quality:correction, information:filtering, linguistic:translation
At constant volume, dV = 0, δQ = dU (isochoric process)

A system undergoing a process at constant volume implies that no expansion work is done, so the heat supplied contributes only to the change in internal energy. The heat capacity obtained this way is denoted C V. {\displaystyle C_{V}.} The value of C V {\displaystyle C_{V}} is always less than the value of C p. {\displaystyle C_{p}.} (C V {\displaystyle C_{V}} < C p. {\displaystyle C_{p}.})

Expressing the inner energy as a function of the variables T {\displaystyle T} and V {\displaystyle V} gives:

δ Q = (∂ U ∂ T) V d T + (∂ U ∂ V) T d V + p d V {\displaystyle \delta Q=\left({\frac {\partial U}{\partial T}}\right)_{V}dT+\left({\frac {\partial U}{\partial V}}\right)_{T}dV+pdV}

For a constant volume (d V = 0 {\displaystyle dV=0}) the heat capacity reads:

C V = δ Q d T | V = const = (∂ U ∂ T) V {\displaystyle C_{V}={\frac {\delta Q}{dT}}{\Bigr |}_{V={\text{const}}}=\left({\frac {\partial U}{\partial T}}\right)_{V}}

The relation between C V {\displaystyle C_{V}} and C p {\displaystyle C_{p}} is then:

C p = C V + ((∂ U ∂ V) T + p) (∂ V ∂ T) p {\displaystyle C_{p}=C_{V}+\left(\left({\frac {\partial U}{\partial V}}\right)_{T}+p\right)\left({\frac {\partial V}{\partial T}}\right)_{p}}