left: Julius Thomsen (1826-1909) right: Marcellin Berthelot (1827-1907)

left: Julius Thomsen (1826-1909) right: Marcellin Berthelot (1827-1907)

Historical Background

In the mid-19th century, chemistry was a predominantly experimental science. There were of course the gas laws, and a handful of other stand-alone laws. But what was missing from chemistry was a theoretical foundation upon which a framework of understanding could be built.

Then around 1850, with the law of energy conservation beginning to be understood thanks to the pioneering work of Robert Mayer and James Joule, progress seemed to be made. Chemists started thinking about the ‘driving force’ causing chemical change, and concluded that the heat produced in chemical reactions provided a good quantitative measure of this force.

According to the Thomsen-Berthelot principle, formulated in slightly different versions by two professors of chemistry – Julius Thomsen in Denmark in 1854 and Marcellin Berthelot in France in 1864 – all chemical changes were accompanied by heat production, and the actual process that occurred was the one which produced most heat.

Right from the start, the Thomsen-Berthelot principle attracted criticism, but more from physicists than chemists. By the late 1870s, most chemists still supported it, but physicists regarded it as ill-conceived because the theory rested solely on the first law of thermodynamics, from which no conclusions could be drawn as to the direction of chemical change. That was the province of the second law.

What finally overthrew the Thomsen-Berthelot principle was a landmark paper “Die Thermodynamik chemischer Vorgänge” (On the Thermodynamics of Chemical Processes), written in 1882 by the 61 year-old German physicist Hermann von Helmholtz. In it, Helmholtz gave physical chemistry a solid theoretical foundation by showing that the driving force of a chemical reaction was measured not by heat production but by the maximum work produced when the reaction was carried out reversibly. Helmholtz also provided a name for this reversible maximum work – he called it “free energy”, for reasons we shall see.


Hermann von Helmholtz (1821-1894)

– – – –

Enthalpy change is a first law concept

Physicists criticized the Thomsen-Berthelot principle on theoretical grounds. They argued that the direction of chemical change must involve the second law of thermodynamics. But heat production during chemical change involves only the first law and so cannot in itself be a measure of the driving force.

The fact that enthalpy change is solely dependent on first law considerations can immediately be seen from the textbook derivation:

Consider a chemical change carried out at constant pressure rather than at constant volume. Under such conditions, the work w done by the system as a result of volume change is not zero.


(U+PV), like U, is a state function, as U, P and V are all state functions. It is called enthalpy and is defined


The enthalpy change of a system is equal to the heat produced at constant pressure, assuming that the system does only PV work.

– – – –

Helmholtz finds a new thermodynamic function

Physicists were unified in their view that the force driving chemical reactions must involve the second law of thermodynamics, which Rudolf Clausius had incorporated into his Mechanical Theory of Heat via the fundamental relation


It was this equation for a reversible process which seems to have served as a starting point for Helmholtz to generate a new state function using a method functionally equivalent to a Legendre transformation (I have not been able to access Helmholtz’s original 1882 paper, so what follows is based on secondary sources and may not be reliable in all details).

Helmholtz substituted TdS in the above equation with




and generating a new state function we now know as the Helmholtz free energy A


The question was, what did this new function represent? Helmholtz recognised that for isothermal reversible processes the new function identified with the maximum work


If during a reversible change the system does work, wrev will be negative. dA will also be negative and A will decrease. Helmholtz saw that under isothermal conditions A was the equivalent in chemical systems of potential energy in mechanical systems: it was a measure of the maximum work the system can do on its surroundings.

Since U was the total energy (gessamt Energie) and A was the maximum work, Helmholtz argued that U–A represented the energy that was unavailable for conversion into work. He called the TS term “bound energy” (gebundene Energie) and therefore by logical extension, A was the “free energy” (freie Energie) for isothermal processes.

And so just as potential energy was the driver of change in mechanical systems, free energy was the driver of change in chemical systems. Heat production, the quantity erroneously used as a measure of driving force of chemical change in classical thermochemistry, identified with the loss of bound energy.

Helmholtz went on to consider the new function A under constant volume conditions, and was the first to derive what is now generically called a “Gibbs-Helmholtz” equation


The cornerstone of the Thomsen-Berthelot principle was that chemical reactions were only possible if they produced heat; endothermic (heat-absorbing) reactions were impossible. Helmholtz was now in a position to challenge this assertion on theoretical grounds: there was nothing in the latter equation to suggest that ΔA could not be less negative than T(∂ΔA/∂T)V, in which case ΔU would be positive corresponding to an absorption of heat from the surroundings. Endothermic reactions were thus compatible with the new concept of free energy as the driver of chemical change.

– – – –

Why is the Gibbs-Helmholtz equation named for Gibbs when he didn’t deduce it?

Good question. It is a matter of historical fact that, although Gibbs was first to state the relation


he did not specifically state the Gibbs-Helmholtz equation


nor did he explore its chemical significance.

As the American physical chemist Wilder Bancroft commented, “This is an equation which Helmholtz did deduce and which Gibbs could have and perhaps should have deduced, but did not.”

wilder bancroft

Wilder Bancroft (1867-1953) Born in Middletown, Rhode Island, he received a BA from Harvard University in 1888 and in 1892 gained a PhD from Leipzig University where Wilhelm Ostwald was professor. Commenting on the apparently inappropriate inclusion of Gibbs’ name in the Gibbs-Helmholtz equation, he explained “The error goes back to Ostwald who was advertising Gibbs at the time.” [Ostwald was the first to translate Gibbs’ milestone monograph on chemical thermodynamics into German, and to promote it in Europe]


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