photo credit: geboren.am

Of all the individuals in the roll-call of classical thermodynamics, Julius Robert Mayer (1814-1878) of Heilbronn, Germany, is surely one of the most unfortunate figures. His life was clouded by personal tragedy, and his highly original ideas – including what amounted to the first statement of the principle of conservation of energy – were received in academic circles with an indifference bordering on hostility, both in his native Germany and in Britain.

It didn’t help that Mayer was slightly ahead of James Joule in propounding the view that heat and work were interconvertible, or that he gave a value for the mechanical equivalent of heat before Joule did. All this served to achieve was an ill-tempered attack on Mayer’s claims to priority, orchestrated by eminent British scientists who should have known better, and to throw into sharper relief Mayer’s shortcomings in experimental technique, incomplete grasp of physical concepts, and obscure metaphysical style.

But then Mayer wasn’t a scientist’s scientist like Joule with his finely calibrated thermometers and carefully engineered experimental apparatus. Mayer was a philosopher’s scientist, a speculative thinker, a man of hypotheses.

One of his hypotheses has always struck me as a truly inspired piece of thinking, and is the subject of the remainder of this post.

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*Mayer’s great idea*

Julius Robert Mayer was adept at making the most of available scientific information, among which was the known fact that the specific heat capacity of a gas at constant pressure (C_{p}) is slightly greater than at constant volume (C_{v}).

Mayer seized upon this small difference and made a giant conceptual leap with it. He reasoned that C_{p} is greater than C_{v} because external work due to expansion is done by the gas in the former case but not in the latter. He then interpreted this external work as the exact mechanical equivalent of the amount of heat represented by the difference between C_{p} and C_{v}:

C_{p} – C_{v} = PΔV

Mayer then applied the equivalence relation in the reverse direction, asserting that the heat evolved from the isothermal compression of a gas is mechanically equivalent to the work done in compressing it.

This was the statement that Britain’s dynamic duo, James Joule and William Thomson (later Lord Kelvin), were to label Mayer’s hypothesis in their epoch-making series of joint papers on the thermal effects of air, which detailed the discovery of the Joule-Thomson effect, while also quietly confirming the approximate validity of Mayer’s hypothesis.

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*Proving Mayer’s relation*

Proofs given can range in length to a surprising degree, according to the assumed level of familiarity with staple formulas. One of the shortest I have seen is this one from E. Brian Smith’s *Basic Chemical Thermodynamics*:

H = U + PV

(q)_{p} = (q)_{v} + PV

For one mole of a perfect gas PV = RT and thus

(q)_{p} = (q)_{v} + RT

Differentiating with respect to temperature and noting that C = (dq/dT) we obtain

C_{p} – C_{v} = R

At the other end of the scale is this one, given by Enrico Fermi during a course of lectures at Columbia University in 1936. Quite a masterclass this – he builds the proof from first principles and doesn’t cut any corners whatsoever in his argumentation. I have tried to preserve the nomenclature he used:

The statement of the first law for the system under consideration is

(1)

If we choose T and V as our independent variables, U becomes a function of these variables so that:

(2)

and the first law statement becomes:

(3)

Similarly, taking T and p as independent variables, we have:

(4)

We define the thermal capacity of a body as the ratio dQ/dT. Let C_{v} and C_{p} be the thermal capacities at constant volume and at constant pressure, respectively. A simple expression for C_{v} can be obtained from (3). For an infinitesimal transformation at constant volume, dV=0; hence,

(5)

Similarly, using (4) we obtain the following expression for C_{p}

(6)

Fermi comments here: “The second term on the right represents the effect on the thermal capacity of the work performed during the expansion. An analogous term is not present in (5), because in that case the volume is kept constant so that no expansion occurs.” This is exactly Mayer’s hypothesis*.

Fermi then asserts on the basis of experimental evidence (namely Joule’s), but without theoretical proof (he does that later), that for an ideal gas U = U(T).

Since U depends only on T, it is not necessary to specify that the volume is to be kept constant in the derivative (5); so that for an ideal gas we may write:

(7)

For an ideal gas, (1) takes on the form

(8)

Differentiating the equation of state for one mole of an ideal gas, pV = RT, we obtain:

(9)

Substituting this for pdV in (8), we find:

(10)

Since dp=0 for a transformation at constant pressure, this equation gives us:

(11)

Which is Mayer’s relation.

*the second term on the right equates to C_{p}–C_{v} since for an ideal gas (∂U/∂T)_{P} = dU/dT = C_{V}

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