Historical view of the second law of thermodynamics as a complete differential

Question

My question sort of has two parts. I've been reading about the foundations of thermodynamics and statistical mechanics, older texts name the fact that $\frac{\delta Q}{T}$ is a complete differential as "the heat theorem", i assume they are just talking about the second law.

That seems to imply that the fact that $\delta Q$ is not a complete differential, but $\frac{\delta Q}{T}$ is the differential of a state function, is a way of understanding the second law, at least a part of what we today know as the second law.

For example Boltzmann, in "Analytical proof of the second law of the mechanical theory of heat..." (1871), claims to prove the second law by showing (using the canonical distribution) that $\frac{\delta Q}{T}$ is a complete differential. The steps can be followed in Klein (1973a), section 3, pages 62-65.

Also, for the second part of the question, Klein stated in a paper about Helmholtz's monocycles (1973b) the following:

Helmholtz’s idea was that [...] a purely mechanical system [the monocycle] [...], might provide an analogy for the complicated systems of thermodynamics with respect to the relationships between heat and work. The essential point was to show that energy provided to the system as heat (that is, as a change in the kinetic energy of the cyclic coordinate) could not be completely converted into work (that is, into a slow change in the auxiliary parameters, the second class of coordinates referred to above). In other words Helmholtz had to show that the analogue of the differential heat had an integrating factor proportional to the kinetic energy of the system.

I'd like to understand how one can go from one sentence to the other.

Also, this PSE post, Can heat be converted into work with 100% efficiency in a non-cyclic, single step process?, names one case where the first sentence of the quote by Klein is contradicted, even if it is for the highly idealized example of the isothermal expansion of an ideal gas.

Summarizing

My two questions are:

1. How much of the second law is related to the fact that $\delta Q/T$ is a complete differential?
2. How the integrating factor $1/T$ implies that not all heat can be converted into work?

References

• The development of Boltzmann's statistical ideas, Klein, M. J., (1973a).
• Mechanical Explanation at the End of the Nineteenth Century, Klein, M. J., (1973b).

Answer 1

I don't know about the history but can discuss the logic.

The Second Law can be asserted in various ways. The various statements each imply the others, by a process of reasoning in part mathematical and in part physical. Therefore any one of them could be called "The Second Law" and then the others are derived. Among these ways are the statement named after Clausius, the statement named after Kelvin, the approach described by Constantin Carathéodory, and an Entropy version. The entropy statement, for example, would take the form of asserting that for every system there exists an extensive state function whose value cannot be reduced by any process in a thermally isolated system.

The connection to $\delta Q/T$ is as follows. By using the Kelvin statement one can deduce Clausius's theorem: $$ \oint \frac{\delta Q}{T} \le 0 $$ (where the sign is such that $\delta Q$ is heat going into the system) and then one finds that for reversible processes $$ \oint \frac{\delta Q_{\rm rev}}{T} = 0 $$ no matter what path was followed. It follows that $\delta Q_{\rm rev}/T$ is a perfect differential, so one can associate with it a function of state whose differential it is. In this line of reasoning, the Second Law is a statement which does not mention this state function (entropy) and one deduces the state function from it.

But one could also argue the other way. If one can manage to prove that $\delta Q_{\rm rev}/T$ is a perfect differential without appealing to a statement of the Second Law, then one will have derived the existence of the state function called entropy. Any such derivation amounts to deriving the Second Law.

Actually, to derive the Second Law from statistical arguments has some subtle issues which do not go away, and it seems that a really watertight derivation is not possible. This is a famous long-standing problem in statistical mechanics. One can show why the Second Law is what one might reasonably expect if the initial conditions are not rather unusual, but it is hard to capture exactly what one means by 'unusual'. Also one cannot by any reasoning process derive a boundary condition, such as the low initial value of the entropy of the universe.

Answer 2

1. How much of the second law is related to the fact that $\delta Q/T$ is a complete differential?

$\frac{\delta Q}{T}$ is a complete differential but only for a reversible transfer of heat which is the definition of a differential transfer of entropy per the second law, per the following

$$dS=\frac{\delta Q_{rev}}{T}$$

The key is the subscript “rev” for $Q$. Note that $\delta Q_{rev}=TdS$.

2. How the integrating factor $1/T$ implies that not all heat can be converted into work?

Not seeing Klein’s quote in complete context, I can’t see how the factor $1/T$ precludes the possibility of theoretically completely converting heat into work, but only for a reversible process (not cycle), as I said in my answer to the PSE post link you provided. This is in fact what happens in the reversible isothermal expansion process of the Carnot cycle.

UPDATE:

The key additional information you provided with the edit is in the following statement;

The essential point was to show that energy provided to the system as heat (that is, as a change in the kinetic energy of the cyclic coordinate) could not be completely converted into work

The Carnot cycle isothermal expansion of an ideal gas involves no change in the kinetic energy of the gas since the temperature is essentially constant (infinitesimally different from the hot reservoir). See my answer here on how that may be theoretically accomplished: Work done by a gas in an isothermal process

So while there is an increase in entropy it is all due to entropy transfer and none due to entropy generation. Only the latter would reduce the amount of available work.

Thus, at least for the reversible isothermal expansion of an ideal gas, all the energy transfer due to heat can be “converted” to work. (Though I dislike the phrase heat converted to work. I prefer to say internal energy is converted to work and heat subsequently replenishes the internal energy).

For an example of how

Hope this helps.

Answer 3

You have started your post with this

I've been reading about the foundations of thermodynamics and statistical mechanics, older texts name the fact that $δQ/T$ is a complete differential as "the heat theorem", I assume they are just talking about the second law. That seems to imply that the fact that $δQ$ is not a complete differential, but $δQ/T$ is the differential of a state function, is a way of understanding the second law, at least a part of what we today know as the second law.

and then

My two questions are:

1. How much of the second law is related to the fact that δQ/T is a complete differential?
2. How the integrating factor 1/T implies that not all heat can be converted into work?

There are many versions of the "2nd law", not all equivalent, so to answer your Q1 one would have to know to which version you are referring. In one thing they all seem to agree is that in thermostatics there is an internal energy function of the empirical temperature $\theta$ and a set of "mechanical" variables, $X_k$, that can be manipulated from the outside and with which one may associate a "mechanically" definable external work. When in a reversible process an infinitesimal amount of external work $\delta w$ produces an infinitesimal amount of internal energy change $dE$ of a function $E=E(\theta, X_k)$ between two nearby equilibrium states then there is a function $T(\theta, X_k)> 0$ such that the difference $dE-\delta w$ when divided by $T$ is also a total differential $d\sigma= \frac{dE-\delta w}{T}$ of some function $\sigma=\sigma(\theta, X_k).$ For this statement to be meaningful physically it is also necessary that this function $T$ be a universal one in the sense that when two systems are in mutual equilibrium then one's $T$ should have the same value as that of the other, in which case we can also write $S(T, X_k)=\sigma(\theta, X_k)$.

Once you accept the existence of these functions $S=S(T, X_k)$ and $T=T(\theta, X_k)$ either by taking them axiomatically, this is also possible, see the "Carnot" tradition, or deriving them from some other axioms, such as those of Kelvin or Clausius or Caratheodory or Sears, etc., you will immediately run into the question as to what about equilibrium states that are connected by an irreversible one. Here the tradition is to postulate separately that in all irreversible processes of an isolated system, $dE=0,\delta w=0,$ that starts and ends in equilibrium states $T^0,X_k^0$ and $T^1,X_k^1$ the entropy is larger in the end state than in the start state, that is it increases: $\Delta S = S(T^1,X_k^1)-S(T^0,X_k^0)>0.$ This cannot be derived from the usual axioms but must be postulated separately and it is the beginning of thermodynamics.

You may call the existence of a positive integrating factor that is also a universal temperature scale as the 1st half and the entropy increase in an irreversible process the 2nd half of the 2nd law.

Now to your Q2 as to how does it follow from these that not all of the heat can be converted to work depends on what you mean by "converting". I do not wish to get into linguistic arguments but note that any time you speak of "heat" that is meant to be understood as a noun and not as a verb (to heat) the concept must involve two (2) quantities: a certain amount of entropy at some temperature. Since entropy is either conserved in a reversible process or increases in an irreversible process, in this context the concept of "conversion" has no obvious meaning.

Contrary to what one may generalize mistakenly from the usual elementary thermodynamics courses regarding ideal gasses, how and what part of the absorbed thermal energy can be used to produce work does not depend on these axioms directly but rather on the particular constitutive relationships defining the body and the processes they go through; to illustrate the issue here is a famous example problem I.6, p347 and Sommerfeld's solution p359 in Sommerfeld: Thermodynamics and Statistical Mechanics

Imagine a Carnot cycle with water as the working substance operating between 2C and 6C so that at 6C there is isothermal expansion and isothermal compression at 2C. It is seen that heat is added during both processes, if the pressure is low enough, and so heat is converted completely into work in violation of the second law. How is it possible to resolve this contradiction?