2.5 Entropy

Take-home message: The second law leads us to define a new function of state, entropy. The entropy of an isolated system can never decrease.

Clausius's theorem says that if a system is taken through a cycle, the sum of the heat added weighted by the inverse of the temperature at which it is added is less than or equal to zero:

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle \oint {{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q\over T }\le 0 $ }}$

This follows from Clausius's statement of the second law. The details of the proof are here. The inequality becomes an equality for reversible systems:

$$\oint {{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}\over T }= 0.$$

We can verify that this holds for a system which is taken through a Carnot cycle, since there heat only enters or leave at one of two temperatures:

$$\oint {{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}\over T }= {Q_H\over T_H}-{Q_C\over T_C}=0.$$

(A less rigorous proof of Clausius's theorem, used in Bowley and Sánchez and in Zemansky, involves approximating any reversible cycle by a large number of Carnot cycles.)

This is interesting because a quantity whose change vanishes over a cycle implies a function of state. We know that heat itself isn't a function of state, but it seems that in a reversible process ``heat over temperature'' is a function of state. It is called entropy with the symbol $S$:

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle dS={{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}\over T } $ }}$

and $\oint dS=0$.

So much for cycles. What about other processes? By considering a cycle consisting of one reversible and one irreversible process, we can show that in general,

$${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q\le T{\rm d}S.$$

(Details here.) This gives rise to the most important result of all. For an isolated system, ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q=0$. So for such a system,

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle {\rm d}S\ge 0. $ }}$

This leads to an alternative statement of the second law: The entropy of an isolated system can never decrease.

A system and its surroundings together (``the universe'') form an isolated system, whose entropy never decreases: any decrease in the entropy of a system must be compensated by the entropy increase of its surroundings.

References

• Bowley and Sánchez 2.5
• Adkins 5.1-3
• Zemansky 8.1-2, 8.9