Application of Law of Equipartition Energy in Specific Heat of a Gas

Meyer's relation. For one mole of an ideal gas, the molar specific heat at constant pressure $C_P$ and at constant volume $C_V$ are always related by Meyer's relation, $C_P - C_V = R$. Combined with the law of equipartition of energy (which fixes the total internal energy $U$ of one mole purely from its degrees of freedom $f$, via $U=\tfrac{f}{2}N_AkT=\tfrac{f}{2}RT$), both $C_V$, $C_P$ and their ratio $\gamma=C_P/C_V$ (the adiabatic exponent) can be worked out exactly for each type of molecule, using $C_V=dU/dT$.

(i) Monatomic molecule ($f=3$). $U=\tfrac32 RT$ per mole, so $C_V=dU/dT=\tfrac32 R$. Then $C_P=C_V+R=\tfrac32 R+R=\tfrac52 R$, and

(ii) Diatomic molecule. At low/normal temperature ($f=5$): $U=\tfrac52 RT$, so $C_V=\tfrac52 R$, $C_P=\tfrac52 R+R=\tfrac72 R$, and $\gamma=\tfrac{7/2}{5/2}=\tfrac75=1.40$. At high temperature ($f=7$, vibrational modes active): $U=\tfrac72 RT$, so $C_V=\tfrac72 R$, $C_P=\tfrac72 R+R=\tfrac92 R$, and $\gamma=\tfrac{9/2}{7/2}=\tfrac97\approx1.28$. Note that both $C_V$ and $C_P$ are larger for a diatomic gas than for a monatomic gas -- a diatomic gas genuinely needs more heat energy to raise its temperature by $1^\circ$C, because it has more degrees of freedom competing to absorb that energy.

(iii) Triatomic molecule. Linear ($f=7$): identical to the high-temperature diatomic case, $C_V=\tfrac72 R$, $C_P=\tfrac92 R$, $\gamma=\tfrac97\approx1.28$. Non-linear ($f=6$): $U=3RT$, so $C_V=dU/dT=3R$, $C_P=3R+R=4R$, and $\gamma=\tfrac43\approx1.33$.

An important caveat. This kinetic-theory model predicts that $C_V$ and $C_P$ are completely independent of temperature (fixed numbers set only by $f$) -- but in reality, specific heat capacities of real gases do vary somewhat with temperature (most visibly as vibrational modes gradually switch on), so the fixed-$f$ picture used here is an idealisation, accurate over a given temperature range but not exact at every temperature. …