download the script: Internal Energy Changes (general expression)

Usually we can directly measure the pressure p, the temperature T and the volume V or the specific volume ν of a system. We choose the internal energy to be a function of T and ν, u=u(T,ν), and its total differential is:

We know that the term (∂u/∂ν)_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, (∂u/∂ν)_T  , does not disappear. Therefore, it is necessary for us to find out some kind of relationship 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:

du=T∙ds – p∙dν                    (1)

What’s more, the entropy is also a function which depends on temperature T and specific volume ν. 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 u can be expressed as:

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