Key Points

The state of an ideal gas is described by the equation $PV = \mu RT$, where P is pressure, V is volume, T is absolute temperature, $\mu$ is the number of moles, and R is the universal gas constant ($8.314 \text{ J mol}^{-1} \text{K}^{-1}$).

In terms of molecules, the ideal gas equation is $PV = N k_B T$, where N is the total number of molecules and $k_B$ is the Boltzmann constant ($1.38 \times 10^{-23} \text{ J K}^{-1}$).

Gases consist of a large number of identical molecules in random motion. Collisions between molecules and with container walls are perfectly elastic, and the volume of molecules is negligible compared to the container volume.

According to kinetic theory, the pressure exerted by a gas is given by $P = \frac{1}{3} n m \overline{v^2}$, where n is the number density of molecules, m is the mass of a molecule, and $\overline{v^2}$ is the mean squared speed.

The average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature T. The relationship is $\frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T$.

The rms speed of gas molecules is the square root of the mean of squared speeds, given by $v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3k_B T}{m}} = \sqrt{\frac{3RT}{M_0}}$, where $M_0$ is the molar mass.

In thermal equilibrium, the total energy of a system is equally distributed among all its degrees of freedom. The average energy associated with each degree of freedom is $\frac{1}{2} k_B T$.

A monatomic gas has 3 translational degrees of freedom. A diatomic gas has 3 translational and 2 rotational degrees of freedom at moderate temperatures, for a total of 5.

Each vibrational mode of a molecule contributes two degrees of freedom (kinetic and potential). Thus, the average energy per vibrational mode is $2 \times (\frac{1}{2} k_B T) = k_B T$.

For a monatomic gas (3 degrees of freedom), the molar specific heat at constant volume is $C_V = \frac{3}{2}R$, at constant pressure is $C_p = \frac{5}{2}R$, and their ratio is $\gamma = \frac{5}{3}$.

For a rigid diatomic gas (5 degrees of freedom), the molar specific heat at constant volume is $C_V = \frac{5}{2}R$, at constant pressure is $C_p = \frac{7}{2}R$, and their ratio is $\gamma = \frac{7}{5}$.

For any ideal gas, the difference between the molar specific heat at constant pressure ($C_p$) and constant volume ($C_V$) is equal to the universal gas constant R: $C_p - C_V = R$.

The mean free path ($l$) is the average distance a molecule travels between two successive collisions. It is given by $l = \frac{1}{\sqrt{2} n \pi d^2}$, where n is the number density and d is the molecular diameter.

The total pressure of a mixture of non-reacting ideal gases is the sum of the partial pressures of the individual gases. $P_{total} = P_1 + P_2 + P_3 + \dots$.

Equal volumes of all gases at the same temperature and pressure contain an equal number of molecules. This number is Avogadro's number, $N_A = 6.02 \times 10^{23} \text{ molecules per mole}$.