Relationships between the molar heat capacities Cp and Cv

Example 1.2 Relationships between the molar heat capacities Cp and Cv The first law of thermodynamics leads to a relation between the molar heat capacities. The change in internal energy expressed in volume and temperature (7= (7(7) F) is 

The right side of Eq. (1.64) shows the energy effect due to the expansion of volume at a constant pressure process. For a mixture of ideal gases, the internal energy is a function of temperature only, and hence Eq. (1.64) and PV=RT yields 

The gas constant R is common to all gases and determined by the product of the Boltzmann constant and the Avogadro number 

In contrast, for a fluid, the enthalpy and entropy may be expressed as a function of temperature and pressure and in terms of thermal expansion/ = (dV/dT)P/V and isothermal compressibility k = — (dVldP)jlV as follows  

For the internal energy and entropy, from Eqs. (1.23) and (1.24) we also have 

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