A brief summary of some useful thermodynamic relations

A brief summary of some useful thermodynamic relations 

Statistical mechanics provides relations between macroscopic thermodynamic quantities and microscopic molecular properties. Thermodynamics, on the other hand, provides only relationships between various thermodynamic quantities. The multitude of these relationships arise from the freedom we have in choosing the independent variables to describe a thermodynamic system. For instance, we can choose the variables T, V, N to describe the system. Hence, all the other variables such as energy, entropy, pressure, etc., are viewed as functions of these independent variables or we could choose T, P, N to describe the system and view all other variables, such as energy, entropy, volume, etc., as functions of T, P, N. 

Many more relationships can be obtained from equations (A.1)-(A.4). Some of the more useful are the heat capacity at constant volume and at constant pressure  

Partial molar quantities are defined for any extensive quantity E, as the partial derivative of E with respect to Nt, at constant T, P, N i.e., 

The chemical potential defined in (A.8) is the partial molar Gibbs energy. 

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