Internal Energy Changes

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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:

delta u general 1We 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:

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

delta u general 3So we compare the equation (*) with equation (3), we have then the following relations: delta u general 4

Using now the Maxwell Relation, we get: delta u general 5

Hence the total differential of internal energy u can be expressed as: delta u general 6

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).