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Changes due to temperature and pressure

When considering fluids or solids, an expression in terms of the temperature and pressure is usually more useful:

where it is assumed that the heat capacity at constant pressure is related to the heat capacity at constant volume according to

The partial derivative of the pressure with respect to temperature at constant volume can be expressed in terms of the coefficient of thermal expansion

and the isothermal compressibility

by writing

and equating d V to zero and solving for the ratio d P /d T. This gives

Substituting (2) and (3) in (1) gives the above expression.