The Second Law of Thermodynamics is commonly stated in the forms of Kelvin’s and Clausius’ postulates. These can be enunciated in the following way (Dittman and Zemansky 2021) 1:
Kelvin’s Postulate. It is impossible to construct an engine that, operating in a cycle, will produce no effect other than the extraction of heat from a reservoir and the performance of an equivalent amount of work.
Clausius’ Postulate. It is impossible to construct a refrigerator that, operating in a cycle, will produce no effect other than the transfer of heat from a lower-temperature reservoir to a higher temperature reservoir.
Either formulation is equivalent to the other and leads to the fundamental Clausius’ theorem. This asserts the existence of a universal state function \(T\), the absolute temperature, defined for any thermodynamic system, that satisfies the Clausius inequality. Concretely, if a system undergoes a cyclic process, during which it absorbs quantities \(\Delta Q _i\) of energy in the form of heat from reservoirs at absolute temperatures \(T_i\), the inequality:
\[ \sum _i\frac{\Delta Q_i}{T_i} \leq 0 \tag{1}\] always holds.
The derivation of Equation 1 from Kelvin’s and Clausius’ postulates, a clever argument that employs ideal Carnot engines, is standard textbook material; see for example (Fermi 1956). On the other hand, I’ve never seen the converse being stressed, that is, that Clausius theorem allows one to recover versions of Kelvin’s and Clausius’ postulates. Here are two (fairly obvious) arguments in this direction.
Consider a cyclic process of a thermodynamic system during which a quantity \(\Delta Q\) of heat is absorbed from a reservoir at constant temperature \(T_0\). Equation 1 applied to this special process implies:
\[ \Delta Q\leq 0. \tag{2}\]
The fact that \(\Delta Q\leq 0\) means that the heat reservoir can only absorb energy during a cycle, which must be supplied by performing a positive work on the system. This is the content of Kelvin’s postulate.
Similarly, if the system performs a cycle exchanging amounts of heat \(\Delta Q_1\) and \(\Delta Q_2\) with two heat sources at temperatures \(T_1\) and \(T_2\) respectively, Equation 1 implies:
\[ \frac{\Delta Q_1}{T_1}+\frac{\Delta Q_2}{T_2}\leq 0 \tag{3}\]
But \(\Delta Q_1 + \Delta Q_2 = \Delta Q =-\Delta W\), the external work performed on the system during a cycle. Hence:
\[ (\frac{1}{T_1}-\frac{1}{T_2})\Delta Q_1\leq \frac{\Delta W}{T_2}. \tag{4}\]
Therefore, \(\Delta Q_1 \geq 0\) with \(T_1 < T_2\) requires \(\Delta W \geq 0\). In other words, in order to perform a cycle in which a positive amount of heat is transferred from a low-temperature reservoir to a high-temperature one, we must necessarily perform some positive work2. This is the content of Clausius’ postulate.
A subtle point that may require some elucidation is that, in the usual logical exposition of Thermodynamics, the temperature to which Kelvin’s and Clausius’ postulates make reference is the empirical temperature, call it \(\theta\). This is the “quantity measured by a thermometer” (Fermi 1956), and is logically distinct from the absolute temperature \(T\), whose existence is a consequence of the second law. What we actually proved here are versions of Kelvin’s and Clausius’ postulates formulated in terms of the absolute temperature, \(T\).
Now, if we take Kelvin’s or Clausius’ postulate (formulated in terms of \(\theta\)) as our logical starting point, we can actually prove that \(T\) is an increasing function of \(\theta\), in which case there is no point in specifying which temperature the postulates refer to. However, if our starting point is
Clausius’ Theorem, there is no a priori logical reason for a relation between \(T\) and \(\theta\), which should be considered as an additional assumption.
Even though this goes a bit beyond the original scope of the post, I’d like to show here how Equation 1 leads the existence of another state function, the entropy \(S\), which satisfies a generalized version of Equation 1, namely:
\[ \sum _i\frac{\Delta Q_i}{T_i} \leq \Delta S \tag{5}\]
where quantities have the same meaning as in Equation 1, but the process is not necessarily cyclic. One can additionally show that the differential of \(S\) is given by:
\[ \text dS = \frac{\delta Q _R}{T}, \tag{6}\]
where \(\delta Q_R = \text d U + \delta W_R\) is the differential heat absorbed by the system in a reversible process, and \(T\) is the system’s temperature.
We start by observing that, for a reversible process, equality must hold in Equation 1. This is so because, for a reversible cycle, the inverse cycle, in which the system absorbs amounts \(-\Delta Q_i\) of heat at temperatures \(T_i\), must also be possible. Altogether, the Clausius inequalities for the direct and inverse cycles thus imply:
\[ \sum _i\frac{\Delta Q_i}{T_i} = 0\quad \text{(reversible process)}. \tag{7}\]
Imagining an ideal cyclic process, in which the system exchanges infinitesimal amounts of heat \(\delta Q(T')\) with a continuous distribution of sources at temperatures \(T'\), we should replace the sum in Equation 7 with an integral:
\[ \intop \frac{\delta Q(T')}{T'} = 0 \quad\text{(reversible process)} \tag{8}\] We now fix a reference state \(\sigma _0\) of our system, and define for any other state \(\sigma\):
\[ S(\sigma;\sigma _0) = \intop _{\sigma_0}^\sigma \frac{\delta Q(T')}{T'} \tag{9}\]
where the integral is taken along any reversible path that connects \(\sigma _0\) and \(\sigma\), and \(\delta Q(T')\) is the amount of heat exchanged at temperature \(T'\) along this representative process. The fact that the integral in Equation 9 depends only upon the final states \(\sigma _0\) and \(\sigma\) is guaranteed by Equation 8.
By construction, we see that Equation 6 must hold with \(T\) being the temperature of a source that, if placed in thermal contact with the system, can produce a reversible exchange of heat. It remains to be shown that this temperature is nothing but the temperature of the system itself. Consider a reversible process in which two systems at temperatures \(T_1\) and \(T_2\) exchange an (infinitesimal) amount of heat. From what we have just said:
\[ \text d S_1 = \frac{\delta Q_1}{T_2},\quad \text d S_2 = \frac{\delta Q _2}{T_1}, \tag{10}\] where \(\delta Q_i\) is the heat absorbed by system \(i\), and \(\text d S_i\) is its corresponding entropy change. However, since the composite system is thermally insulated, we must have \(\delta Q_1 + \delta Q_2=0\) and \(\text d S_1 + \text d S_2 = 0\)3. Equation 10 then implies that, if the process is reversible, we must necessarily have \(T_1 = T_2\). This completes the proof of Equation 6.
References
Footnotes
We may compare these with the corresponding formulations given in Enrico Fermi’s famous book (Fermi 1956). For instance, Kelvin’s postulate reads: “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.”. Even though I’m a big fan of Fermi’s book, I find the more modern formulations given in (Dittman and Zemansky 2021) clearer.↩︎
In fact, Equation 4 tells us a bit more than Clausius postulate, since it gives the maximum theoretical efficiency of a refrigerator operating between temperatures \(T_1 < T_2\): \[ \frac{\Delta Q_1}{\Delta W} \leq \frac{T_1}{T_2-T_1} \]↩︎
The additivity of entropy is a consequence of the additivity of heat, which in turn would require a dedicate discussion. Such a requirement boils down to the additivity of external work, which holds generally if the interaction energies of the systems being composed are negligible. This is always assumed (more or less explicitly) whenever discussing the interaction of a system with a heat reservoir.↩︎
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Citation
@online{gherardi2024,
author = {Gherardi, Valerio},
title = {Statements of the {Second} {Law} of {Thermodynamics}},
date = {2024-06-01},
url = {https://vgherard.github.io/posts/2024-06-01-statements-of-the-second-law-of-thermodynamics/},
langid = {en}
}