15.3.4—PV = 1/3 Nm<c2> with pV = NkT to deduce that the average translational

For N molecules of mass m, pV=⅓Nm⟨c²⟩. Combining with pV=NkT links pressure to mean translational kinetic energy per particle.

Use ⟨c²⟩ as the mean square speed, not the square of the mean speed, and keep N as particle count.

At fixed volume, raising T raises mean molecular kinetic energy and therefore pressure in proportion.

The equation describes a statistical average; not every molecule has the same speed or energy.