# Principle of maximum work

In thermochemistry, the principle of maximum work, or “Thomsen-Berthelot principle”, states that all chemical change occurs in the direction towards which the processes evolve the most of heat. [1] The principle was proposed in simple terms in 1853 by Danish chemist Julius Thomsen and later in more definitive form in circa 1864 by French chemist Marcellin Berthelot, but later found to only hold true only at absolute zero.

Berthelot’s principle of maximum work, was said to have met its demise with the publication of German physicist Herman Helmholtz’ 1882 paper “The Thermodynamics of Chemical Operations”, in which he proved that the affinity is measured not by the heat evolved in a chemical reaction (in a galvanic battery) but by the maximum work produced when the reaction is carried out reversibly. One must distinguish, according to Helmholtz, between that part of energy which appears only as heat and that part which can be freely converted into other kinds of work, i.e. the “free energy” analogous to potential energy in mechanics. Subsequently, the conditions of chemical stability are not determined by heat production but by the production of a decrease in free energy. [2]

Chemical equilibria
It is known that chemical equilibria, at ordinary, temperatures, displace towards the systems which are formed with the evolution of heat, so that it is evident that those chemical changes which occur at ordinary temperatures must in the majority of cases be accompanied by evolution of heat. This effect is only rigorously true at absolute zero of temperature.

The opposite effect occurs, however, at high temperatures. Based on the 'principle of mobile equilibrium', it is found that at very high temperatures the opposite effect must take place, and that the changes which occur under these conditions will in general absorb heat.

These heat dynamic effects, at high temperatures, moderate temperatures, and zero temperatures, are found to be based on the exact laws of thermodynamics, as stated by Dutch chemist Jacobus van’t Hoff in 1896, but do not corroborate with the views expressed by Thomsen and Marcellin Berthelot, whose views were found to contradict with fact and abandoned, but which in 1896 were still being represented in many textbooks as being in accordance with experience. [1] In 1853, Danish chemist Julius Thomsen expressed his view that:

“Every simple or complex change of a purely chemical nature is accompanied by an evolution of heat.”

Berthelot, likewise, in 1867 put forward his views in what he called the ‘principle of maximum work’, which states that:

“Every chemical change, accomplished without intervention of external energy tends towards the production of bodies or of the system which evolves the greatest quantity of heat.”

Van’t Hoff, however, by 1896, had showed, based on thermodynamics, that correctly:

“All chemical equilibria without exception are displaced completely at absolute zero towards those systems which are formed with evolution of heat, and only under these circumstances is the principle of maximum work rigorously true.”

Chemical affinity
Thomsen originally argued, in circa 1853, that heat of a reaction was the true measure of affinity. This same view was expounded on by Berthelot. Berthelot's heat theory of affinity, it seems, was initially criticized by French physicist Pierre Duhem, in a thesis that Berthelot’s colleagues rejected. [2]

Van't Hoff, following Helmholtz, was one of the first to lay to rest Berthelot's theory. According to Van’t Hoff, affinity is the force which produces chemical change, which may be expressed in kilograms or in atmospheres, and the work which it can effect can be expressed in kilogram-meters or in calories. Van’t Hoff showed that Berthelot’s principle was only true in respect to the affinities of the system and the work they can produce, at absolute zero. Van’t Hoff began be noting that in any generic reactions:

$A + B \rightleftharpoons C + D \,$

there will exist a “transition temperature” P or point at which the system on the left equally replaces the system on the right, and vice versa, at temperatures above which the system on the right completely replaces the system on the left, while below it the reverse takes place.

He states that the affinity, i.e. the force which brings about the change, or the difference between the affinities which are at work in the two systems, is zero at the transition temperature; but when passing through the transition temperature the value of the affinity, or of the difference of the affinities, changes sign. Through some derivation, he gives the following expression for the work which can be done by the affinity A:

$A = Q \bigg( \frac{P-T}{P} \bigg) \,$

which states that the work, expressed in calories (joules), which the affinity in a given chemical reaction can perform, when the reaction takes place at a given temperature, is equal to the quantity of heat Q evolved by the reaction, divided by the absolute temperature of the transition point P, and multiplied by the temperature T at which the reaction occurs. The basis of this derivation seems to stem from page 281 of Clausius' The Mechanical Theory of Heat, second English edition. This equation is also similar to another form derived by Helmholtz in 1882:

$A = T \Delta S - \Delta U \,$

In any event, Van't Hoff notes that in the special case of P = T, we have A = 0, meaning that the work done by affinity at the transition temperature, and therefore the force of affinity itself, is zero.

Van’t Hoff then states that considering that the maximum quantity of work which affinity can perform is equal to the quantity of work done by the change when it takes place in a reversible way, it will evidently be possible to obtain the above equation by means of thermodynamics. This derivation was done, it seems, by German chemist Walther Nernst in his 1906 heat theorem.

In any event, Van’t Hoff concludes by noting the application that the work done by affinity at absolute zero, whereby the reaction temperature is zero, T = 0, that A = Q, such that the work which may be obtained from the change is equivalent to the whole of the heat which it evolves. He states, at absolute zero the change could occur either with evolution of Q calories and without doing any work, or evolving no heat and doing a quantity of work equivalent to Q calories. He concludes, this is in agreement with the conclusion arrived at above, where it was pointed out that the ‘principle of maximum work’ is rigorously true at absolute zero, whereby the quantity of heat evolved being then the measure of the work which can be done by the affinities, and its sign therefore the direction in which the change will occur.

At higher temperatures, the work which can be done by the affinity is then equivalent to a part only of the heat evolved, the two have the same sign, however, until the transition temperature is reached, at this point the work which can be done by affinity is zero and at higher temperatures its sign is reversed, that is to say work must be performed in order to bring about the reaction. The change of sign accompanies the change in the direction of the reaction which occurs at the transition point.

References
1. Van’t Hoff, J.H. (1896). Studies in Chemical Dynamics (Chemical Equilibrium, pgs. 143-54; Affinity, pgs. 229-36; Determination of the Work Done by Affinity, pgs. 237-41; Applications, 241-50; Electrical Work Performed by a Chemical Change, pgs. 251-73; Principle of maximum work, pgs. 224-28, 241-42); Revised and Enlarged by Ernst Cohen (trans. Thomas Ewan). 286-pgs. London: Williams & Norgate.
2. Nye, Mary J. (1993). From Chemical Philosophy to Theoretical Chemistry: Dynamics of Matter and Dynamics of Disciplines: 1800-1950 (section: From Chemical Affinity to Chemical Thermodynamics, pgs. 116-20). University of California Press.