Some Elementary Deductions from Kinetic Theory of Gases

Boyle's law. From equation (9.12), $PV=\tfrac23 U$. The internal energy of an ideal gas also equals $N$ times the average kinetic energy $\epsilon$ of a single molecule, $U=N\epsilon$. At a fixed temperature, $\epsilon$ stays constant (since $\epsilon=\tfrac32 kT$ depends only on $T$), so

This is exactly Boyle's law: at constant temperature, the pressure of a fixed amount of gas is inversely proportional to its volume.

Charles' law. From the same relation, $PV=\tfrac23 U$, now hold the pressure fixed instead. Then $V$ is directly proportional to the internal energy $U$ (equivalently, to the average kinetic energy of the gas), and since $U\propto T$, it follows that

This is exactly Charles' law: at constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature.

Avogadro's law. Consider two different gases at the same temperature and pressure, with $N_1$ and $N_2$ molecules respectively (masses $m_1,m_2$, mean square speeds $\overline{v_1^2},\overline{v_2^2}$). Applying equation (9.6) to each gas separately,

Since both gases are at the same temperature, they have the same average kinetic energy per molecule (equation 9.9 does not depend on the type of molecule):

$\frac12 m_1\overline{v_1^2} = \frac12 m_2\overline{v_2^2}. \qquad (9.16)$ …