## 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.