Combining eq.(21.26), TdS-dq 0 and eq.(21.13), dU = dq+dw, we have, since dq TdS
dU TdS + dw
(21.33)
Rearranging,
dwdU – TdS
(21.34)
The most negative value of dw is the maximum work dw max obtained from the system;
dw max = - dw
or dw = -dwmaxdU –TdS
or dw max -dU + TdS = -dA
(21.35)
at constant temperature. The maximum work that can be obtained from the system, dwmax, is equal to the decrease in the Helmholtz free energy of the system. Next, we want to consider the non-expansion work (i.e., work other than the pressure volume work such as electrical work) obtainable from the system. The total work done on the system dw is the sum of expansion work – PdV plus any non expansion work w ne
TdS 
