Kinetic Theory of Gases

Kinetic Theory Assumptions

The kinetic theory of gases models a gas as a large number of tiny molecules in random motion. Key assumptions:

1. Gas consists of a large number $N$ of identical molecules, each of mass $m$.
2. Molecules are point particles — their size is negligible compared to the average intermolecular separation.
3. Molecules move randomly with all directions equally probable (isotropic).
4. Intermolecular forces are negligible except during brief elastic collisions.
5. Collisions with walls and other molecules are perfectly elastic (kinetic energy conserved).
6. The time of collision is much smaller than the time between collisions.

Pressure of an Ideal Gas

Consider $N$ molecules in a cubical container of side $L$. Pressure arises from momentum transfer during wall collisions: $P = \frac{1}{3}\frac{N}{V}m\overline{v^2} = \frac{1}{3}\rho\overline{v^2}$

where $\overline{v^2}$ is the mean square speed and $\rho = Nm/V$ is the gas density. This connects the macroscopic quantity (pressure) to microscopic motion.

Root Mean Square Speed

The root mean square (rms) speed: $v_{\text{rms}} = \sqrt{\overline{v^2}} = \sqrt{\frac{3k_BT}{m}} = \sqrt{\frac{3RT}{M}}$

where $k_B = 1.38 \times 10^{-23}$ J/K is Boltzmann's constant, $m$ is the molecular mass, and $M$ is the molar mass. Key dependence: $v_{\text{rms}} \propto \sqrt{T/M}$.

Kinetic Energy and Temperature

The average translational kinetic energy per molecule: $\overline{KE} = \frac{1}{2}m\overline{v^2} = \frac{3}{2}k_BT$

This is the fundamental link between temperature and molecular motion. Temperature is a measure of the average translational kinetic energy of gas molecules. For $n$ moles: $KE_{\text{total}} = \frac{3}{2}nRT$

Ideal Gas Law from Kinetic Theory

From $P = \frac{1}{3}\frac{Nm\overline{v^2}}{V}$ and $\frac{1}{2}m\overline{v^2} = \frac{3}{2}k_BT$: $PV = Nk_BT = nRT$

The ideal gas law emerges naturally from kinetic theory, validating the model.

Degrees of Freedom and Equipartition of Energy

Each independent mode of energy storage is a degree of freedom. The equipartition theorem states: each degree of freedom contributes $\frac{1}{2}k_BT$ of energy per molecule.

At very high temperatures, vibrational degrees of freedom activate: each vibrational mode contributes 2 degrees of freedom (kinetic + potential), so diatomic gas at high T has $f = 7$.

Specific Heats from Kinetic Theory

Internal energy per mole: $U = \frac{f}{2}RT$

$C_v = \frac{dU}{dT} = \frac{f}{2}R, \quad C_p = C_v + R = \frac{f+2}{2}R, \quad \gamma = \frac{C_p}{C_v} = \frac{f+2}{f}$

Maxwell-Boltzmann Speed Distribution

Not all molecules move at the same speed. The Maxwell-Boltzmann distribution gives the fraction of molecules with speeds between $v$ and $v + dv$: $f(v) = 4\pi n \left(\frac{m}{2\pi k_BT}\right)^{3/2} v^2 e^{-mv^2/(2k_BT)}$

Three Characteristic Speeds

1. Most probable speed: $v_p = \sqrt{\frac{2k_BT}{m}} = \sqrt{\frac{2RT}{M}}$
2. Mean speed: $v_{\text{avg}} = \sqrt{\frac{8k_BT}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}$
3. RMS speed: $v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}} = \sqrt{\frac{3RT}{M}}$

Ratio: $v_p : v_{\text{avg}} : v_{\text{rms}} = 1 : 1.128 : 1.224 = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$

Always: $v_p < v_{\text{avg}} < v_{\text{rms}}$.

Mean Free Path

The average distance a molecule travels between successive collisions: $\lambda = \frac{1}{\sqrt{2}\pi d^2 n}$

where $d$ is the molecular diameter and $n = N/V$ is the number density. Using $PV = Nk_BT$: $\lambda = \frac{k_BT}{\sqrt{2}\pi d^2 P}$

Mean free path increases with temperature and decreases with pressure.

Dalton's Law of Partial Pressures

For a mixture of non-reacting ideal gases in a container: $P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$

Each gas exerts pressure independently as if other gases were absent. Partial pressure: $P_i = n_iRT/V = x_iP_{\text{total}}$, where $x_i = n_i/n_{\text{total}}$ is the mole fraction.

Gas Mixture Properties

For a mixture of ideal gases:

Equivalent molar mass: $M_{\text{mix}} = \frac{n_1M_1 + n_2M_2}{n_1 + n_2}$

Equivalent $\gamma$: From $C_{v,\text{mix}} = \frac{n_1C_{v1} + n_2C_{v2}}{n_1 + n_2}$ and $C_{p,\text{mix}} = C_{v,\text{mix}} + R$: $\frac{n_{\text{total}}}{\gamma_{\text{mix}} - 1} = \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1}$

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