Page 38 - Advanced thermodynamics for engineers

2.9 EQUILIBRIUM 21

FIGURE 2.3
Heat transfer between two blocks.

transfer ‘heat’ to that at lower temperature. If the two blocks together constitute an isolated system the
energy transfers will not affect the total energy in the system. If the high temperature block is at a
temperature T 1 and the other at T 2 and if the quantity of energy transferred is dQ then the change in
entropy of the high temperature block is
dQ
dS 1 ¼ (2.16)
T 1
and that of the lower temperature block is
dQ
dS 2 ¼þ (2.17)
T 2
Both Eqns (2.16) and (2.17) contain the assumption that the heat transfers from block 1, and into
block 2 are reversible. If the transfers were irreversible then Eqn (2.16) would become
dQ
dS 1 > (2.16a)
T 1
and Eqn (2.17) would be
dQ
dS 2 > þ (2.17a)
T 2
Since the system is isolated the energy transfer to the surroundings is zero, and hence the change of
entropy of the surroundings is zero. Hence the change in entropy of the system is equal to the change in
entropy of the universe and is, using Eqns (2.16) and (2.17)

dQ dQ 1 1
dS ¼ dS 1 þ dS 2 ¼ þ ¼ dQ (2.18)
T 1 T 2 T 2 T 1
Since T 1 > T 2 , then the change of entropy of both the system and the universe is
dQ
dS ¼ ðT 1 T 2 Þ > 0:
T 2 T 1
The same solution, viz. dS > 0, is obtained from Eqns (2.16a) and (2.17a). The previous way of
considering the equilibrium condition shows how systems will tend to go towards such a state.
A slightly different approach, which is more analogous to the one used to investigate the equilibrium of