity of incident radiation.

The above observations cannot be explained by the Electromagnetic wave theory. According to this theory, since radiations are continuous, therefore it should be possible to accumulate energy on the surface of the metal, irrespective of its frequency and thus radiations of all frequencies should be able to eject electrons.

Similarly, according to this theory, the energy of the electrons ejected should depend upon the lire intensity of the incident radiation.

If the striking photon of light has energy = hv and the minimum energy required to eject the electron is hv_o then the difference of energy (hv – hv_o) is transferred as the kinetic energy of the photoelectron
\(\frac{1}{2}\)hv – hv_o = h(v – v_o)

where m = mass of the electron and v is the velocity of the ejected electron.

Dual Behaviour of Electromagnetic Radiation: The particle nature of light can explain the black body radiation and photoelectric effect satisfactorily but cannot explain the known wave behavior of light like the phenomenon of interference and diffraction. Therefore, light possesses dual behavior either as a wave or as a stream of particles.

When radiation interacts with matter, it displays particle-like properties. When it propagates, it displays wave-like properties like diffraction and interference. Some microscopic particles like electrons also exhibit this wave-particle duality.

→ Evidence for the Quantized Electronic Energy Levels: Atomic Spectra: Atoms give discontinuous or line spectra. The spectrum given by atoms consists of a series of bright lines or bands separated from each other by a dark space. Each line in the spectrum corresponds to a specific wavelength.

There are two types of atomic spectra

1. Atomic emission spectra,
2. Atomic absorption spectra

1. Atomic emission spectra: A series of bright lines, separated from each other by dark spaces, produced by the excited atoms is called atomic emission spectra.

Each line in the emission spectrum corresponds to a specific wavelength. Therefore, each element gives a unique pattern of lines in the spectrum. No two elements give the same pattern of lines in their spectra.

2. Atomic absorption spectra: When a sample of atomic vapors is placed in the -path of white light from an arc lamp, it absorbs the light of certain characteristic wavelengths, and the light of other wavelengths gets transmitted. This produces a series of dark lines on a white background.

The spectrum of Hydrogen Atom: The spectrum of a hydrogen atom can be obtained by passing an electric discharge through the gas taken in the discharge tube under pressure. The spectrum consists of a large number of lines appearing in different regions of wavelengths. The lines in different regions were grouped into five different series of lines, each being named after the name of its discoverer.

These are the Lyman series. Balmer series, Paschen series, Brackett series and Pfund series. Lyman series appear in the ultraviolet region, Balmer series appear in the visible region while the other three series lie in the infrared region.

A simple relationship between the wavelengths of different lines can be given as
\(\frac{1}{λ}\) = \(\bar{v}\)(in cm^-1)
= R(\(\frac{1}{n_{2}^{2}}-\frac{1}{n_{1}^{2}}\))

where n₁ and n₂ are integers, such that n₁ > n₂. R is a constant, now called the Rydberg constant. The value of R is 109678 cm^-1. This expression is found to be valid for all the lines in the hydrogen spectrum and is also known as Rydberg equation.

For a given spectrum series, n₂ remains constant while n1 varies from line to line in the same series. For example, for Lyman series n₂ = 1 and = 2, 3, 4, 5 and for Balmer series n₂ = 2 and n₁ = 3, 4, 5

…. All the five series, the regions in which lines appear and the values of n₁ and n₂ are given below:


Emission or atomic spectrum of hydrogen

Of all the elements, the hydrogen atom has the simplest line spectrum.

Bohr’s Model For Hydrogen Atom Postulates:
1. The electron in the hydrogen atom can move around the. the nucleus in a circular path of fixed radius and energy. These paths are called orbits, stationary states, or allowed energy states.

2. The energy of an electron in the orbit does not change with time. However, it jumps from a lower energy level to a higher energy level where the requisite amount of energy is supplied to it and jumps from a higher orbit to a lower orbit with the release of energy.
ΔE = E₂ – E₁
where ΔE = change in energy, E₂ = energy of the electron in the higher orbit, E₁ = energy of the electron in the lower orbit.

3. The frequency of the radiation absorbed or emitted is given by
v = \(\frac{\Delta \mathrm{E}}{h}=\frac{\mathrm{E}_{2}-\mathrm{E}_{1}}{h}\)

4. The angular momentum of an electron in a given stationary state can be expressed as
mvr = \(\frac{n h}{2 \pi}\); n = 1, 2,3 2n
Thus an electron can move only in those orbits for which its angular momentum is an integral multiple of \(\frac{h}{2 \pi}\) (Quantization of angular momentum).
That is why only certain fixed orbits are allowed.

(a) The stationary states for electrons are numbered n = 1, 2, 3… . They are called Principal quantum numbers.
(b) the radii of stationary states are expressed as
r_n = n²a₀
where a_o = 52.9 pm

Thus the radius of the first orbit called Bohr radius is 52.9 pm (as n = 1).

(c) Energy of the electron in a given orbit
E_n = – R_H\(\left[\frac{1}{n^{2}}\right]\) where n = 1, 2, 3, ……….
R_H is called Rydberg Constant and its value is 2.18 × 10^-18 J. The energy of the lowest state, also called the ground state is
E₁ = – 2.18 × 10-18(\(\frac{1}{1^{2}}\)) = – 2.18 × 10^-18 J
For n = 2
E₂ = – 2.18 × 10-18(\(\frac{1}{2^{2}}\)) = – 0.545 × 10^-18 J

Significance of the negative sign before the electronic energy E_n: The energy of the electron in a hydrogen atom has a negative sign for all possible orbits. A free-electron at rest far away place from the nucleus has energy = 0, i.e., E_∞ = 0. As the electron gets closer to the nucleus (n decreases) En becomes larger in absolute value and more and more negative. Thus the most negative energy given by n = 1 corresponds to the most stable orbit.

(d) Bohr’s theory can also be applied to ions containing only one ‘ electron like hydrogen. For example He⁺, Li²⁺, Be³⁺ and so on. For them
E_n = – 2.18 × 10^-18(\(\frac{\mathrm{Z}^{2}}{n^{2}}\)) J and radii
r_n = \(\frac{52.9\left(n^{2}\right)}{Z}\) pm
where Z = atomic number. It has a value of 2, 3 for helium and lithium respectively.

(e) Magnitude of the velocity of the electron increases with the increase in nuclear charge and decreases with the increase’ of principal quantum numbers.

→ Explanation of Line Spectrum of Hydrogen: The energy difference between the two orbits is given by
ΔE = E_f – E_i
E_f, E_i energies in final and initial orbits
ΔE = \(\left(\frac{-\mathrm{R}_{\mathrm{H}}}{n_{f}^{2}}\right)-\left(\frac{\mathrm{R}_{\mathrm{H}}}{n_{i}^{2}}\right)\)

n_f, n_i are final and initial orbits

In the case of the absorption spectrum, n_f > n_i energy is absorbed.
In the case of emission spectrum n_i > n_f; ΔE is negative and energy is released.

Advantages of Bohr’s Model:

1. It explains the stability of the atom. An electron can not lose energy as long as it stays in a particular orbit.
2. It explains the line spectrum of hydrogen.

Different series in the hydrogen spectrum

Drawbacks of Bohr’s Model: Bohr’s model of atom suffers from the following weaknesses or limitations.
1. Inability to explain line spectra of multi-electron atoms: Bohr’s theory was successful in explaining the line spectra of the hydrogen atom and hydrogen-like particles, containing a single electron only. However, it failed to explain the line spectra of multi-electron atoms.

When spectroscopes with better resolving powers were used, it was found that even in the case of hydrogen spectrum, each line was split up into a number of closely spaced lines (called fine structure) which could not be explained by Bohr’s model of the atom.

2. Inability to explain Zeeman effect (splitting of lines in the magnetic field and stark effect (splitting of lines in the electric field)

3. Unable to explain the three-dimensional model of the atom. Bohr’s model gives a flat model of the atom with electrons moving in circular paths in one plane.

4. It does not explain the shapes of molecules.

5. It fails to explain de Broglie’s concept of the dual nature of matter and Heisenberg’s uncertainty principle.

Towards Quantum Mechanical Model of the Atom:
1. Dual Behaviour Matter: de-Broglie suggested that matter and hence electron-like radiations have a dual character – wave and particle. In other words, matter also possesses particles as well as Wave characters. This concept of the dual character of matter gave birth to the wave mechanical theory of matter according to which, the electrons, protons, and even atom when in motion possess all wave properties. Mathematically, de Broglie view may be written as below:
λ = \(\frac{h}{m v}\) …(1)

The equation (1) is known as de Broglie equation, m = mass of the particle, v = velocity of the particle, h = Planck’s constant, λ is the wavelength.

Since h is constant, its value is 6.6256 × 10^-34 Js
∴ λ ∝ \(\frac{1}{m v}\)
or
λ ∝ \(\frac{1}{\text { Momentum }}\)
(mv = momentum of a photon) … (2)

Equation (2) is known as de Broglie’s relationship which may be stated as the momentum of a particle in motion is inversely proportional to the wavelength of the waves associated with it.

2. Heisenberg’s Uncertainty Principle: One of the important consequences of the dual nature of an electron is the Uncertainty Principle, developed by Heisenberg. According to the Uncertainty Principle, it is impossible to determine simultaneously at any given moment both the position and momentum (velocity) of an electron with accuracy.

Mathematically, if Δx and Δp are the uncertainties in the position and momentum respectively, then
ΔxΔp ≥ \(\frac{h}{4 \pi}\)

One can see from this equation that if Ap increases, the Ax decreases and vice-versa. Since, Δp = m. Δv, hence the above equation can be written as
Δx × Δv >\(\frac{h}{4 \pi m}\)

Significance of Uncertainty Principle: One of the important implications of the Heisenburg Uncertainty Principle is that it rules out the existence of definite paths or trajectories of electrons and other similar particles.

The effect of the Heisenburg Uncertainty Principle is significant only for the motion of microscopic objects and is negligible for two macroscopic objects.

In dealing with milligram-sized or heavier objects, the value of Δv Δx is extremely small and insignificant and the associated uncertainties are hard of any real consequence. Therefore the precise statements of the position and momentum of electrons have to be replaced by the statements of probability, that the electron has at a given position and momentum. This is what happens in the quantum mechanical model of the atom.

→ Reasons for the failure of the Bohr Model: In the Bohr model, an electron is regarded as a charged particle moving in well-defined circular orbits about the nucleus. The wave character of the electron is not considered.

Bohr’s model of the hydrogen atom, therefore, not only ignores the dual behavior of matter but also contradicts Heisenburg’s Uncertainty Principle.

→ Quantum Mechanical Model of Atom: Quantum mechanics was developed independently by Heisenburg and Schrodinger.

Schrodinger equation is Ĥ φ = Eφ where Ĥ is a mathematical operator called Hamiltonian, E is the total energy of the system and φ is the wave function.

Important features of the quantum mechanical model of the atom:

1. The energy of the electrons in atoms is quantized.
2. The existence of quantized electronic energy levels is allowed solutions of the Schrodinger Wave Equation.
3. Both the exact position and exact velocity of an electron in an atom cannot be determined simultaneously. Therefore, only the probability of finding an electron at different points is required.
4. An atomic orbital is the wave function \p for an electron in an atom.
5. The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function, i.e., |φ|2 at that point. |φ|2 is called probability density and is always positive. From the value of | \p |2 at different points within an atom it is possible to predict the region around the nucleus where the electron will most likely be found.

Orbitals and Quantum Numbers:
→ Atomic Orbital: it is defined as the 3-dimensional region of space around the nucleus where the probability of finding an electron is maximum.

→ Quantum Numbers: The state of an electron in an atom is described by its location with respect to the nucleus and by its energy. Thus, the energy and angular momentum of an electron is quantized, i.e., an electron in an atom can have only certain permissible values of energy and angular momentum. These permissible states of an electron in an atom called Orbitals are identified by a set of four numbers. These numbers are called Quantum Numbers.

The various quantum numbers are
(a) Principal quantum numbers are denoted by n.
(b) Azimuthal or angular momentum quantum number denoted by l.
(c) Magnetic quantum number denoted by m.
(d) Spin quantum number denoted by s.

(a) Principal quantum number (n): This quantum number determines the main energy level or shell in which the electron in an atom is present and also the energy associated with it. In addition, it also determines the average distance of the electron from the nucleus in a particular shell. Starting from the nucleus, the energy shells are denoted as K, L, M, N, … etc., or as 1, 2, 3, 4, … etc: The maximum number of electrons that a shell can accommodate is 2n². Thus, K-shell (n = 1) can have a maximum of two electrons. L- shell (n = 2) can have eight electrons and similarly, eighteen electrons can be accommodated in M-shell (n = 3).

(b) Azimuthal or subsidiary or angular quantum number (l): This, the quantum number determines the angular momentum of the electron. This is denoted by l. The values of l give principal energy- shell in which an electron belongs. It can have positive integer values from zero to (n – 1) where n is the principal quantum number.
That is l = 0, 1,2, 3, …. (n – 1).

(c) Magnetic quantum number: This quantum number describes the behavior of an electron in a magnetic field. The values of ‘m’ are linked to that of l. For a given value of l, the possible values of m vary from -l to 0 and 0 to + l. Thus, the total values of m are (2l + 1). The orbitals are also named after the sub-shell in which these are present. The number of orbitals in different sub-shells are given:

(d) Spin quantum number: This quantum number describes the spin orientation of the .electron. It is designated by ‘s’. Since the electron can spin in only two ways-clockwise or anti-clockwise and, therefore, the spin quantum number can take only two values: + 1/2 or – 1/2. These two values are normally represented by two arrows pointing in the opposite direction i.e.↑ and ↓.

Shapes of atomic orbitals:
1. Shapes of s-orbitals: For s-orbitals, l = 0, hence the orbital angular momentum of an s-orbital is zero. As a result, the distribution of electron density is symmetrical around the nucleus and the probability of finding an electron for a given distance is the same at all angles. As the distribution of electron density is symmetrical, therefore, the most suitable figure to represent an s-orbital is a sphere.

(a) The probability of finding an electron is maximum near the nucleus and decreases with distance. In the case of 2s electrons, the probability is again maximum near the nucleus and then decreases to zero and increases again and then decreases as the distance from the nucleus increases. The intermediate region (a spherical shell) where the probability of finding an electron cloud is zero is called a Nodal surface/node. In general, any n orbital has (n – 1) nodes.

Shapes of is, 2s, and 3s orbitals

(b) The size and energy of the s-orbital increases as the principal quantum number n increases, i.e., size and energy of s-orbital increases in the order 1s < 2s < 3s ……

2. Shape of p-orbitals: For p-orbitals l = 1, so angular momentum of an electron in 2p orbital
= \(\sqrt{l(l+1)} \frac{h}{2 \pi}=\sqrt{2} \frac{h}{2 \pi}\)

As a result, the distribution of electron density around the nucleus is. not spherical. The probability diagram for a p-orbital is dumbbell shape. Such a diagram consists of a distorted sphere of high probability one on each side of the nucleus, concentrated along with N in a particular direction.

Shapes of p-orbital

Different orientatios of p-orbitais

Now, since the electron with l = 1 can have three values for the magnetic quantum number (m), i.e., m = – 1, 0 and + 1, hence there are three p-orbitals. All three 2p-orbitals have the same shape, but their directions are different. The directions are perpendicular to each other. Since these directions can be chosen as the x, y, z axes, hence the p-orbitals along these axes are labeled as p_x, p_y, and p_z respectively. The three p-orbitals of a particular energy level have equal energies and are called degenerate orbitals. 2p has no node, 3p has one 4p has two nodes, and so on. In general, no. of nodes in any orbital = (n – l – 1).

3. Shape of d-orbitals: For d-orbital, 1 = 2. Therefore, the angular momentum of -an electron in d orbitals is not zero. As a result, the d orbitals do not show spherical symmetry. For l = 2, the magnetic quantum number (m) should have five different values i.e., m = – 2, – 1, 0, 1, + 2. Accordingly, there are five different space orientations for d orbitals. These are designated as

The five d-orbitals

4. Energies of orbitals: In atoms, electrons can have only certain permissible energies. These permissible states of electrons are called energy levels.

In hydrogen and hydrogen-like atoms, all the orbitals having the same principal quantum number have the same energy. Thus, 2s and 2p orbitals have equal energies, 3s, 3p, and orbitals have equal energies, and 4s, 4p, and 4f orbitals have equal energies as shown above.

Energy level diagram for the few electronic shells of the hydrogen atom

The atoms containing two or more electrons are called multielectron atoms. In these atoms:
(a) Different orbitals having the same principal quantum number (n) may have different energies.
(b) For a particular main energy level, the orbital having a higher value of the azimuthal quantum number (l) has higher energy. For example, the energy of 2p orbital (l = 1) is higher than that of the 2s (l = 0) orbital, general, energies of the orbitals belonging to the same main energy level follow the order
s < p < d < f

Energy level diagram for the few electronic shells of a multi-electron atom

(c) When n > 3, the same orbitals belonging to a lower main energy level may have higher energy than some orbitals belonging to the higher main energy. For example, in the case of multi-electron atoms, the energy of 3d orbitals is higher than the energy of 4s orbitals.

(d) In multielectron atoms, the energy of any orbital is governed by both the principal quantum number (n) and azimuthal quantum number (l):

• The orbital having leaver (n + l) value has lower energy.
• For the orbitals having equal value of (n + 1), the orbital having lower value of n has lower energy. For example.
  4s orbital has n + l = 4 + 0 = 4 and
  3d orbital has n + l = 3 + 2 = 5.

Since, (n + l) value for 4s orbital is lower than that for 3d, hence 4s orbital has lower energy than 3d.

5. Filling of orbitals in atoms:
Aufbau principle: In the ground state of the atoms, the orbitals are filled in order of their increasing energies. In other words, electrons first occupy the lowest-energy orbital available to them and enter into higher energy orbitals only after the lower energy orbitals are filled.

The order in which the energies of the orbitals increase and hence the order in which orbitals are filled is as follows:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s …….

“n + l” Rule: The lower the value of (n + l) for an orbital, the lower is its energy, and hence earlier it will be filled. If two orbitals have the same value of (n + l), the orbital with the lower value of n will have lower energy. Hence it will be filled first.

Order of Filling Energy Levels (Aufbau Principle)

→ Pauli Exclusion Principle
No two electrons in an atom can have the same set of four quantum numbers.
OR
Only two electrons may exist in the same orbital and these electrons must have opposite spin.

Hund’s rule of maximum multiplicity: According to this rule, electron pairing will not take place in orbitals of the same energy (same subshell) until each orbital is singly filled. This principle is very (important in guiding the filling of p, d, and f orbitals, which have more than one kind of orbitals. For example, we know that there are three p orbitals. (p_x, p_y, and p_z) of the p-subshell in a principal energy level. According-” to Hund’s rule, each o.f the three p orbitals must get one electron of parallel spin before any one of them receives the second electron of opposite spin.

The electronic configuration of some atoms are given below:

After calcium, 3d subshell starts getting filled. The electronic configurations of elements from scandium (Z = 21) to zinc (Z = 30) are given below:

Total number of exchanges = 3+ 2 + l= 6
The number of exchanges that can take place in d5 configuration is as follows:

Gallium (Ga) Z = 31 to Krypton (Kr) Z = 36 (Electronic Configuration)
With Gallium (Ga) onwards, 4p orbitals get filled up as:

Important points to remember:

1. Mass No. (A) = Sum of protons and neutrons.
2. Atomic No. (Z) = No. of protons in the nucleus.
3. No. of neutrons = A – Z.
4. Nucleons are the particles (n + p) present in the nucleus.
5. Max. No. of electrons in a shell that can be present in an atom is given by 2n² where n = no. of the orbit.
6. An s-subshell can contain 2, a p-can contains 6, a d-can contain 10, and an f-subshell can contain 14 electrons, s-subshell has only one orbital; p-can has 3, d has 5; subshell has 7 orbitals.
7. Each orbital can maximum contains two electrons.
8. To form a cation from a neutral atom, electrons are removed equal to the no. of positive charges on the cation, while to form an anion from a neutral atom, electrons are added to the no. of negative charges on an anion.

Table Electronic Configurations of the Elements:

Elements with exceptional electronic configurations


Elements with exceptional electronic configurations.

→ Elements with atomic number 112 amid above have been reported but not yet frilly a the indicated and min med.

→ Electron: It is the fundamental particle that carries one unit negative charge and has a mass nearly equal to \(\frac{1}{1837}\) hydrogen atom.

→ Proton: A proton may be defined as that fundamental particle that carries one unit of positive charge and has a mass nearly equal to that of the hydrogen atom.

→ Neutron: A neutron may be defined as the fundamental particle which carries no charge but has a mass nearly equal to that of a hydrogen atom or proton.

→ Cathode rays: Cathode rays are a stream of electrons.

→ Electrons: Electrons are universal constituents of matter.

→ Mass Number (A): Sum of protons and neutrons.

→ Atomic Number (Z): Number of protons in the nucleus of an atom.

→ Nucleons: Sum of protons and neutrons.

→ Isotopes: Atoms of the same element having the same atomic number, but different mass numbers are called Isotopes.

→ Isobars: Atoms of different elements which have the same mass number, but a different atomic number are Isobars.

→ Isotones: Such atoms of different elements which contain the same number of neutrons are called Isotones.

The wavelength (λ) of a wave is the distance between any two consecutive crests or troughs.
1 Å = 10^-8 cm = 10^-10 m
1 nm = 10^-9 m, 1 pm = 10^-12 m

→ Frequency (v): It is the number of waves passing through a point in space in one second. Its unit is Hertz (Hz).
1 Hz = 1 cycle per second (cps)

→ Velocity (c): The velocity of a wave is defined as the linear distance traveled by the wave in one second. Its unit is cm per second or meters per second.

→ Amplitude (a): It is the height of the crest or depth of the trough of a wave It is expressed in units of length.

→ Wavenumber: It is defined as the number of waves present in one cm length. It is also defined as the reciprocal of the wavelength
\(\bar{v}\) = \(\frac{1}{λ}\)

Relationship between velocity, wavelength, and frequency of a wave
c = v × λ

→ Electromagnetic spectrum: When electromagnetic radiations of different wavelengths are arranged in order of their increasing wavelengths or decreasing frequencies, the complete spectrum obtained is called Electromagnetic Spectrum,

Cosmic rays < y-rays < X-rays < UV rays < visible < Infrared < Microwaves < radiowaves

→ Photon: Each packet of energy is called quantum. In the case of light, such a quantum is called Photon.

→ Black Body Radiation: If the substance being heated is a black body (which is a perfect absorber and perfect radiator of energy) the radiation emitted is called blackbody radiation.

→ Zeeman Effect: Splitting of spectral lines in the magnetic field.

→ Stark Effect: Splitting of spectral lines in the electric field.

→ Probability.: It is the best possible description of a situation that cannot be exactly described.

→ Orbit: It, is a well-defined circular path around the nucleus with . a fixed energy in which the electrons revolve.

→ Orbital: The three-dimensional region of space around the nucleus where there is a maximum probability of finding an electron.

→ Quantum Numbers may be defined as a set of four numbers that give complete information about the electron in an atom, i.e., energy, orbital occupied, size, shape, and orientation of that orbital, and the direction of electron spin.

Some Important Formulae:
→ c = v × λ
c = velocity;
λ = wavelength,
v = Frequency

→ E = hv
h = Planck’s coristt. = 6.625 × 10^-34 J sec
E = Energy of a photon

→ \(\bar{v}\) = \(\frac{1}{λ}\)
\(\bar{v}\) = wavenumber .
Charge on 1 electron = – 1
Change on 1 proton = 1 +
Charge on 1 neutron = 0
Mass of one electron = 9.1 × 10^-31 kg
Mass of a proton = 1.67 × 10^-27 kg .
Mass of neutron = 1.67 × 10^-27 kg
One unit charge = 4.8 × 10^-10 e.s.u.
= 1.6 × 10^-19 coulomb

Work of R.A. Millikan ,
Charge on one electron = 1.6 × 10^-19 coulomb
e/m for electron = 1.76 × 10⁸

∴ Mass of an electron = \(\frac{e}{e / m}=\frac{1.60 \times 10^{-19}}{1.76 \times 10^{8}}\)
= 9.11 × 10^-28 g

If X is an atom of an element

Mass of 1 Mole of electron
. = 9.11 × 10^-28 × 6.022 × 10²³ g = 0.55 mg

→ Photoelectric Effect: The phenomenon of the emission of electrons from the surface of certain metals (usually potassium, cesium, rubidium). When they are exposed to a team of light with certain minimum frequency called threshold frequency.
hv = hv₀ + \(\frac{1}{2}\) mv²

K.E. imparted to the ejected electron .
= \(\frac{1}{2}\) mv² = hv – hv₀

→ Line spectrum of hydrogen
\(\bar{v}\) = 109677(\(\frac{1}{2^{2}}-\frac{1}{n^{2}}\)) cm^-1
where n > 3, i.e. n = 3, 4, 5, ….
The value 109677 cm-1 is called Rydburg Constant.

→ de Broglie Equation
λ = \(\frac{h}{m \times v}=\frac{h}{p}\)
where p = momentum of the particle.

→ Heisenburg’s Uncertainty Principle

It is impossible to determine simultaneously both the position as well as momentum (or velocity) of a moving particle like an electron with absolute accuracy.

→ Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers.
Or
Only two electrons may exist in the same orbital and these electrons must have opposite spin.

→ The maximum number of electrons in the shell with principal quantum number n is equal to 2n².

→ Hund’s Rule of Maximum Multiplicity: Pairing of electrons in the orbitals belonging to the same subshell (p, d, or f) does not take place until each orbital belonging to that subshell has got one electron each, i.e., it is singly occupied.

→ Schrodinger Wave Equation: It is applicable to the wave nature of electrons.
Ĥ φ = E φ
where Ĥ is a mathematical operator called Hamiltonian operator.
or
\(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}}+\frac{8 \pi^{2} m}{h^{2}}\)(E – V)φ = 0

where φ is the amplitude of the wave, x, y, z are space coordinates E is the total energy of the electron, V is its potential energy m is the mass of the electron.

Studying from CBSE Class 11th Biology Revision Notes helps students to prepare for the exam in a well-structured and organised way. Making Biology Class 11 NCERT Notes saves students time during revision as they don’t have to go through the entire textbook. In CBSE Notes, students find the summary of the complete chapters in a short and concise way. Students can refer to the NCERT Solutions for Class 11 Biology, to get the answers to the exercise questions.

Class 11 Biology NCERT Notes | Notes of Biology Class 11

Class 11 Bio Notes | Bio Notes Class 11 | Bio Class 11 Notes

1. The Living World Class 11 Biology Notes
2. Biological Classification Biology Class 11 Notes
3. Plant Kingdom Biology Notes Class 11
4. Animal Kingdom Class 11 Bio Notes
5. Morphology of Flowering Plants Class 11 Notes
6. Anatomy of Flowering Plants Class 11 Notes
7. Structural Organisation in Animals Class 11 Notes
8. Cell: The Unit of Life Class 11 Notes
9. Biomolecules Bio Notes Class 11
10. Cell Cycle and Cell Division Class 11 Notes
11. Transport in Plants Bio Class 11 Notes
12. Mineral Nutrition Class 11 Notes Biology
13. Photosynthesis in Higher Plants Class 11 Notes
14. Respiration in Plants Class 11 Notes
15. Plant Growth and Development Class 11 Notes
16. Digestion and Absorption Class 11 Notes
17. Breathing and Exchange of Gases Class 11 Notes
18. Body Fluids and Circulation Class 11 Notes
19. Excretory Products and their Elimination Class 11 Notes
20. Locomotion and Movement Class 11 Notes
21. Neural Control and Coordination Class 11 Notes
22. Chemical Coordination and Integration Class 11 Notes

We hope students have found these CBSE Revision Notes of Class 11 Biology useful for their studies. If you have any queries related to Biology Class 11 NCERT Notes, drop your questions below in the comment box.

Studying from CBSE Class 11th Chemistry Revision Notes helps students to prepare for the exam in a well-structured and organised way. Making NCERT Notes Class 11 Chemistry saves students time during revision as they don’t have to go through the entire textbook. In CBSE Notes, students find the summary of the complete chapters in a short and concise way. Students can refer to the NCERT Solutions for Class 11 Chemistry, to get the answers to the exercise questions.

NCERT Notes for Class 11 Chemistry | Notes of Chemistry Class 11

Notes of Class 11 Chemistry | Chemistry Class 11 NCERT Notes

1. Some Basic Concepts of Chemistry Class 11 Notes
2. Structure of Atom Class 11 Notes
3. Classification of Elements and Periodicity in Properties Class 11 Notes
4. Chemical Bonding and Molecular Structure Class 11 Notes
5. States of Matter Class 11 Notes
6. Thermodynamics Notes of Class 11 Chemistry
7. Equilibrium Chemistry Notes Class 11
8. Redox Reactions Chemistry Class 11 Notes
9. Hydrogen Class 11 Chem Notes
10. The s-Block Elements Class 11 Notes
11. The p-Block Elements Class 11 Notes
12. Organic Chemistry Some Basic Principles and Techniques
13. Hydrocarbons Notes of Chemistry Class 11
14. Environmental Chemistry Class 11 Notes

We hope students have found these CBSE Class 11 Chemistry NCERT Notes useful for their studies. If you have any queries related to NCERT Notes of Chemistry Class 11, drop your questions below in the comment box.

By going through these CBSE Class 11 Physics Notes Chapter 15 Waves, students can recall all the concepts quickly.

Waves Notes Class 11 Physics Chapter 15

→ A wave is a form of disturbance that transmits energy from one place to another without the actual flow of matter as a whole.

→ Waves are of three types:

1. Mechanical waves,
2. e.m. waves,
3. matter waves.

→ Water waves or sound waves are called mechanical or elastic waves as they require a material medium for their propagation.

→ A material medium possesses both elasticities as well as inertia.

→ Light waves don’t require any material medium for their propagation.

→ Light waves are electromagnetic waves or non-mechanical waves which can propagate through a vacuum.

→ Matter waves are associated with moving electrons, protons, neutrons and other fundamental particles and even atoms and molecules.

→ The matter is constituted by electrons, protons, neutrons and other fundamental particles.

→ The waves associated with matter particles are called matter waves.

→ Matter waves arise in the quantum mechanical description of nature.

→ Wave motion is a form of disturbance that is due to the repeated periodic vibrations of the particles of the medium about their mean positions.

→ The motion is handed over from one medium particle to another without any net transport of the medium during wave motion.

→ Mechanical waves are of two types

1. transverse waves and
2. longitudinal waves.

→ A wave is said to be a progressive or travelling wave if it travels from one point of the medium to another.

→ The waves on the surface of the water are of two types: capillary waves and gravity waves.

→ The restoring force that produces capillary waves is the surface tension of water.

→ The restoring force that produces gravity waves is the pull of gravity which tends to keep the water surface at its lowest level.

→ The oscillations of the particles in gravity waves are not confined to the surface only but extend with diminishing amplitude to the very bottom.

→ The particle motion in water waves involves a complicated motion, they not only move up and down but also back and forth.

→ The waves in an ocean are a combination of both longitudinal and transverse waves.

→ Transverse and longitudinal waves travel at different speeds in the same medium.

→ k is called propagation constant or angular wavenumber.

→ S.I. unit of k is radian (rad) per metre of rad m^-1.

→ The speed of transverse waves in a string is determined by two factors:

1. Linear mass density i.e. mass per unit length (m),
2. Tension (T) in the string.

→ Positions of zero amplitude are called nodes.

→ Positions of maximum amplitude are called antinodes.

→ Nodes and antinodes are separated by \(\frac{λ}{4}\).

→ Two successive nodes or antinodes are separated by \(\frac{λ}{2}\).

→ Audible sound waves have a frequency between 20 Hz to 20,000 Hz.

→ The equation of a simple harmonic wave travelling in the positive X-direction is given by
y = A sin (ωt – kx)
where ω = \(\frac{2 \pi}{\mathrm{T}}\) = 2πv
k = \(\frac{2 \pi}{\lambda}\)

→ The particle velocity in a wave is given by v = \(\frac{\mathrm{dy}}{\mathrm{dt}}\)

→ Wave velocity is given by C = \(\frac{\mathrm{dx}}{\mathrm{dt}}\).

→ A wave travelling in negative x-direction is given by
y = A sin (ωt + kx)

→ The speed of sound does not depend on the frequency or wavelength.

→ Sound waves are mechanical waves that can’t propagate in a vacuum.

→ Sound waves can’t travel in sawdust or dry sand because the medium is not continuous.

→ The damping of sound in wood is much larger as compared to that in metals.

→ The higher the frequency of sound greater is the pitch of the sound.

→ The voice of ladies and children is of higher pitch than that of men.

→ Unit of loudness is bell (B) = log\(\frac{\mathrm{I}}{\mathrm{I}_{0}}\).

→ The sound is reflected and refracted according to the same laws as the light does.

→ The wavelength for ultrasonics is very small, therefore they are not diffracted by the ordinary objects or holes etc.

→ The speed of mechanical waves is determined by the properties of the medium i.e. elasticity and inertia and not by the nature, intensity, amplitude or shape of the wave.

→ The velocity of sound is the largest in hydrogen among the gases.

→ Monosyllabic sound is produced in about 0.2 s.

→ The vibrations of the prongs of a tuning fork are transverse and that of the stem are longitudinal.

→ The point where the stem of the tuning fork is connected to the prongs is an antinode.

→ The ends of the prongs are also antinodes.

→ There is a node between them that is nearer to the stem than the ends of the prongs.

→ The speed of sound in the air is not affected by the changes in pressure.

→ For every 1°C rise in temperature, the speed of sound increases by 0. 61 ms^-1.

→ Due to a change in temperature, the wavelength of sound waves is affected.

→ Beats are not audible if the beat frequency is more than 10 Hz.

→ If the prong of a tuning fork is loaded near the stem its frequency increases and when it is filled near the stem, the frequency decreases.

→ The number of beats produced per second is equal to the difference in the frequencies of the superposing notes.

→ In the progressive wave, the crest and troughs or compressions and rarefactions move with the speed of the wave.

→ When there is no relative motion between the source and listener, the Doppler’s effect is not observed.

→ When a source of sound moves, it causes a change in the wavelength of k the sound received by the listener.

→ When the listener moves, it causes a change in the number of waves ( received by the listener.

→ If source and listener move in mutually perpendicular .directions, no Doppler’s effect is observed.

→ Not Doppler’s effect is produced when only the medium moves.

→ A musical sound consists of a quick, regular and periodic succession of compressions and rarefactions without a sudden change in amplitude.

→ Pitch, loudness and quality are the characteristics of musical sound.

→ Pitch depends on frequency, loudness depends on intensity and quality depends on the number and intensity of overtones.

→ Pitch increases with an increase in frequency.

→ The ratio of the frequencies of the two nodes is called the interval between them. e.g. interval between 256 and 512 Hertz is 1: 2.

→ Two nodes are said to be in unison if their frequencies are equal i. e. if the interval between them is 1: 1.

→ Some other common intervals found useful in producing musical founds are as follows: octave (1: 2), major tone (8: 9), minor tone (9: 10), semitone (15: 16).

→ The fundamental note is called the first harmonic.

→ If n, be the fundamental frequency, then 2n₁, 3n₁, 4n₁, …. are respectively called second, third, fourth,…. harmonics respectively.

→ Harmonics are the integral multiples of the fundamental frequency.

→ Overtones are the notes of frequency higher than the fundamental frequency actually produced by the instrument.

→ In the strings, all harmonics are produced.

→ In the open organ pipe, all the harmonics are produced.

→ In the closed organ pipe, only the odd harmonics are produced.

→ In an open organ pipe as well as the string the second harmonics is the first overtone.

→ In the closed organ pipe, the third harmonic is the first overtone.

→ The ratio of the frequencies of the overtones in an open organ pipe is 2: 3: 4: 5:…

→ The ratio of the frequency of the overtones in the closed organ pipe is 3: 5: 7: …..

→ The frequency of the notes produced by the organ pipe varies:

1. directly as \(\sqrt{λ}\) , where λ, is a constant.
2. ∝ \(\sqrt{T}\) where T is the absolute temperature of the gas.
3. inversely as \(\sqrt{ρ}\) where ρ is the density of the gas.
4. inversely as length (l) of the tube.

→ The sound produced by the open organ pipe is comparatively pleasant as compared to that produced by the closed organ pipe.

→ The rarefactions are the regions of decrease in density or pressure and compressions are the regions of increase in density or pressure in air, gas when wave propagates through it.

→ Two waves travelling along the same path in the same or opposite direction superpose.

→ Superposition of waves gives rise to the phenomenon of interference, stationary waves and beats.

→ Interference of waves: Superposition of two waves of the same frequency and same wavelength travelling in the same direction with the same speed results in interference of waves.

→ Constructive interference: Interference is said to be constructive if two waves of the same frequency travelling in the same direction with the same speed superpose on each other such that the resultant displacement is more than the individual displacements.

→ Destructive interference: If the resultant displacement due to the superposition of two waves is less than the individual displacements then it is called destructive interference.

→ The wavelength of a wave: It is defined as the distance between two consecutive points (i.e. two consecutive troughs or crests) in the same phase of wave motion.

→ The fundamental mode of the first harmonic: It is defined as the oscillation mode with the lowest frequency.

→ Infrasonics: Sound waves of frequency less than 20Hz are called infrasonics. They can’t be heard by the human ear.

→ Beats: They are defined as the periodic variations in the intensity of sound due to the superposition of two sound waves of slightly different frequencies.

→ Mechanical or elastic waves: The waves set up and propagated due to the presence of a material medium and its properties of elasticity and inertial are called mechanical waves.

→ Electromagnetic waves: They are defined as the waves set up by the variation in electric and magnetic fields of an oscillating charge.

→ Transverse wave: It is defined as the wave motion set up due to vibrations of medium particles perpendicular to the direction of propagation of the wave.

→ Longitudinal wave: It is defined as the wave motion set up due to the vibrations of the medium particles along the direction of wave propagation.

→ Phase (Φ): It is defined as the argument of sine or cosine function representing a wave. It is the angular periodic position of a wave.

→ Time period (T): It is defined as the time taken by the medium particles to complete one oscillation.

→ Velocity of wave motion (v): It is defined as the ratio of wavelength to the time period i.e. v = \(\frac{λ}{T}\) = vλ, (∵ v = \(\frac{1}{T}\))

→ Stationary wave: It is defined as the wave due to the superposition of two progressive waves of the same frequency and amplitude but travelling in the opposite directions along the same line. It is also called a standing wave.

→ Harmonics: The wave of frequencies having integral multiples of a fundamental frequency are called harmonics of the fundamental wave including itself.

→ Overtones: They are defined as the waves of frequencies having integral multiples of a fundamental frequency but excluding it.

→ The 2nd harmonics is the first overtone, the third harmonics is 2nd overtone and so on.

→ Taut string: It is defined as a string vibrating in any mode/modes fixed at one end and loaded at the other end.

→ Musical sound: It is defined as a sound having series of harmonic waves following each other rapidly at regular intervals of time without a Sudden change in their amplitude. It produces a pleasant effect on the ear of the listener.

→ Noise: It is defined as a sound having series of harmonic waves following each other at irregular intervals of time with a sudden change in their amplitude. It produces a displeasing effect on the ear of the listener.

→ The intensity of sound at a point (I): It is defined as the amount of energy passing per unit time per unit area held perpendicular to the incident sound waves at that point.

→ Temperature coefficient of the velocity of sound (α): It is defined as the change in velocity of sound per Kelvin change in temperature.

→ Capillary waves: They are defined as the ripples of a fairly short wavelength not more than a few centimetres.

→ Gravity waves: They are defined as waves that have wavelengths typically ranging from several metres to several hundred metres.

→ Superposition Principle: It states that when two or more waves of the same nature travel in a medium, then the resultant displacement at a point is the vector sum of the displacement due to the individual waves.

→ Threshold of hearing or zero levels (I₀): It is defined as the lowest intensity of sound that can be heard by the human ear. It is about 10^-12 Wm^-2 for a sound of frequency I KHz.

→ The loudness of a sound: It ¡s defined as the degree of sensation of sound produced ¡n the car. It distinguishes between a loud and a faint sound.

→ Weber Fechner Law : It states that the loudness of sound is proportional to the logarithm of its intensity i.e. L = log₁₀ \(\left(\frac{\mathrm{I}}{\mathrm{I}_{0}}\right)\)

→ Bel (B): Loudness is said to be one bel if the intensity of sound is 10 times the threshold of hearing.

→ Pitch: It ¡s defined as that characteristic of musical sound which helps one to classify a note as a high note or low note.

→ Quality or Timber: Ills defined as that characteristic of musical sound which helps us to distinguish between.two sounds of the same intensity and pitch.

→ Musical scale: It consists of a series of flotes (frequencies) separated by definite and simple intervals so as to produce a musical effect when played in Succession.

→ Decibel (dB): \(\frac{1}{10}\)th of bel is called decible i.e. 1 dB = \(\frac{1}{10}\)B.

→ Keynote: The first note of the lowest frequency is called keynote. Octave: Two notes are said to be octave if the ratio of their frequencies is 2. It is also a musical scale called the diatomic scale which has 8 intervals (octave + 7 other intervals).

→ Shock wave: It is defined as the wave produced by a body moving with a speed greater than the speed of sound. Shock waves carry a large amount of energy and when strike a building rattling sound due to the vibration of the building is produced.

→ Mach number: It is defined as the ratio of the velocity of the body producing shock waves to the velocity of sound.
∴ Mach number = \(\frac{\mathrm{V}_{\mathrm{s}}}{\mathrm{v}}\)

→ Echo: It is defined as the repetition of the sound of short duration. It (echo) is heard if the minimum distance between the obstacle reflecting sound waves and the source of sound is 17 m.

→ Reverberation: It is defined as the persistence or prolongation of audible sound after the source has stopped emitting sound. It is due to multiple reflections of sound waves.

→ Reverberation time: It is defined as the time during which the intensity of sound falls to one million of its original value after the source has stopped producing it.

→ The acoustics of Building: It is that branch of science which deals with the design of big halls and auditoriums so that a speech delivered or music produced in them is distinctly and clearly heard at all places in the building.

Important Formulae:
→ Velocity of wave: v = vλ
v = frequency of oscillator generating the wave
λ = wavelength of the wave
v = velocity of wave

→ Velocity of transverse wave in a string:
v = \(\sqrt{\frac{T}{m}}=\sqrt{\frac{T}{\pi r^{2} \rho}}\), where

ρ = density of the material of string
r = radius of string
T = tension Applied on the string
m = mass per unit length of the string

→ Newton’s form ula for velocity of sound in air:
v = \(\sqrt{\frac{P}{\rho}}\)
P = air pressure
ρ = density of air

→ Velocity of elastic waves or longitudinal waves in a medium is:
v = \(\sqrt{\frac{E}{\rho}}\)
E = coefficient of elasticity of the medium
ρ = density of the medium

→ Leplace’s formula for velocity of sound is air/gases:
v = \(\sqrt{\frac{\gamma \mathrm{P}}{\rho}}\) where

E = γP = adiabatic elasticity of air/gas
ρ = density of air/gas
γ = C_P/C_V.

→ Velocity of wave in gas/liquid medium (Longitudinal wave):
V = \(\sqrt{\frac{Y}{\rho}}\), where

Y = Young’s modulus
ρ = coefficient of rigidity

→ Velocity as a function of:
1. temperature, \(\frac{v_{1}}{v_{2}}=\sqrt{\frac{T_{1}}{T_{2}}}\)

2. density, \(\frac{v_{1}}{v_{2}}=\sqrt{\frac{\rho_{1}}{\rho_{2}}}\)

→ The equation of a plane simple harmonic wave (progressive wave) travelling from left to right is:
y = A sin 2π(\(\frac{\mathrm{t}}{\mathrm{T}}-\frac{\mathrm{x}}{\lambda}\))
= A sin \(\frac{2 \pi}{\lambda}\)(vt – x)
= A sin (ωt – kx)
and from right to left i.e. along – X axis is obtained by replacing
x = -x, i.e. y = A sin \(\frac{2 \pi}{\lambda}\)(vt – x)

→ Phase difference = \(\frac{2 \pi}{\lambda}\) × path difference
or
ΔΦ = \(\frac{2 \pi}{\lambda}\) × Δx

→ Total energy transmitted per Unit volume in waves is given by
E = 2π² ρ v² A²
= \(\frac{2 \pi^{2} \rho v^{2} A^{2}}{\lambda^{2}}\)

→ Intensity of wave = \(\frac{2 \pi^{2} \rho v^{2} A^{2}}{\text { area } \times \text { time }}\)

→ I_max = (A₁ + A₂)².

→ I_min = (A₁ – A₂)².

→ Apparent frequency of sound when:
1. Source moves towards listener at rest is
ν’ = \(\frac{v}{v-v_{s}}\)ν

2. When source moves away from listener at rest is
ν’ = \(\frac{v}{v+v_{s}}\)ν

3. When listener moves towards source at rest is
ν’ = \(\frac{\mathbf{v}+\mathbf{v}_{0}}{\mathbf{v}}\)ν

4. When listener moves away from source at rest is
ν’ = \(\frac{\mathbf{v}-\mathbf{v}_{0}}{\mathbf{v}}\)ν

5. When both source and listener move towards each other
ν” = \(\frac{v-v_{0}}{v+v_{s}}\)ν

6. If both move away from each other, then
ν” = \(\frac{v-v_{0}}{v+v_{s}}\)ν

→ Sabine’s formula for reverberation time is
t = \(\frac{0.166 \mathrm{~V}}{\sum \alpha \mathrm{s}}\), Where
k = constant
V = volume of the hall
α = coefficient of absorption
s = area exposed to sound

→ Particle velocity at any instant in a progressive wave is
v = v_o cos 2π (\(\frac{t}{T}-\frac{x}{\lambda}\))
Where v_o = \(\frac{2 \pi}{\lambda}\) A = 2πAv
= velocity amplitude.

→ Particle acceleration at any instant of time in a progressive wave is
where a_o = a_o sin 2π (\(\frac{t}{T}-\frac{x}{\lambda}\))
where a_o = 4π² v² = ω²
= acceleration amplitude.

By going through these CBSE Class 11 Physics Notes Chapter 14 Oscillations, students can recall all the concepts quickly.

Oscillations Notes Class 11 Physics Chapter 14

→ All oscillatory motions are periodic motions but all periodic motions may not be oscillatory.

→ Oscillatory or Vibratory motions are harmonic motions of the simplest type, so they are called simple harmonic motions (S.H.M.).

→ Simple Harmonic Motion is defined as the projection of a uniform circular motion on any diameter of a cycle of reference.

→ The periodic motions are described by fundamental concepts of period$, frequency, amplitude, and displacement.

→ v is the number of oscillations per second.

→ A measurable physical quantity that changes with time is called displacement.

→ The phase difference between displacement and velocity is \(\frac{π}{2}\)

→ The phase difference between displacement and acceleration is π.

→ The S.H.M. is characterized by displacement, amplitude, period, frequency, velocity, acceleration, vibration, and phase.

→ Angular frequency (ω) is related to the period and frequency of the motion by: ω = 2πv = \(\frac{2π}{T}\) .

→ One full oscillation back and forth is known as a cycle or a vibration.

→ A liquid in a U-tube set in oscillations executes S.H.M. with a period T = \(\sqrt{\frac{h}{g}}\), where h is the rise or depression of liquid from the mean position in one limb,

→ The velocity amplitude (v_o) of S.H.M. and the acceleration amplitude (a_o) are related as follows:
a_o = ω v_o

→ The necessary and sufficient condition for a particle to execute S.H.M. is that the acceleration is directly proportional to the displacement and is always directed towards the mean position i.e. opposite to the displacement.

→ The work done by a simple pendulum in one complete oscillation is zero.

→ The total energy of S.H.M. is directly proportional to the square of the amplitude.

→ The total energy of S.H.M. is directly proportional to the square of the frequency.

→ The simple pendulum cannot oscillate in weightlessness but the spring can do so.

→ The driving force is a time-dependent force and can be represented by F(t) = f_o cos ωt = f_o cos 2πvt, v = frequency of driving force.

→ Restoring force must act on the particle executing S.H.M.

→ S.H.M. is represented by y = r sin ωt, where y = displacement of the particle, r = amplitude of oscillation of the particle.

→ Velocity of a particle executing S.H.M. is v = rω cosωt = ω\(\sqrt{r^{2}-y^{2}}\)

→ The maximum velocity of the particle is called velocity amplitude (v_o) which is equal to rω.

→ Acceleration of a particle executing a = – ω²y.

→ Acceleration amplitude (i.e. maximum acceleration), a_o = ω²r.

→ The velocity of a particle executing S.H.M. is zero at the extreme position and maximum at the mean position.

→ Acceleration is maximum at the extreme position and zeroes at the mean position.

→ The phase difference between velocity and acceleration is \(\frac{π}{2}\)

→ The time period of a simple pendulum is independent of its mass.

→ The graph between l and T² is a straight line in the case of a simple pendulum.

→ When length of the spring is made n times, its spring constant becomes \(\frac{1}{n}\) times and hence time period will increase \(\sqrt{n}\) times

→ When spring is cut into n equal pieces, the spring constant of each piece will become n times and hence time period will become \(\frac{1}{\sqrt{n}}\) times.

→ The time period of a simple pendulum is oo at the center of the earth because g = 0 at the center of the earth.

→ The time period of a simple pendulum decreases if it accelerates upward with an acceleration a.

→ The time period of a simple pendulum increase if it accelerates downward with an acceleration ‘a’.

→ The time period of the pendulum increases with an increase in length. If its length is increased n times, its time period becomes \(\sqrt{n}\) times.

→ The time period of a simple pendulum increase when it is immersed in a liquid of density σ.

→ The time period of a simple pendulum increase when the temperature of the wire of the bob is increased.

→ The length of a second’s pendulum is 99.3 cm ≈ 1 m.

→ The time period of a simple pendulum of infinite length is 84.6 minutes. In a medium, all oscillations are damped oscillations as their, amplitude decreases with time.

→ Oscillations of a simple pendulum in a room are damped ones.

→ For resonance, frequency of an applied periodic force = natural frequency of the body.

→ The energy-time graph of damped oscillations is shown in the figure here.

→ Periodic Motion: A motion that repeats itself after regular intervals of time is called periodic motion.

→ Oscillatory or Vibratory Motion: A periodic motion in which a body moves to and fro about a central fixed point (called mean position) is called the oscillatory or vibratory motion of the body of the particle.

→ Driving Force: A time-dependent force applied on an oscillator to increase its vibrations is called a driving force.

→ Second, ’s Pendulum: A pendulum whose time period is 2 seconds is called a second’s pendulum.

→ Undamped Oscillations: The oscillations whose amplitude does not change with time are called undamped oscillations. Such oscillations exist only in a vacuum.

→ Restoring Force: It is defined as the periodic force which comes into play when an object moves away from its equilibrium position while executing S.H.M.

→ Phase: The phase of a vibrating particle at any instant is its state regarding its displacement and direction of vibration at that particular instant. It is denoted by Φ. It is a function of time and is expressed as
Φ = ωt + Φ_o = \(\frac{2π}{T}\)t + Φ_o ………..(1)

→ Epoch: It is defined as the initial phase of the vibrating particle i. e. phase at f = 0. It is denoted by Φ_o. From (1), at t = 0, Φ = Φ_o.

→ Free Vibrations: When a body vibrates with its own natural frequency, it is said to execute free vibrations.

→ Forced Vibrations: When a body is maintained in a state of vibration by a strong periodic force of frequency other than the natural frequency of the body, the vibrations are said to be forced vibrations.

→ Resonant Vibrations: When a body vibrates with a frequency equal to its natural frequency of vibration, then the vibrations are called resonant vibrations.

→ Resonant vibrations are merely a special case of forced vibrations.

→ Coupled system: A system of two or more oscillators linked together in such a way that there is a mutual exchange of energy between them is called a coupled system.

→ Coupled Oscillations: The oscillations of a coupled system are called coupled oscillations.

→ Force Constant of Spring Constant (k): It is defined as the restoring force per unit displacement, i.e. k = \(\frac{F}{x}\), when F = force, x = displacement of particle executing S.H.M.

→ Phase Difference: The difference in phase angles of two positions of a body or oscillator in periodic motion is called phase difference.

→ Amplitude: The maximum displacement on either side of the mean position of the particle executing S.H.M. is known as the amplitude (A) of the particle.

→ Displacement: Displacement is the change in the position with time from the mean position of oscillatory motion.

→ Period of periodic motion: The smallest time interval after which the process repeats itself is called the period of the periodic motion (T) i.e. It is the time required for one complete cycle or oscillation.

→ Frequency: The reciprocal of the Time Period of motion is known as the frequency. It is the number of oscillations per second, v = \(\frac{1}{T}\).

Important Formulae:
→ Angular frequency/Angular velocity, ω = 2πv = \(\frac{2π}{T}\)

→ Displacement in S.H.M., y(t) = r sin (ωt + Φ_o).

→ Velocity of particle in S.H.M., v(t) = rωcos (ωt + Φ_o) = ω\(\sqrt{r^{2}-y^{2}}\)

→ Acceleration in S.H.M., a(c) = – rω² sin (ωt – Φ_o) = – ω²y.

→ Time period of a particle in S.H.M. is
T = 2π\(\sqrt{\frac{y}{a}}\) = 2π\(\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\)

= 2π\(\sqrt{\begin{array}{c}
\text { inertia factor } \\
\hline \text { spring constant }
\end{array}}\)

→ Time period of a mass m suspended by two springs connected in parallel.
T = 2π\(\sqrt{\frac{\mathrm{m}}{\mathrm{k}_{\mathrm{I}}+\mathrm{k}_{2}}}\)

→ Time period of a mass m suspended by two springs connected in . series.
T = 2π\(\sqrt{\left(\frac{1}{k_{1}}+\frac{1}{k_{2}}\right) m}\)

= 2π\(\sqrt{\frac{m\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}}\)

→ Time period of the simple pendulum, T = 2π\(\sqrt{\frac{l}{\mathrm{~g}}}\)

→ Time period of pendulum decreases if it accelerates upward with an acceleration ‘a’ s.t.
T = 2π\(\sqrt{\frac{l}{g+a}}\)

→ Time period of a simple pendulum increases if it accelerates downward with an acceleration ‘a’ s.t.
T = 2π\(\sqrt{\frac{l}{g-a}}\)

→ Time period of a simple pendulum immersed in a liquid of density σ is T’ = T\(\sqrt{\frac{\rho}{\rho-\sigma}}\) = 2π\(\sqrt{\frac{l}{g} \frac{\rho}{(\rho-\sigma)}}\) , ρ = density of the material of the bob.

→ Increase in time period of a simple pendulum with increase in temperature = \(\frac{\alpha \mathrm{d} \theta \mathrm{T}}{2}\), where T = 2π\(\sqrt{\frac{l}{g}}\)

→ Time period of a cylinder floating in a liquid of density ρ_l, is T = 2π\(\sqrt{\frac{h \rho}{\rho_{l} g}}\) , where h = height of the cylinder, ρ = density of the material of the cylinder, ρ_l = density of the liquid.

→ Time period of a liquid in a U-tube is
T’ = 2π\(\sqrt{\frac{h}{g}}\) where h = height of liquid column.

→ Energy of a particle in S.H.M. = 2π² mv² r² = \(\frac{1}{2}\) m ω²r² = \(\frac{1}{2}\) kr², where ω = angular frequency, r = amplitude, m = mass of particle.

→ Time period of oscillation of a ball in the neck of an air chamber under isothermal conditions is:
T = 2π\(\sqrt{\frac{\mathrm{mV}}{\mathrm{EA}^{2}}}=\frac{2 \pi}{\mathrm{A}} \sqrt{\frac{\mathrm{mV}}{\mathrm{P}}}\)

Where E = coefficient of elasticity
P = atmospheric pressure
m = mass of ball
V = volume of air in the chamber
A = area of cross-section of the neck of the air chamber
Under isothermal conditions E = P.

→ The mechanical energy of damped oscillations for small damping is given by
E(t) = \(\frac{1}{2}\)kx_m² e^bt/m
where x_m = r = amplitude. It is obtained by replacing r by x_me^bt/2m in the equation of energy of a particle in S.H.M.

→ The frequency of damped oscillations is given by
W’ = \(\sqrt{\frac{\mathrm{k}}{\mathrm{m}}-\frac{\mathrm{b}^{2}}{4 \mathrm{~m}^{2}}}\)

By going through these CBSE Class 11 Physics Notes Chapter 13 Kinetic Theory, students can recall all the concepts quickly.

Kinetic Theory Notes Class 11 Physics Chapter 13

→ The molecules of the ideal gas are point masses with zero volume.

→ P.E. for the molecules of an ideal gas is zero and they possess K.E. only.

→ There is no. intermolecular force for the molecules of an ideal gas.

→ An ideal gas cannot be converted into solids or liquids which is a consequence of the absence of intermolecular force.

→ No gas in the universe is ideal. Gases such as H₂, N₂, O₂, etc. behave very similarly to ideal gases.

→ The behavior of real gases at high temperatures and low pressure is very similar to ideal gases.

→ NTP stands for normal temperature and pressure.

→ STP stands for standard temperature and pressure.

→ STP and NTP both carry the same meaning and they refer to a temperature of 273 K or 0°C and 1 atm pressure.

→ The kinetic theory of an ideal gas makes use of a few simplifying assumptions for obtaining the relation:
P = \(\frac{1}{3}\)ρC² = \(\frac{1}{3} \frac{\mathrm{M}}{\mathrm{V}}\)C² = \(\frac{1}{3} \frac{\mathrm{mn}}{\mathrm{V}}\)C²

where m = mass of each molecule,
n = no. of molecules in the gas.

→ Combined with the ideal gas equation, it yields a kinetic interpretation of temperature
\(\frac{1}{2}\) mC² = \(\frac{3}{2}\) k_BT.

→ Using the law of equipartition of energy, the molar specific heats of gases can be predicted as:
For Monoatomic gases: C_V = \(\frac{3}{2}\) R, C_P = \(\frac{5}{2}\) R, γ = \(\frac{5}{2}\)

For Diatomic gases: C_V = \(\frac{5}{2}\) R, C_P = \(\frac{7}{2}\) R, γ = \(\frac{7}{5}\)

For Polyatomic gases: C_V = 3R, C_P = 4R, γ = \(\frac{4}{3}\)

→ These predictions are in agreement with the experimental values of the specific heat of several gases.

→ The agreement can be improved by including vibrational modes of motion.

→ The mean free path λ is the average distance covered by a molecule between two successive collisions.

→ Brownian motion is a striking confirmation of the kinetic molecular picture of matter.

→ Any layer of gas inside the volume of a container is in equilibrium because the pressure is the same on both sides of the layer.

→ The intermolecular force is minimum for the real gases and zero for ideal gases.

→ Real gases can be liquified as well as solidified.

→ The internal energy of real gases depends on volume, pressure as well as temperature.

→ Real gases don’t obey the gas equation PV = nRT.

→ The volume and pressure of ideal gas become zero at the absolute zero.

→ The molecules of a gas are rigid and perfectly elastic spheres.

→ The molecules of each gas are identical but different from that of the other gases.

→ The molecules of the gases move randomly in all directions with all possible velocities.

→ The molecules of the gas continuously collide with one another as well as with the walls of the containing vessels.

→ The molecular collisions are perfectly elastic.

→ The total energy of the molecules remains constant during collisions.

→ The molecules move with constant velocity along a straight line between the two successive collisions.

→ The density of the gas does not change due to collisions.

→ 1 atm pressure =1.01 × 10⁵ Pa.

→ Maxwell’s law proved that the molecules of a gas move with all possible speeds from 0 to ∞.

→ The no. of molecules having speeds tending to zero or infinity is very very small (almost tending to zero).

→ There is a most probable speed (C_mp) which is possessed by a large number of molecules.

→ C_mp increases with the increase in temperature.

→ C_mp varies directly as the square root of the temperature i.e.
C_mp ∝ \(\sqrt{T}\)

→ Absolute temperature can never be negative.

→ The peak of the no. of molecules (n) versus speed (C) curve corresponds to the most probable speed (C_mp).

→ The number of molecules with higher speeds increases with the rise in temperature.

→ At the constant temperature of the gas, λ decreases with the increase in pressure because the volume of the gas decreases.

→ At constant pressure, the λ increases with an increase in temperature due to the increase in volume.

→ The numerical value of the molar mass in grams is called molecular weight.

→ Law of Gaseous Volumes: It states that when gases react together, they do so in volumes which will be a simple ratio to one another and also to the volumes of product.

→ Law of equipartition of energy: It states that the energy for each degree of freedom in thermal equilibrium is \(\frac{1}{2}\)K_BT.

→ Monoatomic gases: The molecule of a monoatomic gas has three translational degrees of freedom and no other modes of motion.
Thus the average energy of a molecule at temperature T is \(\frac{3}{2}\) K_BT.

→ The total internal energy of a mole of such a gas is
U = \(\frac{3}{2}\)K_B T × N_A = \(\frac{3}{2}\)RT

→ Diatomic Gases: The molecule of a diatomic gas has five translational and two rotational degrees of freedom. Using the law of equipartition of energy, the total internal energy of a mole of such a diatomic gas is
U = \(\frac{5}{2}\) K_BT × N_A = \(\frac{5}{2}\) RT

→ Polyatomic Gases: In general, a polyatomic molecule has three translational and three rotational degrees of freedom and a certain number (0 of vibrational modes. According to the law of equipartition of energy, one mole of such gas has
U = [\(\frac{3}{2}\)K_BT + \(\frac{3}{2}\)KBT + fK_BT]N_A

→ Mean Free Path: Mean free path is the average distance covered between two successive collisions by the gas molecule moving along a straight line.

→ Degree of freedom: It is defined as the number of ways in which a gas molecule can absorb energy.
Or
It is the number of independent quantities that must be known to specify the position and configuration of the system completely.

→ Molar mass: It is defined as the mass of 1 mole of a substance. Molar mass = Avogadro’s no. × mass of one molecule.

→ The law of equilibrium of energy states that if a system is in equilibrium at absolute temperature T, the total energy is distributed equally in different energy modes of absorption, the energy of each mode being equal to \(\frac{1}{2}\)K_BT. Each translational and rotational degree of freedom corresponds to one energy model of absorption and has energy \(\frac{1}{2}\)K_BT. Each vibrational frequency has two modes of energy (Kinetic and Potential) with corresponding energy equal to 2 × \(\frac{1}{2}\)K_BT = K_BT.

Important Formulae:
→ K.E./mole of a gas = \(\frac{1}{2}\)MC² = \(\frac{3}{2}\)RT
K.E./molecule = \(\frac{1}{2}\)mC² = \(\frac{3}{2}\)k_B T

C_rms = \(\sqrt{\frac{3 P}{\rho}}=\sqrt{\frac{3 P V}{M}}\)
γ = \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{v}}}\)

→ PV = nRT is ideal gas equation.

→ PV = rT is gas equation for one gram of gas.
where r = \(\frac{\mathrm{R}}{\mathrm{M}}\),
M = molecular weight of the gas.

→ The gases actually found in nature are called real gases.

→ Real gases don’t obey Boyle’s law at all temperature.

→ The mean free path is given by:
γ = \(\frac{1}{\sqrt{2 \pi n d^{2}}}\)
= \(\frac{\mathrm{m}}{\sqrt{2} \pi \mathrm{d}^{2} \mathrm{mn}}=\frac{\mathrm{m}}{\sqrt{2} \pi \mathrm{d}^{2} \rho}\)
Where ρ = mn = mass/volume of the gas
= density of gas.

d = diameter of molecule.
n = number densisty = \(\frac{\mathrm{N}}{\mathrm{V}}\)

Also P = \(\frac{\mathrm{RT}}{\mathrm{V}}=\frac{\mathrm{N}}{\mathrm{V}} \frac{\mathrm{R}}{\mathrm{N}}\)T = nkT

∴ n = \(\frac{\mathrm{P}}{\mathrm{kT}}\)

∴ λ = \(\frac{\mathrm{kT}}{\sqrt{2} \pi \mathrm{d}^{2} \mathrm{P}}\)

→ Graham’s law of diffusion:
\(\frac{\mathrm{R}_{1}}{\mathrm{R}_{2}}=\sqrt{\frac{\mathrm{M}_{2}}{\mathrm{M}_{1}}}\)
where R₁ and R₂ are diffusion rates of gases 1 and 2 having molecular masses M₁ and M₂.

By going through these CBSE Class 11 Physics Notes Chapter 12 Thermodynamics, students can recall all the concepts quickly.

Thermodynamics Notes Class 11 Physics Chapter 12

→ A thermodynamic system is a collection of a large no. of atoms or molecules confined within the boundaries of a closed surface so that it has definite values of P, V, and T.

→ Work is done during expansion or contraction of the system and is given by dW = PdV where dV = change in volume at constant pressure P.

→ The temperature of the system decreases during expansion and increases during contraction.

→ The slope of the adiabatic curve is steeper than that of the isothermal curve.

→ W_iso > W_adia during expansion if the initial (V_t) and final (V_f) volumes are the same in both the cases.

→ Work done during isothermal compression is less than that during adiabatic compression if V_t and V_f are the same in both cases.

→ Δp = 0 in isobaric process and ΔV = 0 for an isochoric process.

→ Heat engines are devices that convert heat into work.

→ The refrigerator is regarded as a heat engine in the reverse direction.

→ 1 litre = 10^-3 m³.

→ SI and G.G.S. unit of heat capacity is JK^-1 and cal/°C respectively.

→ η of Carrot heat engine is independent of the nature of the working substance.

→ C_P – C_V is constant for all gases.

→ C_P/C_V is not constant for all gases.

→ C_P/C_V has different values for mono, di, and triatomic gases.

→ U for a system is the unique function of the state of the system i.e. U is a unique function of P, V, T.

→ The refrigerator absorbs heat from the cold reservoir and rejects the heat to the hot reservoir..

→ The liquid used as a working substance in the refrigerator is called refrigerant.

→ The most commonly used refrigerants are pheon (dichlorodifluoromethane), SO₂ and ammonia.

→ Freon or SO₂ are used in household refrigerators.

→ NH₃ is used for large-scale refrigeration.

→ U for real gas depends on T and V i.e. U = f (T, V).

→ U for ideal gas depends only on T i.e. U = f (T).

→ For isothermal process, dU = 0 and dQ = dW.

→ For an adiabatic process, dQ = 0 and dU = – dW.

→ PV^γ = constant for an adiabatic process.

→ Open system: The system which can exchange energy with the surroundings is called an open system.

→ Closed system: The system which cannot exchange energy with its surroundings is called a closed system.

→ The first law of thermodynamics: According to this law, the total energy of an isolated system remains the same. However, it can change the form, Mathematically,
ΔQ = ΔU + ΔW

where ΔQ = amount of heat supplied,
ΔU = change in the internal energy and
ΔW = the amount of work done by the system
ΔW = ΔQ – ΔU.

→ Zeroth law of thermodynamics: If two given bodies are in thermal equilibrium with a third body individually, then the given bodies will also be in thermal equilibrium with each other.

→ The second law of thermodynamics:

1. It is impossible to get a continuous supply of work from a body by cooling it to a temperature lower than that of its surroundings.
2. In other words, a perpetual motion of the second kind is impossible without doing anything else.
3. It is impossible to make heat flow from a body at a lower temperature to a body at a higher temperature without doing any work.
4. It is impossible to construct a device that can without other effect lift one object by extracting internal energy from another.

→ Isothermal process: The variation of P with V at T remaining constant is called the isothermal process.

→ Isobaric process: A process in which volume (V) and temperature (T) vary but the pressure (P) remains constant is known as the isobaric process.

→ Isochoric process: A process in which volume remains constant but P and T can change is known as the isochoric process.

→ Adiabatic process: A process in which the total heat content of the system (Q) remains conserved when it undergoes various changes is called an adiabatic process.

→ Indicator diagram: The graph between (P) and volume (V) of a thermodynamic system undergoing certain changes is called a P-V diagram or an indicator diagram as it is drawn with the help of a device called an indicator.

→ Non-cyclic process: A process in which a system after undergoing certain changes does not return to its initial state is called a non-cyclic process.

→ Cyclic process: A process in which a system after undergoing certain changes returns to its initial state is called a cyclic process.

→ External combustion engine: An engine in which fuel is burnt in a separate unit than the main engine is called an external combustion engine.

→ Internal combustion engine: An engine in which the fuel is burnt within the working cylinder of the engine is called an internal combustion engine.

→ Heat engine: A device that uses thermal energy to deliver mechanical energy is called a heat engine.

→ Heat reservoir: A source of heat at constant temperature is called a heat reservoir.

→ Heat sink: A sub-system of the engine in or out of it in which unspent heat is rejected at constant temperature for use is called a heat sink.

→ Working substance: A substance that receives some heat from a source and after converting a part of it into work rejects the remaining heat into the sink. Gas, steam are usual working substances in an engine.

→ Critical pressure: It is the pressure that is necessary to produce liquefaction at the critical temperature.

→ Critical volume: It is the volume of 1 mole of a gas at the critical temperature and critical pressure.

→ Real gas: The real gases are those in which the molecular energy is both kinetic and potential due to attraction between the molecules.

→ Ideal or perfect gas: A gas in which intermolecular attractive force is zero and energy of molecules is only kinetic are called ideal or perfect gases.

→ The internal energy of a perfect gas depends only on its temperature and not on its volume.

→ Phases: The existence of a substance in liquid, vapor, or solid-state are known as three phases of a substance on a given pressure-temperature graph.

→ Phase diagram: The way of showing different phases of substance on a pressure-temperature graph is known as a phase diagram.

→ Reversible process: A process is said to be reversible when the various stages of an operation to which it is subjected can be traversed back in the opposite direction in such a way that the substance passes through exactly the same conditions at every step in the reverse process as in the direct process.

→ Irreversible process: A process in which any one of the conditions stated for the reversible process is not fulfilled is called an irreversible process.

Important Formulae:
→ Equation of state for an ideal gas of μ moles is
PV = μRT

→ Equation of state for a real gas is
(P + \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\))(V – b) = RT

→ Internal energy of the gas molecules is given by
U = KE. + P.E.

→ First law of thermodynamics is the law of conservation of energy and is mathematically expressed as
dQ = dU + dW
= dU + PdV.

→ Work done during isothermal and adiabatic processes are given by

1. W_iso = 2.303 RT log10(\(\frac{\mathrm{V}_{2}}{\mathrm{~V}_{1}}\))
2. W_adia = \(\frac{R}{\gamma-1}\)(T₁ – T₂)

→ Efficiency of heat engine is given by
η = \(\frac{\mathrm{W}}{\mathrm{Q}_{1}}=\frac{\mathrm{Q}_{1}-\mathrm{Q}_{2}}{\mathrm{Q}_{1}}\)

= 1 – \(\frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}}\) = 1 – \(\frac{\mathrm{T}_{2}}{\mathrm{T}_{1}}\)

where Q₁ = heat absorbed from the source at temperature T₁
Q₂ = heat rejected to the sink at temperature T₂.

→ P₁V₁ = P₂V₂ for an isothermal process.

→ P₁V₁^r = P₂V₂^r for an adiabatic process.

→ Coefficient of performance of refrigerator is given by
β = \(\frac{\text { Heat extracted from cold body }}{\text { Work doneon the refrigerant }}\)
= \(\frac{\mathrm{Q}_{2}}{\mathrm{~W}}=\frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}-\mathrm{Q}_{2}}\)

→ In a true camot cycle,
\(\frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}}=\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}\)

∴ β = \(\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}-\mathrm{T}_{2}}\)

→ C_P – C_V = \(\frac{\mathrm{r}}{\mathrm{J}}\)

→ Work done is given by
dW = PdV(J) = \(\frac{\mathrm{PdV}}{4.2}\)cal.

→ Internal energy gained or lost by a perfect gas is
ΔU = n C_VΔT.

→ For isochoric process,
ΔQ = n C_u ΔT.

→ For isobaric process,
ΔQ = n C_P ΔT.

By going through these CBSE Class 11 Physics Notes Chapter 11 Thermal Properties of Matter, students can recall all the concepts quickly.

Thermal Properties of Matter Notes Class 11 Physics Chapter 11

→ Heat is the thermal energy that transfers from a body at a higher temperature to the other body at a lower temperature.

→ Temperature is the property of a body that determines whether or not it is in thermal equilibrium with its surroundings.

→ Thermometry is the branch of heat that deals with the measurement of temperature.

→ S.I. Unit of coefficient of thermal expansion in K^-1.

→ The volume of water decreases with the increase in temperature from 0°C to 4°C. It is called the anomalous expansion of water.

→ The density of water is maximum at 4°C and. its maximum value is 1 g cm^-3 or 10³ kg m^-3.

→ Water (0° to 4°C) and silver iodide (80°C to 141°C) contract on heating.

→ Quartz, pyrex glass, fused silica and invar neither expand nor contract on heating.

→ On a freezing, the volume of ice becomes more than that of water in cold countries when the temperature goes below 0°C and thus the pipe expands and may burst.

→ The principle of Calorimetry is:
Heat gained = Heat lost.

→ A sensitive thermometer is one that shows a large change in the position of mercury meniscus for a small change in temperature.

→ The critical temperature is that temperature up to which gas can be liquified by applying pressure alone.

→ Vapour is a gas above the critical temperature and gas is a vapour below the critical temperature.

→ ΔC = ΔK.

→ In order to convert the temperature from one scale to another, the following relation is used :
\(\frac{\mathrm{C}-0}{100}=\frac{\mathrm{F}-32}{180}=\frac{\mathrm{R}-0}{80}=\frac{\mathrm{K}-273.15}{100}\)

→ Ideal gas equation is PV = nRT.

→ Heat Capacity = mC = W = water equivalent.

→ There are three modes of transfer of heat i.e. conduction, convection and radiation.

→ Radiation mode is the fastest mode of heat transfer.

→ A body that neither reflects nor transmits any heat radiation but absorbs all the radiation is called a perfectly black body.

→ Q = mL
where Q = quantity of heat required for a change from one state to another.
L = Latent heat, m = mass of substance.

→ Melting point is a characteristic of the substance and it also depends 7 on the pressure.

→ Skating is possible on snow due to the formation of water below the skates. It is formed due to the increase of pressure and it acts as a lubricant.

→ The change from solid-state to vapour state without passing through the liquid state is called sublimation and the substance is said to be sublime.

→ Solid CO₂ is called dry ice and it sublimes.

→ During the sublimation process, both the solid and the vapour states of a substance coexist in thermal equilibrium.

→ Melting: The change of state from solid to liquid is called melting.

→ Fusion: The change of state from liquid to solid is called fusion.

→ Melting point: The melting point is the temperature at which the solid and the liquid states of the substance co-exist in thermal equilibrium with each other.

→ Regelation: Regelation is the process of refreezing.

→ Vaporisation: Change of state from liquid to vapour is called vaporisation.

→ Boiling point: Boiling point is the temperature at which the liquid and the vapour states of the substance co-exist in thermal equilibrium with each other.

→ Normal melting point: The melting point of a substance at standard atmospheric pressure is called its normal melting point.

→ Normal boiling point: The boiling point of a substance at standard atmospheric pressure is called its normal boiling point.

Important Formulae:
→ \(\frac{T}{T_{t r}}=\frac{P}{P_{t r}}\)

→ Change in length is given by
Δ l = l_o α Δθ

→ Change in area is given by
Δ S = S_o β Δθ.

→ Change in volume is given by
ΔV = V_o Y Δθ.
l_t = l_o(1 + α Δθ).
S_t = S_o (1 + β Δθ).
V_t = V_o (1 + γ Δθ).
where α, β & γ are called coefficient of linear, superficial and volume expansion respectively.

→ Thermal conductivity of a composite rod made of two conductors. of equal lengths and joined in series is given by
K = \(\frac{2 \mathrm{~K}_{1} \mathrm{~K}_{2}}{\mathrm{~K}_{1}+\mathrm{K}_{2}}\)

→ Temperature of the interface connecting two rods of different lengths d₁ and d₂ is given by
T_o = \(\frac{\mathbf{K}_{1} \mathrm{~d}_{2} \theta_{1}+\mathrm{K}_{2} \mathrm{~d}_{1} \theta_{2}}{\mathbf{K}_{2} \mathrm{~d}_{1}+\mathrm{K}_{1} \mathrm{~d}_{2}}\)
and
T_o = \(\frac{\mathrm{K}_{1} \theta_{1}+\mathrm{K}_{2} \theta_{2}}{\mathrm{~K}_{1}+\mathrm{K}_{2}}\) if their lengths are equal i.e. if d₁ = d₂.

→ If areas of the cross-section are equal, Then
K = \(\frac{\mathrm{K}_{1}+\mathrm{K}_{2}}{2}\)

→ \(\frac{\text { Change in temperature }}{\text { Time }}\) = KΔθ
where Δθ = difference of average temperature and room temperature.

→ Specific heat capacity of a substance is given by
C = \(\frac{\Delta \mathrm{Q}}{\mathrm{M} \Delta \theta}\)
or
ΔQ = MCΔθ.

→ The relation between Kelvin temperature (T) and the celcius temperature t_c is
T = t_c + 273.15.

→ Resistance varies with temperature as:
R_t = R_o(l + α Δθ)
where R_o = resistance at 0°C
R_t = resistance at t°C
α = temperature coefficient of resistance
Δθ = change in temperature.

→ Q = mL, where L = latent heat.

→ Temperature difference Δ°F equivalent to Δ°C is
ΔF = \(\frac{9}{5}\) × ΔC

→ Temperature difference ΔK equivalent to ΔF is
ΔF = \(\frac{9}{5}\)ΔK.

→ T_K = T_c + 273.15

By going through these CBSE Class 11 Physics Notes Chapter 10 Mechanical Properties of Fluids, students can recall all the concepts quickly.

Mechanical Properties of Fluids Notes Class 11 Physics Chapter 10

→ Fluids are substances that can flow e.g. liquids and gases. Fluids don’t possess a definite shape.

→ When a liquid is in equilibrium, the force acting on its surface is perpendicular everywhere.

→ In a liquid, the pressure is the same at the same horizontal level.

→ The pressure at any point in the liquid depends on the depth (h). below the surface, the density of liquid and acceleration due to gravity.

→ Pressure is the same in all directions.

→ If two drops of the same volume but different densities are mixed together, then the density of the mixture is the arithmetic mean of their densities i.e. ρ = \(\frac{\rho_{1}+\rho_{2}}{2}\)

→ The upthrust on a body immersed in a liquid depends only on the volume of the body and is independent of the mass, density or shape of the body.

→ The weight of the plastic bag full of air is the same as that of the empty bag because the upthrust is equal to the weight of the air enclosed.

→ The wooden rod can’t float vertically in a pond of water because the centre of gravity lies above the metacentre.

→ The cross-section of the water stream from a tap decreases as it goes down in accordance with the equation of continuity.

→ The loss in weight of a body = Weight of the fluid displaced by the body.

→ Upthrust = Weight of the liquid displaced.

→ The floating body is in stable equilibrium when the metacentre is above the C.G. (C.G. is below the centre of buoyancy).

→ The floating body is in unstable equilibrium when the metacentre lies below the C.G. (i.e. C.G. is above the centre of buoyancy).

→ The floating body is in the neutral equilibrium when the C.G. coincides with the metacentre {i.e. C.G. coincides with the C.B.).

→ When a gale blows over a roof, the force on the roof is upwards.

→ If a beaker is filled with a liquid of density ρ up to height h, then the mean pressure on the walls of the beaker is \(\frac{\mathrm{h} \rho \mathrm{g}}{2}\)

→ The viscosity of liquids decreases with the rise in temperature i.e.
η ∝ \(\frac{1}{\sqrt{\mathrm{T}}}\)

→ The viscosity of gases increases with the rise in temperature i.e.
η ∝ \(\sqrt{T}\)

→ The streamlined or turbulent nature of flow depends on the velocity of flow of the liquid.

→ Streamline flow is also called laminar flow.

→ Reynolds number is low for liquids of higher viscosity.

→ N_R < 2000 for streamline flow.

→ N_R > 3000 for turbulent flow.

→ N_R lies between 2000 and 3000 for unstable flow.

→ Viscosity is due to the transport of momentum.

→ Bernoulli’s theorem is based on the conservation of energy.

→ Bernoulli’s theorem is strictly applicable to non-viscous fluids.

→ Viscosity arises out of tangential dragging force acting on the fluid layer.

→ Grease is more viscous than honey.

→ The coefficient of viscosity is measured in Nm^-2.

→ Stake’s law can be used to find the size of tiny spherical objects.

→ The flow of fluid under pressure may be zig-zag or in parallel layers of slow velocity in which the velocity vector is parallel at each point of the fluid.

→ The flow of fluid whose velocity varies from point to point is called turbulent flow.

→ The flow of fluid whose velocity at every point remains constant is called streamline flow.

→ For streamline flow, conservation of energy law-holds good and this law is known as Bernoulli’s Theorem.

→ A large number of phenomena like the flow of fluids through constructed pipes, the flight of planes, birds, burners, filter pumps and many other devices work on the principle of Bernoulli’s theorem.

→ The flow of fluids through pipes and capillaries is described by Poiseuille’s formula.

→ Pascal’s law accounts for the Principle of transmission of pressure in fluids.

→ The equation of continuity always holds good which is A₁v₁ = A₂v₂. . The force acting per unit length of the imaginary line drawn on the liquid surface parallel to the surface is called the force of surface tension.

→ Due to surface tension, free surfaces of fluids tend to have minimum surface and so, the liquid drops tend to be spherical and also bubbles are formed in such a film.

→ The free surface has surface energy per unit area equal to surface tension.

→ Free surfaces in tubes, pipes of negligible bore tend to be concave sides which forces the liquid to rise in the capillary.

→ There is the force of pressure inside a soap bubble equal to \(\frac{4 \mathrm{~T}}{\mathrm{R}}\) due to two surfaces in the bubble.

→ Practical use of surface tension made in the capillary rise of liquids f (rise of ink in fountain pen) and cleaning of other stains by detergents.

→ Molecular forces don’t obey the inverse square law.

→ Molecular forces are of electrical origin.

→ Work done in forming a soap bubble of radius R is 8πR²T, where T = surface tension.

→ The angle of contact increases with the rise in temperature and it decreases with the addition of soluble impurities.

→ The angle of contact is independent of the angle of inclination of the walls.

→ The materials used for waterproofing increase’s the angle of contact e as well as the surface tension.

→ Detergents decrease both the angle of contact as well as surface tension.

→ Surface tension does not depend on the area of the surface.

→ When there is no external force, the shape of a liquid is determined by the surface tension of the liquid.

→ Soap helps in better cleaning of clothes because it reduces the surface tension of the liquid.

→ A liquid having an obtuse angle of contact does not wet the walls of containing vessel.

→When force of adhesion is less than \(\frac{1}{\sqrt{2}}\) times the force of cohesion (F_A < \(\frac{\mathrm{F}_{\mathrm{c}}}{\sqrt{2}}\)) the liquid does not wet the walls of vessel and meniscus is convex.

→ The height of a liquid column in a capillary tube is inversely proportional to acceleration due to gravity.

→ Energy is released when the liquid drops merge into each other to form a larger drop.

→ The liquid rises in a capillary tube when angle of contact is acute and F_A > \(\frac{\mathrm{F}_{\mathrm{c}}}{\sqrt{2}}\)

→ The surface tension of molten cadmium increases with the increase in temperature.

→ Surface tension is numerically equal to surface energy.

→ Surface energy is the potential energy of the surface molecules per unit area.

→ The surface tension of lubricants, paints, antiseptics should below so that they may spread easily.

→ C.G.S. and S.L units of rare poise (dyne s cm^-2 or g cm^-1 s^-1 ) and decompose (Nsm^-2 or kg m^-1 s^-1) respectively.

→ 1 decapoise= 10 poise..

→ Thrust: It ¡s defined as the total force exerted by the fluid on any surface in contact.

→ Atmospheric Pressure: It is defined as the weight of a column of air of unit cross-sectional area extending from that point to the top of the atmosphere.
= 1.013 × 10⁵ Pa = 76cm of Hg column.

→ Gauge pressure: It is the difference between absolute pressure and atmospheric pressure.

→Archimede’s Principle: It states that when a body is dipped wholly or partially in a fluid, it loses its weight.

→Surface Tension: It is the property of the liquid by virtue of which the free surface of the liquid at rest tends to have minimum area and as such ft behaves like a stretched elastic membrane.

→ Poiseuille’s Formula: According to it, the volume of the fluids flowing through ¡h pipe-isdireçly pLoportona1 to the pressure difference across the ends of the pipe and fourth power of the radius, it is inversely proportional to the coefficient of viscosity and length of the pipe.
i.e. mathematically. V = \(\frac{\pi}{8} \frac{\mathrm{pr}^{4}}{\eta l}\)

→ 1 Torr: It is the pressure exerted by a mercury column of 1 mm in height.

→ Law of Floatation: It states that a body floats in a fluid if the weight of the fluid displaced by the immersed portion of the body is equal to the weight of the body.
i.e V₁ ρ₁ g =V₂ ρ₂ g
or
\(\frac{\rho_{1}}{\rho_{2}}=\frac{V_{2}}{V_{1}}\)
or
\(\frac{\text { density of solid }}{\text { density of liquid }}=\frac{\text { Volume of immersed part of solid }}{\text { Total volume of solid }}\)

→ Force of Cohesion: It is the force of attraction between the molecules of the same substance or the same kind.

→ Force of adhesion: It is the force of attraction between the molecules of different substances.

→ The angle of Contact: It is defined as the angle at which the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid.

→ Capillarity: It is the phenomenon of rising or fall of a liquid in a capillary tube.

→ Jurin’s Law: It states that the liquid rises more in a narrow tube and lesser in a wider tube.

→ Viscosity: It is the property of fluid layers to oppose the relative motion among them.

→ Coefficient of Viscosity: It is defined as the tangential force required per unit area of the fluid surface to maintain a unit velocity gradient between two adjacent layers.

→ Stoke’s Law: It states that the viscous drag on a spherical body of radius r moving with terminal velocity v_T in a fluid of viscosity r| is given by F – 6πηrv_T.

→ Central line: The line joining the C.G. and centre of buoyancy is called the Central line.

→ Metacentre: It is defined as the point where the vertical line through the centre of buoyancy intersects the central line.

→ Terminal Velocity: It is defined as the constant velocity attained by a spherical body falling through a viscous medium when the net force on it is zero.

→ Pascal’s Law: It states that in an enclosed fluid, the increased pressure is transmitted equally in all possible directions if the effect of gravity is neglected.

→ Streamline: It is defined as the path straight or curved, the tangent to which at any point gives the direction of flow of the liquid at ‘ that point.

→ Tube of flow: It is a bundle of streamlines having the same velocity of liquid elements over any cross-section perpendicular to the direction of flow.

→ Streamline flow: The flow of a liquid is said to streamline flow or steady flow if all its particles pass through a given point with the same velocity.

→ Turbulent flow: The flow of a liquid in which the velocity of all particles crossing a given point is not the same and the motion of fluid becomes disorderly is called turbulent flow,

→ Laminar flow: The flow is said to be laminar if the liquid flows over a horizontal surface in the form of layers of different velocities.

→ Critical velocity: It is defined as the maximum velocity of a liquid or fluid up to which the flow is streamlined and above which it is turbulent.

→ Reynolds’ number: It is a pure number that tells about the type of flow. It is the ratio of inertial force and the viscous force for a fluid in motion.

→ Equation of Continuity: It expresses the law of conservation of ‘ mass in fluid dynamics.
i. e. a₁v₁ =a₂v₂ .

→ Bernoulli’s Theorem: It states that the total energy (sum of pressure energy, K..E. and P.E.) per unit mass is always constant for an ideal fluid.
i.e. \(\frac{\mathrm{P}}{\mathrm{\rho}}\) + gh + \(\frac{1}{2}\) v² = constant

→ Surface film: It is the topmost layer of the liquid at rest with a thickness equal to the molecular range.

Important Formulae:
→ Pressure is given by P = \(\frac{F}{A}\).

→ Pressure exerted by a liquid column.
P = hρg

→ Downward acceleration of a body falling down in a fluid
(i.e. effective value of g) is
a = (\(\frac{\text { density of body }-\text { density of fluid }}{\text { density of body }}\))g

→ Pascal’s law, \(\frac{\mathrm{F}_{1}}{\mathrm{a}_{1}}=\frac{\mathrm{F}_{2}}{\mathrm{a}_{2}}\) = Constant.

→ Surface tension, T = \(\frac{F}{l}=\frac{\text { Force }}{\text { Length }}\).

→ Excess of pressure inside an air bubble is
p_i – p_o = \(\frac{2 \mathrm{~T}}{\mathrm{R}}\)

→ Excess of pressure inside a soap bubble is
p_i – p_o = \(\frac{4 \mathrm{~T}}{\mathrm{R}}\)
And inside a liquid drop,
p_i – p_o = \(\frac{2 \mathrm{~T}}{\mathrm{R}}\)

→ Ascent formula is h = \(\frac{2T cosθ}{rρg}\)

→ Shape of drops is decided by using
cos θ = \(\frac{\mathrm{T}_{\mathrm{SA}}-\mathrm{T}_{\mathrm{SL}}}{\mathrm{T}_{\mathrm{LA}}}\)

→ Viscous force is given by
F = – η A \(\frac{\mathrm{d} \mathrm{v}}{\mathrm{dx}}\)

→ Volume of liquid flowing per second is given by
V = \(\frac{\pi \mathrm{pr}^{4}}{8 \eta l}\)

→ Terminal velocity is given by
VT = \(\frac{2}{9} \frac{r^{2}}{\eta}\)(ρ – σ)g
where ρ = density of body
σ = density of liquid (fluid).

→ P + ρgh + \(\) ρv² = constant.
If h = constant, Then
P₁ + \(\) ρv₁² = P₂ + \(\) ρv₂².

→ The weight of the aircraft is balanced by the upward lifting force due to pressure difference
Let mg = Δp × A
or
mg = \(\frac{1}{2}\) ρ(v₁² – v₂²) × A.

→ Inertial force = (avρ)v = av²ρ

→ Viscous force = \(\frac{ηav}{D}\).

By going through these CBSE Class 11 Physics Notes Chapter 9 Mechanical Properties of Solids, students can recall all the concepts quickly.

Mechanical Properties of Solids Notes Class 11 Physics Chapter 9

→ Young’s modulus is defined only for solids.

→ Bulk modulus is defined for all types of materials solids, liquids, and gases.

→ Hook’s law is obeyed only for small values of strain (say of the order of 0.01).

→ Reciprocal the bulk modulus is called compressibility.

→ Within elastic limits, force constant for a spring is given by
k = \(\frac{\mathrm{YA}}{l}\)

→ Higher values of elasticity mean the greater force is required for producing a given change.

→ The deformation beyond the elastic limit is called plasticity.

→ The materials which don’t break well beyond the elastic limitary called ductile.

→ The materials which break as soon as stress exceeds the elastic limit are called brittle.

→ Rubber sustains elasticity even when stretched several times its length However it is not ductile. It breaks down as soon as the elastic limit is crossed.

→ Quartz is the best example of a perfectly elastic material.

→ Stress and pressure have the same units and dimensions, but the pressure is always normal to the surface while the stress may be parallel or perpendicular to the surface.

→ When a body is sheared two mutually perpendicular strains are produced which are called longitudinal strain and compressional strain. Both are of equal magnitude.

→ Thermal stress in a rod = Y ∝ Δθ. It is independent of the area of cross-section or length of wire.

→ Breaking force depends on the area of the cross-section of the wire. Breaking stress per unit area of cross-section is also called tensile strength of a wire.

→ Elastic after effect is a temporary absence of the elastic properties.

→ Temporary loss of elastic properties due to continuous use for a long time is called elastic fatigue.

→ Normal stress is also called tensile stress when the length of the body tends to increase.

→ Normal stress is also called compressional stress when the length of the body tends to decrease.

→ Tangential stress is also called shearing stress.

→ When the deforming force is inclined to the surface, both tangential stress, as well as normal stress, are produced.

→ Diamond and Carborundum are the nearest approaches to a rigid body. Elasticity is the property of non-rigid bodies.

→ A negative value of Poisson’s ratio means that if length increases then the radius decreases.

→ If a beam of rectangular cross-section is loaded its depression is inversely proportional to the cube of thickness of the beard.

→ If a beam of circular cross-section is loaded, its depression is -inversely proportional to the cube of the radius.

→ If we double the radius of a wire its breaking load becomes four times and the breaking stress remains unchanged.

→ S.I. unit of stress is Nm^-2 or Pascal (Pa).

→ The strain has no unit.

→ Breaking stress is independent of the length of the wire.

→ When a beam is bent, both extensional as well as compressional strain is produced.

→ Y is infinity for a perfectly rigid body and zero for air.

→ K for a perfectly rigid body is infinity and its compressibility is zero.

→ The modulus of rigidity of water is zero.

→ Solids: They are defined as substances that have definite shape and volume and have close packing of molecules.

→ Compression Strain: The reduction in dimension to the original dimension is called compression strain.

→ Compression Stress: The force per unit area which reduces the dimension of the body is called compression stress. It is maximum stress which the body can withstand on or before breaking.

→ Brittle: The materials in which yield point and breaking points are very close are called brittle.

→ Compressibility: It is defined as the reciprocal bulk modulus of elasticity.

→ Elastic limit: It is defined as the maximum stress on the removal of which, the body regains its original configuration.

→ Modulus of elasticity: It is defined as the ratio of stress to strain.

Important Formulae:
→ Young’s Modulus is given by
Y = \(\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{L} / \mathrm{L}}=\frac{\mathrm{FL}}{\mathrm{A} \Delta \mathrm{L}}\)

→ Bulk Modulus is given by
K = \(\frac{\frac{p}{-\Delta V}}{V}=\frac{p V}{-\Delta V}=-\frac{F V}{A \Delta V}\)
when -ve sign shows that volume decreases when pressure is applied.

→ Compressibility = \(\frac{1}{K}=\frac{\Delta V}{p V}\)

→ Modulus of rigidity is given by
η = \(\frac{T}{\theta}=\frac{T}{\left(\frac{x}{L}\right)}=\frac{T L}{x}=\frac{F L}{A x}\) where T = \(\frac{F}{A}\) = tangential stress.

→ Work done per unit volume of wire = \(\frac{1}{2}\) stress × strain.

→ Work done to stretch a wire = \(\frac{1}{2}\) × stretching force × extension
= \(\frac{1}{2} \frac{\mathrm{YAl}^{2}}{\mathrm{~L}}\)
l = extension.

By going through these CBSE Class 11 Physics Notes Chapter 8 Gravitation, students can recall all the concepts quickly.

Gravitation Notes Class 11 Physics Chapter 8

→ Gravitation is a central force.

→ It acts along the line joining the particles.

→ Gravitation is the weakest force in nature.

→ It is about 10³⁸ times smaller than the nuclear force and 10³⁶ times smaller than the electric force.

→ Gravitation is the conservative force.

→ Gravitation is caused by gravitational mass.

→ Gravitation acts in accordance with Newton’s third law of motion. That is F₁₂ = – F₂₁.

→ Gravitation is independent of the presence of the other bodies in the surroundings.

→ Acceleration due to gravity is 9.81 ms^-2 on the surface of Earth.

→ Its value on the moon is about one-sixth of that on Earth.

→ Its value on Sun, Jupiter and Mercury is about 27 times, 2.5 times and 0.4 times that on the Earth. i.e. g_moon = g_e/6. g_sun = 27g, g_mercury = 0.4g, g_jupiter = 2.5g.

→ The value of g (acceleration due to gravity) does not depend upon the mass, shape or size of the falling body.

→ Inside the Earth, the value of g decreases linearly with distance from the centre of Earth.

→ Above the surface of Earth, the value of g varies inversely as the square of the distance from the centre of Earth.

→ The value of g decreases faster with altitude than with depth.

→ For small values of height (h), the value of g at a height ‘h’ is the same as the value of g at a depth d (= 2h).

→ Decrease in g at a height h = x (very near the Earth’s surface i.e. h << R) is twice as compared to the decrease in g at the same depth d = x.

→ g is maximum on Earth’s surface and decreases both when we go above or below the Earth’s surface.

→ The value of g is zero at the centre of Earth.

→ The rate of variation of g with height (near the surface of Earth, when h << R) is twice the rate of variation of g with depth i.e.
\(\frac{\Delta \mathrm{g}_{\mathrm{h}}}{\Delta \mathrm{h}}\) = 2 \(\frac{\Delta \mathrm{g}_{\mathrm{d}}}{\Delta \mathrm{d}}\)

→ With the increase in latitude, g decreases.

→ Its value at latitude Φ is given by
g_Φ = g_p – Rω² cos²Φ

→ The decrease in ‘g’ with latitude is due to the rotation of Earth about its own axis.

→ The decrease in ‘g’ with latitude on rotation is because a part of the weight is used to provide centripetal force for the bodies rotating with the Earth.

→ The g is maximum at poles (g_p) and minimum at the equator (g_e).

→ g_p = 9.81 ms^-2, g_c = 9.78 m^-2.

→ The decrease in g from pole to equator is about 0.35%.

→ If the Earth stops rotating about its own axis, the value of g on the poles will remain unchanged but at the equator, it will increase by about 0.35%. If the rotational speed of Earth increases, the value of g decreases at all places on its surface except poles.

→ The gravitational pull of Earth is called true weight (w_t) of the body i.e. W_t = mg.

→ The true weight of the body varies in the same manner as the /acceleration due to gravity i.e. it decreases with height above and depth below the Earth’s surface.

→ Also, Wt changes with latitude. Its value is maximum at the poles and minimum at the equator.

→ S.I. unit of weight is Newton (N) and it is often expressed in kilogram weight (kg wt) or kg f (kilogram-force).
i. e. 1 kg wt = 1 kg f = 9.8 N

→ The reaction of the surface on which a body lies is called apparent weight (W_a) of the body and ga = W_a/M is called apparent acceleration due to gravity.

→ If a body moves with acceleration a, then the apparent weight of the body of mass M is given by W_a = M|g – a| = apparent weight of the body when it falls with acceleration ‘a’ and it decreases.

→ When a body rises with acceleration, then W_a = M(g + a) i.e. it increases.

→ If the body is at rest or moving with uniform velocity, then W_a = Mg = true weight of the body.

→ For free-falling body, W_a = 0 (∵ a = g here).

→ The spring balance measures the apparent weight of the body.

→ The apparent weight provides restoring force to the simple pendulum i.e. the time period of the simple pendulum depends on the apparent value of the acceleration due to gravity i.e.
T = 2π\(\sqrt{\frac{l}{\mathrm{~g}_{\mathrm{a}}}}\)

→ In a freely falling system, the time period of a simple pendulum is infinity.

→ If a simple pendulum is suspended from the roof of an accelerating or retarding train, the time period is given by
T = 2π\(\sqrt{g^{2}+a^{2}}\)

→ The Earth is ellipsoid. It is flat at poles and bulges out at the equator. Consequently, the distance of the surface from the centre is more at the equator than on the poles. So g_pole > g_equator.

→ All bodies fall freely with the same acceleration = g.

→ The acceleration of the falling body does not depend on its mass.

→ If two bodies are dropped from the same height, they reach the ground at the same time with the same velocity.

→ If a body is thrown upward with velocity u from the top of a tower and another is thrown downward from the same point and with the same velocity, then both reach the ground with the same speed.

→ If a body is dropped from a height h, it reaches the ground with speed v = \(\sqrt{2 \mathrm{gh}}\) = gt. Time taken by it to reach the ground is t = \(\sqrt{\frac{2 h}{g}}\)

→ When a body is dropped, then initial velocity i.e. u = 0.

→ If a body is dropped from a certain height h, then the distance covered by it in nth second is \(\frac{1}{2}\) g(2n -1).

→ The greater the height of a satellite, the smaller is the orbital velocity. Work done to keep the satellite in orbit is zero.

→ When a body is at rest w.r.t. the Earth, its weight equals gravity and is known as its true or static weight.

→ The centripetal acceleration of the satellite = acceleration due to gravity.

→ Orbital velocity is independent of the mass of the satellite.

→ Orbital velocity depends on the mass of the planet as well as the radius of the orbit.

→ All communication satellites are geostationary satellites.

→ Escape velocity (V_e) from the surface of Earth = 11.2 km s^-1.

→ The body does not return to the Earth when fired with V_e irrespective of the angle of projection.

→ When the velocity of the satellite increases, its kinetic energy increases and hence total energy becomes less negative i.e. the satellite begins to revolve in orbit of greater radius.

→ If the total energy of the satellite becomes +ve, the satellite escapes from the gravitational pull of the Earth.

→ When the satellite is taken to a greater height, the potential energy increases (becomes less negative) and the K.E. decreases.

→ For the orbiting satellite, the K.E. is less than the potential energy. When K.E. = P.E., the satellite escapes away from the gravitational pull of the Earth.

→ The escape velocity from the moon is 2.4 km s^-1.

→ The ratio of the inertial mass to gravitational mass is one.

→ If the radius of a planet decreases by n% keeping mass constant, then g on its surface decreases by 2n%.

→ If the mass of a planet increases by m% keeping the radius constant, then g on its surface increases by m%.

→ If the density of the planet decreases by x% keeping the radius constant, the acceleration due to gravity decreases by x%.

→ The intensity of the gravitational field inside a shell is zero.

→ The weight of a body in a spherical cavity concentric with the Earth is zero.

→ Gravity holds the atmosphere around the Earth.

→ The reference frame attached to Earth is non-inertial because the Earth revolves about its own axis as well as about the Sun.

→ When a projectile is fired with a velocity less than the escape velocity, the sum of its gravitational potential energy and kinetic energy is negative.

→ If the Earth were at one fourth the present distance from Sun, the duration of the year will be one-eighth of the present year.

→ The tail of the comets points away from the Sun due to the radiation pressure of the Sun.

→ Even when the orbit of the satellite is elliptical, its plane of rotation passes through the centre of the Earth.

→ If a packet is just released from an artificial satellite, it does not fill the Earth. On the other hand, it will continue to orbit with the satellite.

→ Astronauts orbiting around the Earth cannot use a pendulum clock.

→ However, they can use a spring clock.

→ To the astronauts in space, the sky appears black due to the absence of the atmosphere above them.

→ The duration of the day from the moment the Sun is overhead today to the moment the Sun is overhead tomorrow is determined by the revolution of the Earth about its own axis as well as around the Sun.

→ If the ratio of the radii of two planets is r and the ratio of the acceleration due to gravity on their surface is ‘a’ then the ratio of escape velocities is \(\sqrt{ar}\).

→ Two satellites are orbiting in circular orbits of radii R₁ and R₂. Their orbital speeds are in the ratio: \(\frac{V_{1}}{V_{2}}=\left(\frac{R_{2}}{R_{1}}\right)^{\frac{1}{2}}\). It is independent of their masses.

→ An object will experience weightlessness at the equator if the angular speed of the Earth about its axis becomes more than (\(\frac{1}{800}\)) rad/s.

→ If a body is orbiting around the Earth then it will escape away if its velocity is increased by 41.8% or when its K.E. is doubled.

→ If the radius of Earth is doubled keeping mass unchanged, the escape \(\left(\frac{1}{\sqrt{2}}\right)\) times the present value.

→ V_o close to Earth’s surface = 7.9 km s^-1.

→ The time period of the satellite very near the surface of Earth is about 107 minutes.

→ No energy is dissipated in keeping the satellite in orbit around a planet.

→ If a body falls freely from infinite height, then it reaches the surface of Earth with a velocity of 11.2 km s^-1.

→ A body in a gravitational field has maximum binding energy when it is at rest.

→ Acceleration due to gravity: The acceleration with which bodies fall towards the Earth is called acceleration due to gravity.

→ Newton’s law of gravitation: It states that everybody in this universe attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres i.e.
F = \(\frac{\mathrm{Gm}_{1} \mathrm{~m}_{2}}{\mathrm{r}^{2}}\)

→ Gravitation: Force of attraction between any two bodies.

→ Gravity: Force of attraction between Earth and any other body.

→ Inertial mass: The resistance to the acceleration caused by a force is called inertial mass. It is the measure of its inertia in linear motion
i.e. m1 = \(\frac{\mathrm{F}}{\mathrm{a}}\)

→ Gravitational mass: The resistance to the acceleration caused by gravitational force i.e.
mg = \(\frac{\mathrm{F}}{\mathrm{g}}\)

→ Kepler gave threads of planetary motion.

→ Kepler’s 1st law of planetary motion: Every planet revolves around Sun in an elliptical orbit with Sun at one of its foci.

→ Second law: The radius vector joining the centre of Sun and planet sweeps out equal areas in equal intervals of time i.e. areal velocity of the planet around the Sun always remain constant.

→ Third law: The square of the time period T of revolution of a planet around the Sun is proportional to the cube of the semi-major axis R of its elliptical orbit i.e.
T² ∝ R³

→ Satellite: It is a body that constantly revolves in an orbit around a body of relatively much larger size.

→ A geostationary satellite is a satellite that appears stationary to the observer on Earth. It is also called a geosynchronous satellite.

• Its orbit is circular and in the equatorial plane of Earth.
• Its time period = time period of rotation of the Earth about its own axis i.e. one day or 24h = 86400s.
• Its height is 36000 km.
• Its orbital velocity is about 3.08 km s^-1.
• Its angular velocity is equal and is in the same direction as that of Earth about its own axis.

→ Latitude at a place: Latitude at a place on the Earth’s surface is the angle at which the line joining the place to the centre of Earth makes with the equatorial plane. It is denoted by X.

→ At poles, X = 90°.

→ At equator, X = 0.

→ Polar satellites: They are positioned nearly 450 miles above the Earth. Polar satellites travel from pole to pole in nearly 102 minutes.

→ In each successive orbit, the satellite scans a strip of the area towards the West.

→ Orbital Velocity: It is the velocity required to put the satellite in a given orbit around a planet. It is denoted by V0.

→ Escape Velocity: It is the minimum velocity with which a body be thrown upwards so that it may just escape the gravitational pull of Earth or a given planet. It is denoted by Ve.

→ The intensity of Gravitational field at a point: It is the force experienced by a unit mass placed at that point.

→ The gravitational potential energy of a body at a point in a gravitational field is the amount of work done in bringing the body from infinity to that point without acceleration.

→ Gravitational Potential: It is the amount of work done in bringing a body of unit mass from infinity to that point without acceleration.

Important Formulae:
→ The universal force of gravitation,
F = \(\frac{\mathrm{Gm}_{1} \mathrm{~m}_{2}}{\mathrm{r}^{2}}\)

→ Invector from F = \(\frac{\mathrm{Gm}_{1} m_{2} \hat{r}}{r^{3}}\)

→ G = 6.67 × 10^-11 Nm² kg^-2.

→ Weight of the body, W = mg.

→ Mass of Earth, Me = \(\frac{\mathrm{g} \mathrm{R}_{\mathrm{e}}^{2}}{\mathrm{G}}\)

→ Gravitational mass, mg = \(\frac{\mathrm{F}}{\mathrm{g}}=\frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}_{\mathrm{g}}}{\mathrm{g} \mathrm{R}_{\mathrm{e}}^{2}}\)

→ Variation of g:

1. At height h: gh = g\(\left(1+\frac{\mathrm{h}}{\mathrm{R}}\right)^{2}\) ≈ (1 – \(\frac{2 \mathrm{~h}}{\mathrm{R}}\))
2. At depth d: g_d = g(1 – \(\frac{\mathrm{d}}{\mathrm{R}}\))
3. At latitude λ: g_λ = g – Rω²cos²λ
   (a)At poles: λ = 90°, cosλ = 0 ∴ g_λpole = g
   (b) At equator: λ = 0, cosλ = 1, ∴ g_λ = g Rω²

→ g_pole– g_equator = Rω²

→ Gravitation field intensity: (I) = \(\frac{\mathrm{GM}}{\mathrm{R}^{2}}\)
or
|I| = g

→ Orbital velocity in orbit at a height h,
v_o = \(\sqrt{\frac{\mathrm{gR}^{2}}{\mathrm{R}+\mathrm{h}}}=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}+\mathrm{h}}}\)

→ If h = 0 i.e. close to Earth’s surface, v_o = \(\sqrt{gR}\).

→ Time period of the satellite,
T = \(\frac{2 \pi(\mathrm{R}+\mathrm{h})}{\mathrm{v}_{0}}=\frac{2 \pi}{\mathrm{R}} \sqrt{\frac{(\mathrm{R}+\mathrm{h})^{3}}{\mathrm{~g}}}\)

→ Escape velocity, V_e = \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}}=\sqrt{2 \mathrm{gR}}=\sqrt{2}\)V_o
= \(\sqrt{\frac{8}{3} \pi R^{3} G \rho}\)

→ Gravitational potential energy,
E = – \(\frac{\mathrm{GMm}}{\mathrm{r}}\)

→ Self energy of Earth = – \(\frac{3}{5} \frac{\mathrm{GM}^{2}}{\mathrm{R}}\)

→ Increase in gravitational RE. when the body is moved from surface of Earth to a height h,
ΔE = E_h – E_e
= – \(\frac{\mathrm{GMm}}{\mathrm{R}+\mathrm{h}}-\frac{\mathrm{GMm}}{\mathrm{R}}\)

= \(\frac{\text { GMmh }}{R(R+h)}\)

→ Gravitational potential, V = – \(\frac{\mathrm{GM}}{\mathrm{r}}\)

→ Time period of motion of the satellite:
T² = \(\frac{4 \pi^{2}}{\mathrm{GM}_{\mathrm{E}}}\) r³ = \(\frac{4 \pi^{2}}{\mathrm{gR}_{\mathrm{E}}^{2}}\) r³

→ Angular velocity (ω) of a satellite in an orbit at a height h above Earth.
ω = \(\frac{2 \pi}{T}=\sqrt{\frac{G M}{(R+h)^{3}}}=\sqrt{\frac{g_{h}}{R+h}}\)

→ Shape of the orbit of a satellite having velocity v in the orbit:

1. If v < v_o, the satellite falls to the Earth following a spiral path.
2. If v = v_o, the satellite continues to move in orbit.
3. If v_o < v < v_es, then the satellite moves in an elliptical orbit,
4. If v = v_es, then it escapes from Earth following a parabolic path,
5. If v > v_es, then the satellite “escapes from Earth following a hyperbolic path.