Page 41 - Advanced thermodynamics for engineers

24 CHAPTER 2 THE SECOND LAW AND EQUILIBRIUM

dQ R
and the change in entropy of system A during this process, because it is reversible, is dS ¼ .
T
Hence dW net ¼ dE þ T 0 dS ¼ dðE T 0 SÞ; because T 0 is constant: (2.24)
The expression E T 0 S is called the Helmholtz energy or Helmholtz function. In the absence of
motion and gravitational effects the energy, E, may be replaced by the intrinsic internal energy, U,
giving
dW net ¼ dðU T 0 SÞ (2.25)
The significance of dW net will now be examined. The changes executed were considered to be
reversible and dW net was the net work obtained from system B (i.e. system A þ heat engine E R ). Thus
dW net must be the maximum quantity of work that can be obtained from the combined system.
The expression for dW net is called the change in the Helmholtz energy, where the Helmholtz energy is
defined as
(2.26)
F ¼ U TS
Helmholtz energy is a property which has the units of energy and indicates the maximum work that
can be obtained from a system. It can be seen that this is less than the internal energy, U, and it will be
shown that the product TS is a measure of the unavailable energy.

2.11 GIBBS ENERGY
In the previous section the maximum work that can be obtained from system B, comprising system A
and heat engine E R was derived. It was also stipulated that system A could change its volume by dV,
and while it is doing this it must perform work on the atmosphere equivalent to p 0 dV, where p 0 is the
pressure of the atmosphere. This work detracts from the work previously calculated and gives the
maximum useful work, dW u as
dW u ¼ dW net p 0 dV (2.27)
if the system is in pressure equilibrium with surroundings,

dW u ¼ dðE T 0 SÞ p 0 dV ¼ d E þ p 0 V T 0 S because p 0 is constant:
Hence,
dW u ¼ dðH TSÞ (2.28)
The quantity H TS is called the Gibbs energy, Gibbs potential or the Gibbs function, G.
Hence,
(2.29)
G ¼ H TS
Gibbs energy is a property which has the units of energy, and indicates the maximum useful work
that can be obtained from a system. It can be seen that this is less than the enthalpy, H, and it will be
shown that the product TS is a measure of the unavailable energy.