Note 50 — Chapter VIII · SCIENTIFIC WORK—MOLECULAR THEORY.

The proof is as follows:—

If σ be the specific heat at constant volume, σ′ at constant pressure, and consider a unit of mass of gas at pressure p and volume v, let the volume increase by an amount dv, while the temperature dy.

Thus σ′dT = σdT + pdv

But pv = ⅔T/m

Hence p being constant,

pdv = ⅔ dT/m Therefore σ′ = σ + ⅔ 1/m

Now suppose an amount of heat, dH, is given to a single molecule and that its temperature is T. Its specific heat is σ, and

dH = σmdT But dH = βdT Therefore β = σm

Hence 1/m = σ/β

Thus σ′ = σ(1 + 2/(3β))

And σ′/σ = γ

Therefore γ = 1 + 2/(3β)

Or β = 2/(3(γ-1))