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Similar with the internal energy, we choose the enthalpy to be a function of T and p, h=h(T,p), and its total differential is:

We know that the term (∂h/∂p)_{T} equals to zero when the gas is assumed to be ideal gas. However, if the properties of a gas do not follow the ideal gas law, that is, p∙ν=R∙T+f(p,T,ν), then this term, (∂h/∂p)_{T} , does not disappear. Therefore, it is necessary for us to find out some kind of relation concerning this term which can be determined by the pressure, temperature or volume.

For a compressible system, we know from the definition of entropy that:

dh=T∙ds + ν∙dp (1)

What’s more, the entropy is also a function which depends on temperature T and pressure p. So the total differential of entropy function can be expressed as:

Now we eliminate the ds in equation (1) by using equation (2), we obtain:

So we compare the equation (*) with equation (3), we have then the following relations:

Using now the Maxwell Relation, we get:

Hence the total differential of internal energy h can be expressed as:

It is obvious to notice now that the change in enthalpy of a system can be calculated by pressure, temperature and volume (or specific volume).