Physics: Class 11 (Book 2) — Chapter 12: Kinetic Theory

• Atomic Size: The typical size of an atom is of the order of 1 \text{ \AA} ($10^{-10}\text{ m}$).
• Interatomic Distances: In solids and liquids, atoms are tightly packed with a spacing of about 2 \text{ \AA} . In gases, the interatomic distances are much larger, typically in tens of angstroms ( 20 \text{ \AA} to 30 \text{ \AA} ).
• Intermolecular Forces: Molecules experience long-range attraction at larger separations and short-range repulsion when squeezed close together. In gases, because molecules are far apart, these forces are negligible except during collisions.

The state of an ideal gas is governed by the relation:

Where:

• $P$ is the pressure of the gas.
• $V$ is the volume.
• $T$ is the absolute temperature in Kelvin.
• $\mu$ is the number of moles, defined as $\mu = \frac{M}{M_0} = \frac{N}{N_A}$.
• $R$ is the universal gas constant ($R = 8.314\text{ J mol}^{-1}\text{K}^{-1}$).

Alternative forms of the ideal gas equation:

Where $k_B = \frac{R}{N_A} = 1.38 \times 10^{-23}\text{ J K}^{-1}$ is the Boltzmann constant , $N$ is the total number of molecules, and $n = \frac{N}{V}$ is the number density.

• Boyle's Law: For a fixed mass of gas at constant temperature, the volume is inversely proportional to pressure ($PV = \text{constant}$).
• Charles's Law: For a fixed mass of gas at constant pressure, the volume is directly proportional to absolute temperature ($V \propto T$).
• Dalton's Law of Partial Pressures: The total pressure exerted by a mixture of non-interacting ideal gases is equal to the sum of the partial pressures of individual gases: $$P = P_1 + P_2 + \dots = (\mu_1 + \mu_2 + \dots)\frac{RT}{V}$$

Consider a gas enclosed in a cube of side $l$ (volume $V = l^3$). Let a molecule of mass $m$ move with velocity $(v_x, v_y, v_z)$.

1. When the molecule collides elastically with the wall parallel to the $yz$-plane, its $x$-component of velocity reverses from $v_x$ to $-v_x$, while $v_y$ and $v_z$ remain unchanged.
2. The change in momentum of the molecule is: $$\Delta p_x = -m v_x - m v_x = -2 m v_x$$
3. By conservation of momentum, the momentum imparted to the wall in a single collision is $2 m v_x$.
4. In a small time interval $\Delta t$, the molecules that can hit the wall of area $A = l^2$ must be within a distance $v_x \Delta t$. The volume of this region is $A v_x \Delta t$.
5. On average, only half of the molecules in this volume are moving towards the wall. Thus, the number of collisions in time $\Delta t$ is: $$N_{\text{collisions}} = \frac{1}{2} A v_x \Delta t n$$ where $n$ is the number density of molecules.
6. The total momentum transferred to the wall in time $\Delta t$ is: $$Q = (2 m v_x) \left(\frac{1}{2} n A v_x \Delta t\right) = n m A v_x^2 \Delta t$$
7. The force exerted on the wall is the rate of momentum transfer: $$F = \frac{Q}{\Delta t} = n m A v_x^2$$
8. The pressure $P$ is force per unit area: $$P = \frac{F}{A} = n m v_x^2$$
9. Since molecules have a distribution of velocities, we replace $v_x^2$ with its average value $\bar{v_x^2}$: $$P = n m \bar{v_x^2}$$
10. Assuming the gas is isotropic, there is no preferred direction of motion: $$\bar{v_x^2} = \bar{v_y^2} = \bar{v_z^2} = \frac{1}{3} \bar{v^2}$$ where $\bar{v^2}$ is the mean square speed.
11. Substituting this gives the fundamental pressure equation: $$P = \frac{1}{3} n m \bar{v^2} = \frac{1}{3} \rho \bar{v^2}$$ where $\rho = n m$ is the mass density of the gas.

The pressure equation can be rewritten as:

Comparing this with the ideal gas equation $PV = N k_B T$, we get:

This shows that the average translational kinetic energy of a molecule is directly proportional to the absolute temperature of the gas and is independent of its pressure, volume, or nature.

The total internal energy $E$ of a mole of ideal gas (purely translational) is:

The root mean square speed $v_{\text{rms}}$ is defined as:

The number of independent coordinates required to completely specify the position and configuration of a system is called its degrees of freedom .

• Monatomic Gas: Has only 3 translational degrees of freedom ($f = 3$). Example: Helium, Argon.
• Diatomic Gas (Rigid Rotator): Has 3 translational and 2 rotational degrees of freedom ($f = 5$). Example: $\text{O}_2$, $\text{N}_2$, $\text{CO}$.
• Diatomic Gas (Non-Rigid): At high temperatures, vibrational modes are excited. A single vibrational mode contributes 2 quadratic terms (kinetic energy and potential energy), adding 2 degrees of freedom ($f = 7$).

In a solid, each atom is held in a lattice and vibrates about its mean position in 3 dimensions. Each 1D oscillation has 2 quadratic energy terms (kinetic and potential), so a 3D oscillation has 6 quadratic terms.

• Average energy per atom = $3 k_B T$.
• Total internal energy of 1 mole of solid ($N = N_A$): $$U = 3 R T$$
• Molar specific heat capacity of solids (Dulong-Petit Law): $$C = \frac{dU}{dT} = 3 R \approx 24.9\text{ J mol}^{-1}\text{K}^{-1}$$

1. Assume molecules are spheres of diameter $d$. Let a single molecule move with average speed $\langle v \rangle$ while all other molecules are temporarily assumed to be at rest.
2. In time $\Delta t$, the moving molecule sweeps out a collision cylinder of volume: $$V_{\text{cylinder}} = \pi d^2 \langle v \rangle \Delta t$$
3. If $n$ is the number density of molecules, the number of collisions in time $\Delta t$ is: $$N_{\text{collisions}} = n \pi d^2 \langle v \rangle \Delta t$$
4. The mean free path $l$ is the total distance traveled divided by the number of collisions: $$l = \frac{\langle v \rangle \Delta t}{n \pi d^2 \langle v \rangle \Delta t} = \frac{1}{n \pi d^2}$$
5. When the motion of all molecules is taken into account using Maxwell's velocity distribution, the relative velocity $\langle v_r \rangle = \sqrt{2}\langle v \rangle$ is used, yielding the exact formula: $$l = \frac{1}{\sqrt{2} n \pi d^2}$$