Kinetic Theory Class 11 Notes Physics Chapter 13

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.