Question

derive the relationship between the heat capacity at constant pressure and at constant temperature​

Answer 1

Answer:

The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23):

{\displaystyle C_{P}-C_{V}=VT{\frac {\alpha ^{2}}{\beta _{T}}}\,} C_{{P}}-C_{{V}}=VT{\frac {\alpha ^{{2}}}{\beta _{{T}}}}\,

{\displaystyle {\frac {C_{P}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}\,} {\frac {C_{{P}}}{C_{{V}}}}={\frac {\beta _{{T}}}{\beta _{{S}}}}\,

Here {\displaystyle \alpha } \alpha is the thermal expansion coefficient:

{\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\,} \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{{P}}\,

{\displaystyle \beta _{T}} \beta _{{T}} is the isothermal compressibility (the inverse of the bulk modulus):

{\displaystyle \beta _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\,} \beta _{{T}}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{{T}}\,

and {\displaystyle \beta _{S}} \beta _{{S}} is the isentropic compressibility:

{\displaystyle \beta _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\,} \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:

{\displaystyle c_{p}-c_{v}={\frac {T\alpha ^{2}}{\rho \beta _{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.

The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size-dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Thus:

{\displaystyle {\frac {c_{p}}{c_{v}}}={\frac {\beta _{T}}{\beta _{S}}}\,} {\frac {c_{{p}}}{c_{{v}}}}={\frac {\beta _{{T}}}{\beta _{{S}}}}\,

The difference relation allows one to obtain the heat capacity for solids at constant volume which is not readily measured in terms of quantities that are more easily measured. The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio.