Kinetic Theory of Gases

The atomic theory proposed by John Dalton and consolidated through Brownian motion (Robert Brown, 1827; Albert Einstein, 1905) affirms that matter is not continuous, but particulate. A macroscopic gas sample consists of an astronomical number of microscopic entities called molecules. The molecular diameter is of the order of $10^{-10}\,\text{m} = 1\,\text{\AA}$, whereas the average intermolecular separation under standard conditions ($1\,\text{atm}, 0\,^\circ\text{C}$) is roughly $3 \times 10^{-9}\,\text{m}$, approximately 30 times the molecular diameter.

Avogadro's Constant ($N_A$): One mole of any substance contains precisely the same number of elementary entities:

Centuries of experimental investigations yielded four empirical gas laws governing the macroscopic parameters: Pressure ($P$), Volume ($V$), Absolute Temperature ($T$), and Amount of substance ($n$):

• Boyle's Law (Robert Boyle, 1662): For a fixed mass of gas at constant temperature, volume is inversely proportional to pressure: $P V = \text{constant} \iff P_1 V_1 = P_2 V_2$.
• Charles's Law (Jacques Charles, 1787): For a fixed mass of gas at constant pressure, volume is directly proportional to absolute temperature: $\frac{V}{T} = \text{constant} \iff \frac{V_1}{T_1} = \frac{V_2}{T_2}$.
• Gay-Lussac's Law / Pressure Law (1802): For a fixed mass of gas at constant volume, pressure is directly proportional to absolute temperature: $\frac{P}{T} = \text{constant} \iff \frac{P_1}{T_1} = \frac{P_2}{T_2}$.
• Avogadro's Law (Amedeo Avogadro, 1811): Equal volumes of all gases under identical temperature and pressure conditions contain equal numbers of molecules: $V \propto n$.

Synthesizing these four empirical laws yields the unified Ideal Gas Equation of State:

where:

• $n = \frac{m}{M} = \frac{N}{N_A}$ is the number of moles ($m = \text{mass of gas}$, $M = \text{molar mass}$).
• $R = 8.314\,\text{J}/(\text{mol}\cdot\text{K})$ is the Universal Gas Constant.
• $k_B = \frac{R}{N_A} = 1.3806 \times 10^{-23}\,\text{J}/\text{K}$ is Boltzmann's Constant (the gas constant per single molecule).
• $N$ is the total number of molecules in the volume $V$.

The kinetic theory formulated by Clausius, Maxwell, and Boltzmann relies on six core simplifying assumptions:

1. Molecular Composition: Any gas consists of an immense number of identical, tiny, hard, spherical, perfectly elastic particles called molecules.
2. Point Masses (Negligible Volume): The actual volume occupied by the gas molecules themselves is negligible compared to the total volume enclosing the gas ($V_{\text{molecules}} \ll V_{\text{container}}$).
3. Chaotic Motion: Molecules are in perpetual, chaotic, straight-line random motion in all possible directions with a broad distribution of speeds.
4. Perfect Elasticity: Collisions between molecules and with the container walls are instantaneous and perfectly elastic; linear momentum and kinetic energy are conserved during collisions.
5. Zero Intermolecular Forces: Molecules exert no forces on one another (neither attractive nor repulsive) except during instantaneous contact ($U_{\text{potential}} = 0$).
6. Negligible Gravitational Influence: Due to extremely small molecular masses and very high speeds, gravitational effects on molecular trajectories are completely negligible.

Consider a cubical container of side $L$ (volume $V = L^3$) containing $N$ identical molecules, each of mass $m$. Let a molecule possess velocity vector $\vec{v}_1 = (v_{x1}, v_{y1}, v_{z1})$.

1. When this molecule collides elastically with the wall perpendicular to the $x$-axis (at $x = L$):

By Newton's third law, the momentum transferred to the wall per collision is $+2 m v_{x1}$.

2. Time between two successive collisions of this molecule with the same wall:

3. Force exerted on the wall by molecule 1 (rate of momentum transfer):

4. Total force on the wall from all $N$ molecules:

where $\overline{v_x^2} = \frac{1}{N}\sum_{i=1}^N v_{xi}^2$ is the mean square velocity in the $x$-direction.

5. Pressure $P$ is force divided by wall area $A = L^2$:

6. Since the molecular motion is completely isotropic (no preferred direction in space):

where $\overline{v^2} = \overline{v_x^2} + \overline{v_y^2} + \overline{v_z^2}$ is the mean square speed.

Multiplying both sides of the kinetic pressure equation by volume $V$:

where $E_{\text{trans}} = N \left(\frac{1}{2} m \overline{v^2}\right)$ is the total translational kinetic energy of the gas. Comparing with the ideal gas law $P V = N k_B T$:

For 1 mole of gas ($N = N_A, N_A k_B = R$):

The average translational kinetic energy per single molecule is:

Due to ceaseless collisions, gas molecules exhibit a wide statistical spread of speeds described by the Maxwell-Boltzmann distribution. Three distinct speeds characterize the distribution:

Ratio of Molecular Speeds:

Therefore, for any gas at any temperature: $v_{\text{rms}} > \overline{v} > v_{\text{mp}}$.

The Degrees of Freedom ($f$) of a dynamical system is the total number of independent coordinates or velocity components required to completely specify the position and configuration of all particles in the system in space.

If a molecule contains $N_a$ atoms and is subject to $k$ independent geometric constraints (such as fixed bond lengths):

• Monoatomic Gas (He, Ne, Ar, Kr): Point mass moving freely in 3D space. No rotational inertia about its center. Constraints $k = 0$.
  Degrees of freedom: $f = 3 \text{ (3 translational: } v_x, v_y, v_z)$.
• Diatomic Gas (rigid rotator, e.g., $\text{O}_2, \text{N}_2, \text{CO}$ at room temperature): 2 atoms connected by a fixed bond length ($k = 1$). Total $3 \times 2 - 1 = 5$.
  Degrees of freedom: $f = 5$ (3 translational + 2 rotational about axes perpendicular to the internuclear axis). Rotation about the bond axis has negligible moment of inertia.
• Diatomic Gas at High Temperatures: Interatomic vibration is activated, adding 1 vibrational mode with 2 energy terms (kinetic + potential):
  Degrees of freedom: $f = 3 \text{ trans} + 2 \text{ rot} + 2 \text{ vib} = 7$.
• Polyatomic Non-linear Gas ($\text{H}_2\text{O}, \text{NH}_3, \text{CH}_4$): 3 or more atoms not in a straight line.
  Degrees of freedom: $f = 6$ (3 translational + 3 rotational about three mutually orthogonal principal axes).

Total internal energy per molecule having $f$ degrees of freedom:

For 1 mole of gas ($N = N_A$), the total internal energy is:

From thermodynamics, the molar heat capacity at constant volume is defined as $C_v = \left(\frac{dU}{dT}\right)_v$. Substituting $U = \frac{f}{2} R T$:

Using Mayer's relation for an ideal gas ($C_p - C_v = R$):

The Adiabatic Index (Specific Heat Ratio $\gamma$) is:

When $n_1$ moles of gas 1 (with molar heat capacity $C_{v1}$) are mixed with $n_2$ moles of gas 2 (with $C_{v2}$):

Between successive collisions, a gas molecule travels in a straight line with constant velocity. The distance traversed between two consecutive collisions is the free path. Because collision intervals vary randomly, we define the Mean Free Path ($\lambda$):

Definition: The average distance traversed by a molecule between two successive collisions with other gas molecules.

Let the molecules be spheres of collision diameter $d$. Consider a target molecule moving with average speed $\overline{v}$ while all other molecules are temporarily assumed stationary.

1. In time $\Delta t$, the molecule sweeps out an imaginary collision cylinder of cross-sectional area $\pi d^2$ and length $\overline{v}\Delta t$, giving cylinder volume:$$V_{\text{cyl}} = \pi d^2 \overline{v}\Delta t$$
2. If $n_0 = \frac{N}{V}$ is the number density of molecules, the number of collisions in time $\Delta t$ is:$$N_{\text{coll}} = n_0 V_{\text{cyl}} = n_0 \pi d^2 \overline{v}\Delta t$$
3. The collision frequency (collisions per second) is $\nu = n_0 \pi d^2 \overline{v}$, and the mean free path is:$$\lambda = \frac{\text{Total Distance}}{\text{Number of Collisions}} = \frac{\overline{v}\Delta t}{n_0 \pi d^2 \overline{v}\Delta t} = \frac{1}{\pi n_0 d^2}$$
4. Accounting for the relative motion of all molecules (Maxwellian velocity distribution), the average relative speed between colliding molecules is $\overline{v}_{\text{rel}} = \sqrt{2}\,\overline{v}$.

In 1873, Johannes Diderik van der Waals modified the ideal gas equation by introducing two corrective factors addressing the flaws in the kinetic postulates:

1. Volume Correction: Real molecules possess finite physical size. The free volume available for motion is reduced from $V$ to $(V - n b)$, where $b = 4 N_A \left(\frac{4}{3}\pi r^3\right)$ is the co-volume or excluded volume per mole.
2. Pressure Correction: Real molecules exert attractive van der Waals forces on one another. A molecule near the container wall experiences an inward net attraction from interior molecules, reducing impact momentum. The measured pressure $P$ is lower than ideal pressure by $\frac{a n^2}{V^2}$, where $a$ is a measure of intermolecular attraction forces.

For 1 mole of real gas, the van der Waals Equation of State is:

For $n$ moles: