<P> One can similarly rewrite the partial derivative (∂ S ∂ T) V (\ displaystyle \ left ((\ frac (\ partial S) (\ partial T)) \ right) _ (V)) by expressing dV in terms of dS and dT, putting dV equal to zero and solving for the ratio d S d T (\ displaystyle (\ frac (dS) (dT))). When one substitutes that expression in the heat capacity ratio expressed as the ratio of the partial derivatives of the entropy above, it follows: </P> <Dl> <Dd> C P C V = (∂ P ∂ T) S (∂ P ∂ S) T (∂ V ∂ S) T (∂ V ∂ T) S (\ displaystyle (\ frac (C_ (P)) (C_ (V))) = (\ frac (\ left ((\ frac (\ partial P) (\ partial T)) \ right) _ (S)) (\ left ((\ frac (\ partial P) (\ partial S)) \ right) _ (T))) (\ frac (\ left ((\ frac (\ partial V) (\ partial S)) \ right) _ (T)) (\ left ((\ frac (\ partial V) (\ partial T)) \ right) _ (S))) \,) </Dd> </Dl> <Dd> C P C V = (∂ P ∂ T) S (∂ P ∂ S) T (∂ V ∂ S) T (∂ V ∂ T) S (\ displaystyle (\ frac (C_ (P)) (C_ (V))) = (\ frac (\ left ((\ frac (\ partial P) (\ partial T)) \ right) _ (S)) (\ left ((\ frac (\ partial P) (\ partial S)) \ right) _ (T))) (\ frac (\ left ((\ frac (\ partial V) (\ partial S)) \ right) _ (T)) (\ left ((\ frac (\ partial V) (\ partial T)) \ right) _ (S))) \,) </Dd> <P> Taking together the two derivatives at constant S: </P>

Relation between cp and cv for ideal gas