28 The Transformations of U and H
In a isochoric process ΔQ' = ΔH' holds. The heat energy added contributes to the internal energy U' and the thermodynamic work P' · V' . Even if V' is constant, pressure P' increases when the gas is heated up in a isochoric process.
In the system at rest we have ΔE = ΔQ = ΔH . From 26 we deduce
ΔH' = ΔQ' = ΔQ · √ = ΔH · √
Enthalpy H transforms by multiplication by the root term.
From U = H – P · V and U' = H' — P' · V' we finally get the transformation of internal energy U :
U' = H' — P' · V' = H · √ – P · V · √ = ( H – P · V ) · √ = U · √
Internal energy U transforms by multiplication by the root term, too. We will derive this result on a totally different route in section 30 .
By the way, the same type of transformation applies for Helmholtz free energy F = U – T · S and Gibbs free energy G = H – T · S . According to 8, 12 and 14 the product of T and S has to be transformed by multiplication by the root.
In the next section we will show that these results are consistent with the transformation rule of total Energy E . Total Energy E ist transformed 'the other way round', we have E' = E / √ . However, our pieces fit perfectly into one another.