Relations between heat capacities

In thermodynamics, the heat capacity at constant volume, C V {\displaystyle C_{V}} , and the heat capacity at constant pressure, C P {\displaystyle C_{P}} , are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): C P − C V = V T α 2 β T {\displaystyle C_{P}-C_{V}=VT{\frac {\alpha ^{2}}{\beta _{T}}}\,} C P C V = β T β S {\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}\,} Here α {\displaystyle \alpha } is the thermal expansion coefficient: α = 1 V ( ∂ V ∂ T ) P {\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\,} β T {\displaystyle \beta _{T}} is the isothermal compressibility (the inverse of the bulk modulus): β T = − 1 V ( ∂ V ∂ P ) T {\displaystyle \beta _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\,} and β S {\displaystyle \beta _{S}} is the isentropic compressibility: β S = − 1 V ( ∂ V ∂ P ) S {\displaystyle \beta _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\,} A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: c p − c v = T α 2 ρ β T {\displaystyle c_{p}-c_{v}={\frac {T\alpha ^{2}}{\rho \beta _{T}}}} where ρ is the density of the substance under the applicable conditions.

Source: Wikipedia — Relations between heat capacities (CC BY-SA 4.0)