# Kinetic Theory Class 11 Notes 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 H_{2}, N_{2}, O_{2}, 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 = \(\frac{1}{3}\)ρC^{2} = \(\frac{1}{3} \frac{\mathrm{M}}{\mathrm{V}}\)C^{2} = \(\frac{1}{3} \frac{\mathrm{mn}}{\mathrm{V}}\)C^{2}

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

\(\frac{1}{2}\) mC^{2} = \(\frac{3}{2}\) k_{B}T.

→ 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 × 10^{5} 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_{B}T.

→ 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_{B}T.

→ 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_{B}T × 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_{B}T + \(\frac{3}{2}\)KBT + fK_{B}T]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_{B}T. Each translational and rotational degree of freedom corresponds to one energy model of absorption and has energy \(\frac{1}{2}\)K_{B}T. Each vibrational frequency has two modes of energy (Kinetic and Potential) with corresponding energy equal to 2 × \(\frac{1}{2}\)K_{B}T = K_{B}T.

**Important Formulae:**

→ K.E./mole of a gas = \(\frac{1}{2}\)MC^{2} = \(\frac{3}{2}\)RT

K.E./molecule = \(\frac{1}{2}\)mC^{2} = \(\frac{3}{2}\)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 R_{1} and R_{2} are diffusion rates of gases 1 and 2 having molecular masses M_{1} and M_{2}.