5 The Second Law of Thermodynamics

Nowhere in the discussion above has the Second Law of thermodynamics appeared. I hid it. It was really there all the time. In the second section, when I discussed the steam engine, I assumed that some heat will be lost to a cold reservoir. Why must this be true? Kelvin postulated the second law of thermodynamics in the following form:²¹

A transformation whose only final result is to transform into work heat extracted from a source which is at the same temperature throughout is impossible.

This doesn't look at all obvious. The key is the word only. A steam engine can absorb heat and do work, but the process leaves the piston pushed out. This is a result in addition to the doing of work. Kelvin is saying that you cannot take heat from a hot object and convert it all to work without producing some other change in the universe. Because of this I included a low-temperature constant temperature bath in our Carnot cycle.

The statement of the Second Law that I prefer is one that agrees with everyone's experience:²²

If heat flows by conduction from body $A$ to another body $B$, then a transformation whose only final result is to transfer heat from $B$ to $A$ is impossible.

Heat will, of course, only flow by conduction from A to B if A is at a higher temperature than B. Then, of course, you cannot get the heat to flow back from B to A spontaneously. You will have to produce some other change in the universe as well.

These two statements of the Second Law, the first due to Kelvin, the second to Clausius, are equivalent. Fermi gives a non-mathematical argument that is quite convincing:²³He argues:

"Let us first suppose that Kelvin's postulate were not valid. Then we could perform a transformation whose only final result would be to transform completely into work a definite amount of heat taken from a single source at the temperature $t_1$. By means of friction we could then transform this work into heat again and with this heat raise the temperature of a given body, regardless of what its intitial temperature $t_2$, might have been. In particular, we could take $t_2$ to be higher than $t_1$. Thus the only final result of this process would be the transfer of heat from one body (the source at temperature $t_1$) to another body at a higher temperature, $t_2$. This would be a violation of the Clausius postulate."

That rather convincingly demonstrates that if Kelvin was wrong, Clausius must also be wrong. To fully establish that the two statements are the same, we must also show that if Clausius is wrong, Kelvin must be wrong.

Again, quoting Fermi:

"Let us assume, in contradiction to Clausius' postulate, that it were possible to transfer a certain amount of heat $Q_2$ from a source at the temperature $t_1$ to a source at a higher temperature $t_2$ in such a way that no other change in the state of the system occured. With the aid of a Carnot cycle, we could then absorb this amount of heat $Q_2$ and produce an amount of work $L$. Since the source at the temperature $t_2$ receives and gives up the same amount of heat, it suffers no final change. Thus the process just described would have as its only final result the transformation into work of heat extracted from a source which is at the same temperature $t_1$ throughout. This is contrary to the Kelvin postulate." ²⁴

Thus the two statements are equivalent. Either can be taken as a statement of the Second Law.