Internal Energy and Chemical Potential

Atmospheric Chemistry and Physics From Air Pollution to Climate Change, Second Edition, by John H. Seinfeld and Spyros N. Pandis. Copyright 2006 John Wiley Sons, Inc. 

The infinitesimal work done on the system by its surroundings is equal to 

A thermodynamically reversible process is defined as one in which the system changes infinitesimally slowly from one equilibrium state to the next. According to the second law of thermodynamics, the heat added to a system during a reversible process dQKV is given by 

In our discussion so far we have assumed that the system is closed. According to (10.4), if the number of moles of all system species ti, n2, , remains constant, the change in internal energy U of the system depends only on changes in 5 and V. However, for variable composition we must have 

In this expression, the subscript n, in the first two partial derivatives implies that the amounts of all species are constant during the variation in question. On the other hand, the last partial derivative assumes that all but the ith substance are constant. Note that for a closed system drij = 0 and from (10.5) 

Comparing (9.4) and (9.6) that are both valid for a closed system, we obtain 

---