Heat capacity and definition of enthalpy

In general, the change in internal energy or simply the energy of a system U may now be written as 

The temperature dependence of the internal energy is given by the heat capacity at constant volume at a given temperature, formally defined by 

For a constant-volume system, an infinitesimal change in temperature gives an infinitesimal change in internal energy and the constant of proportionality is the heat capacity at constant volume 

For processes taking place at constant pressure it is convenient to introduce the enthalpy function, H, defined as 

When only work against a constant external pressure is done  

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