Page 39 - Advanced thermodynamics for engineers

22 CHAPTER 2 THE SECOND LAW AND EQUILIBRIUM

mechanical systems, is to consider the two blocks of metal to be in equilibrium and for heat transfer to
occur spontaneously (and reversibly) between one and the other. Assume the temperature change in
each block is dT, with one temperature increasing and the other decreasing, and the heat transfer is dQ.
Then the change of entropy, dS, is given by

dQ dQ dQ
dS ¼ ¼ ðT dT T dTÞ
T þ dT T dT ðT þ dTÞðT dTÞ
(2.19)
dQ dT
¼ 2 2 ð 2dTÞ z 2dQ 2
T þ dT T
This means that the entropy of the system would have decreased. Hence maximum entropy is
obtained when the two blocks are in equilibrium and are at the same temperature. The general criterion
of equilibrium according to Keenan (1963) is
For stability of any system it is necessary and sufficient that, in all possible variations of the state of
the system which do not alter its energy, the variation of entropy shall be negative.
This can be stated mathematically as

DSÞ < 0 (2.20)
E
It can be seen that the statements of equilibrium based on energy and entropy, viz.
DEÞ > 0 and DSÞ < 0, are equivalent by applying the following simple analysis. Consider the
S E
marble at the base of the bowl, as shown in Fig. 2.2(a): if it is lifted up the bowl its potential energy will
be increased. When it is released it will oscillate in the base of the bowl until it comes to rest as a result
of ‘friction’, and if that ‘friction’ is used solely to raise the temperature of the marble then its tem-
perature will be higher after the process than at the beginning. A way to ensure the end conditions, i.e.
the initial and final conditions, are identical would be to cool the marble by an amount equivalent to the
increase in potential energy before releasing it. This cooling is equivalent to lowering the entropy of
the marble by an amount DS, and since the cooling has been undertaken to bring the energy level back
to the original value this proves that DEÞ > 0 and DSÞ < 0.
S E
Equilibrium can be defined by the following statements:
i. If the properties of an isolated system change spontaneously there is an increase in the entropy of
the system.
ii. When the entropy of an isolated system is at a maximum the system is in equilibrium.
iii. If for all the possible variations in state of the isolated system there is a negative change in
entropy then the system is in stable equilibrium.
These conditions may be written mathematically as

i. DS) E > 0, spontaneous change (unstable equilibrium)
ii. DS) E ¼ 0, equilibrium (neutral equilibrium)
iii. DS) E < 0, criterion of stability (stable equilibrium)

2.10 HELMHOLTZ ENERGY (HELMHOLTZ FUNCTION)
There are a number of ways of obtaining an expression for Helmholtz energy, but the one based on the
Clausius derivation of entropy gives the most insight.