Consider a rod, at an angle to the normal of a wall, with its centre-of-mass velocity along the normal direction, of mass m and angular velocity about an axis parallel to the wall but perpendicular to the rod. Suppose one end of the rod, with normal component of velocity
is infinitesimally close to the wall and undergoes an instantaneous energy conserving collision; the momentum transfer from the rod to the wall can be easily found to be:
Since both and independently obey Gaussian distribution with their variances given by the equipartition of the thermal energy,
it follows, from Eq.(40), that also obeys a Gaussian distribution:
with a variance . The pressure can then be integrated to give:
which is the required result. Although derived by kinetic theory, this result must of course correspond to that found from an appropriate free energy derivative. Accordingly it remains applicable to rods in solution (rather than in vacuo), so long as p is interpreted as an osmotic pressure.