Concise Physical Chemistry

c05 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

72 ENTROPY AND THE SECOND LAW

HOT COLD

WORK

FIGURE 5.1 An Engine.

thermodynamics, which, like the first law, can be stated in many ways. It follows that


S=

∫b

adS=

∫b

adq/Tis the entropy change of a system carried reversibly over an

arbitrary path fromatoband isindependent of the path. This powerful definition

constitutes the second of the two great pillars of thermodynamics. If we can devise

a way of calculating
Sfor a reversible chemical reaction, we shall know it for all

chemical reactions having the same initial and final states (reactants and products)

because of path independence.

Clausius expanded upon the concept of entropy by writing the complete statement

as

dS≥

dq

T

which takes both reversible andirreversiblechanges into account. The irreversible

changedSirr>dqirr/Tis the real case, a change that takes place in finite time.

If we attempt to take an engine around an irreversible cycle to reproduce its initial

state, we shall fall short. We have received a certain amount of work from the engine,

but when it comes to the payback (in heat) we see the following with regard to the

second law:

dS≥

dq

T

implies that dqirr<TdSirr

The system will not be returned to its original state, violating the principal stipulation

that the system operate around a cyclic path. We shall have to take some heat from

the hot reservoir in Fig. 5.1 to complete the cycle and bring the entropy back to its

initial value. Where does the extra heat over and above the reversible heat eventually

end up? It can go only one place. Since it hasn’t done any work, it must have passed

through the engine and gone directly to the low-temperature reservoir. The efficiency,

work out relative to heat in, of a real engine operating irreversibly is less than 1.0

because some heat is doing work and some is not. The important concept is that, of

the heat taken from the hot reservoir, not all of it can do work. Some heatmustpass

through the engine from the hot reservoir directly to the cold reservoir doing nothing