If we introduce Eq.s (V-4) and (V-5) in (IV-3), the term involving is eliminated and we obtain
However, from (II-1) we have
. Eq. (110) then simplifies to
. Finally, it is noted that
Elimination of between Eqs. (112) and (113) leads to
This represents an important form of the first law of thermodynamics for irreversible processes, involving the rate of change of entropy. This equation is a special case of a much more general relation formulated by Eckart (1940).