Close-up on the equivalence between Kelvin’s/Clausius’ postulates and
Clausius’ *theorem*.

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 Eq. (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, (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 work^{2}. 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 (1) leads the existence of another state
function, the *entropy* \(S\), which satisfies a generalized version of (1), namely:

\[ \sum _i\frac{\Delta Q_i}{T_i} \leq \Delta S\tag{5} \]

where quantities have the same meaning as in Eq. (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 Eq. (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 Eq. (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 (9) depends only upon the final states \(\sigma _0\)
and \(\sigma\) is guaranteed by (8).

By construction, we see that Eq. (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}. Eq.
(10) then implies that, if the process is
reversible, we must necessarily have \(T_1 = T_2\). This completes the proof of (6).

Dittman, Richard H, and Mark W Zemansky. 2021. “Heat and Thermodynamics SEVENTH EDITION.”

Fermi, E. 1956. *Thermodynamics*. Dover Books in Physics and Mathematical Physics. Dover Publications. https://books.google.es/books?id=VEZ1ljsT3IwC.

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, Eq. (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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Gherardi (2024, June 1). vgherard: Statements of the Second Law of Thermodynamics. Retrieved from https://vgherard.github.io/posts/2024-06-01-statements-of-the-second-law-of-thermodynamics/

BibTeX citation

@misc{gherardi2024statements, author = {Gherardi, Valerio}, title = {vgherard: Statements of the Second Law of Thermodynamics}, url = {https://vgherard.github.io/posts/2024-06-01-statements-of-the-second-law-of-thermodynamics/}, year = {2024} }