Relation between Pressure and Mean Kinetic Energy

Pressure in terms of internal energy. From the internal energy result (9.11), $U=\tfrac32 NkT$, and since the ideal gas equation gives $NkT=PV$, this can be written $U=\tfrac32 PV$, or rearranged as

where $u=U/V$ is the internal energy per unit volume, i.e. the internal energy density. Equation (9.12) says the pressure of a gas is always exactly two-thirds of its internal energy density -- a compact, purely thermodynamic restatement of the kinetic-theory result.

Pressure in terms of mean kinetic energy density. Equation (9.6) can also be written using the mass density $\rho=nm$ (mass per unit volume) as

Multiplying and dividing the right-hand side by 2,

$P = \frac23\left(\frac12\rho\overline{v^2}\right). \qquad (9.14)$ …