**History**

Le Châtelier’s principle is unusual in that it was conceived as a generalization of a principle first stated by someone else.

In 1884, the Dutch theoretician JH van ‘t Hoff published a work entitled *Etudes de Dynamique Chimique* [Studies in Chemical Dynamics]. In it, he stated a principle drawn from observations of different forms of equilibrium:

*“Lowering the temperature displaces the equilibrium between two different conditions of matter (systems) towards the system whose formation produces heat.”*

The converse statement was also implied, leading van ‘t Hoff to the realization that application of the principle made it possible *“to predict the direction in which any given chemical equilibrium will be displaced at higher or lower temperatures.”*

A few months after the publication of the *Etudes*, the following note appeared on page 786 of volume 99 of Comptes-rendus de l’Academie des Sciences:

The note covers two pages, but the crucial paragraph is the one shown immediately above, in which Le Châtelier extends van ‘t Hoff’s recently published principle to include pressure and (in modern terms) chemical potential. Rendered in English, the paragraph reads

*“Any system in stable chemical equilibrium, subjected to the influence of an external cause which tends to change either its temperature or its condensation (pressure, concentration, number of molecules in unit volume), either as a whole or in some of its parts, can only undergo such internal modifications as would, if produced alone, bring about a change of temperature or of condensation of opposite sign to that resulting from the external cause.”*

Just as van ‘t Hoff used inductive reasoning to relate temperature change to displacement of equilibrium, so Le Châtelier adopts the same technique to extend the principle to changes of pressure and potential.

Having arrived at a generalized principle – that systems in stable equilibrium tend to counteract changes imposed on them – Le Châtelier then sought to deduce this result mathematically from equations describing systems in equilibrium. During this quest, he discovered that the American physicist Josiah Willard Gibbs had done a good part of the groundwork in his milestone monograph *On The Equilibrium of Heterogeneous Substances (1876-1878)*. In 1899, Le Châtelier translated this hugely difficult treatise into French, thereby helping many scientists in France and beyond to access Gibbs’ powerful ideas.

**– – – –**

**Early misunderstandings**

Le Châtelier’s principle, first stated in 1884 and extended as the Le Châtelier-Braun principle in 1887, has stood the test of time. Today we view it as a very useful law, but that was not how it was viewed by some of the academic establishment in the early 20th century. Critics including the illustrious Paul Ehrenfest and Lord Rayleigh regarded the principle as vaguely worded and impossible to apply without ambiguity. As late as 1937, Paul Epstein in his *Textbook of Thermodynamics* wrote that this criticism *“has been generally accepted since”*.

This was news to me; when I was taught Le Châtelier’s principle at school, the wording was the same as in Epstein’s day but we had no issues with vagueness or ambiguity. I wondered what this criticism was all about, so I delved into the online archive of ancient journals. And came up with this:

It is clear from the first paragraph that the charge of ambiguity by Ehrenfest and Rayleigh arose from a failure to distinguish between cause and effect. Perturbations of systems in stable chemical equilibrium are caused by changes in generalized forces which, as Le Châtelier documents, are intensive variables. The ‘response of the system’, or generalized displacements, are the extensive conjugates. This answers Rayleigh’s question as to why we are to choose the one (pressure) rather than the other (volume) as the independent variable.

What surprised me was that this misunderstanding persisted for three decades. It just goes to show that in thermodynamics, even the most perspicacious individuals can have enduring blind spots.

– – – –

**The Principle behind the Principle**

In the *Etudes* of 1884, van ‘t Hoff stated his principle on the basis of different observations of equilibrium displacement with temperature. But while reaching his conclusion inductively, he still managed to give a precise mathematical expression of the principle. In modern notation it reads:

This famous equation, sometimes called the van ‘t Hoff isochore, was stated without proof in the 1884 edition, but in the second edition of 1896 a proof was provided which is based – as with many proofs of that era – on a reversible cycle of operations involving heat and work.

Although thermodynamically exact, the equation provides little insight into why a system in stable equilibrium tends to resist actions which alter that state. Not that this would have bothered van ‘t Hoff, who was much more interested in practicality than philosophical pondering.

But in the early 1900s, physical chemists began to look for an explanation. In *A Textbook of Thermodynamics with special reference to Chemistry (1913)*, J.R. Partington remarked that Le Châtelier’s principle is an expression of *“a very general theorem … called the Principle of Least Action. We can state it in the form that, if the system is in stable equilibrium, and if anything is done so as to alter this state, then something occurs in the system itself which tends to resist the change, by partially annulling the action imposed on the system.”*

Partington was hinting at a more general notion underlying Le Châtelier’s original description. That notion was more concisely expressed in another volume entitled *A Textbook of Thermodynamics*, written by Frank Ernest Hoare in 1931, in which he stated *“every system in equilibrium is conservative”*.

– – – –

**Interlude : Mapping chemical reactions**

It is one the conditions of stable equilibrium in thermodynamic systems that for a given temperature and pressure, the Gibbs free energy is a minimum. In the context of a chemical reaction, it means that the Gibbs free energy of the reaction mixture will decrease in the manner shown above, where the difference between P (pure products) and R (pure reactants) is the standard free energy of reaction and E is the equilibrium point at the minimum point of the curve.

If the reactants are initially present in stoichiometric proportions, the x-axis represents the mole fraction of products in the reaction mixture. In 1920, a Belgian mathematician and physicist called Théophile de Donder proposed another name for this dimensionless extensive variable. He called it “the degree of advancement of a chemical reaction”, and represented it by the Greek letter ξ (xi).

– – – –

**Defining conservative behavior**

In 1937, Professor Mark Zemansky – at the time an associate professor of physics at what was then called the College of the City of New York – published a textbook entitled *Heat and Thermodynamics*.

In the last section of the last chapter of the book, Zemansky turns his attention to Le Châtelier’s principle. He considers a heterogeneous chemical reaction which is in phase equilibrium but not chemical equilibrium; under these circumstances the Gibbs free energy G is a function of temperature T, pressure P and degree of advancement ξ.

When the chemical reaction reaches stable equilibrium at temperature T and pressure P, it follows that ∂G/∂ξ = 0. Zemansky then considers a neighboring equilibrium state at temperature T+dT and pressure P+dP. The new degree of reaction will be ξ+dξ, but the change in the slope of the curve during this process is zero. Therefore

Zemansky thus arrives at a mathematical definition of conservative behavior for a thermodynamic system consisting of a reaction mixture in stable equilibrium with respect to the reaction to which ξ refers.

The next task is to use the operations of calculus to find expressions for the derivatives ∂ξ/∂T and ∂ξ/∂P in terms of ΔS (=ΔH/T) and ΔV respectively. The first step is to write out fully the condition on dT, dP and dξ required to maintain conservative behavior:

Zemansky then employs a neat device to introduce S and V into the calculation. The order of differentiation of a state function is immaterial, so he reverses the order of differentiation in the first two terms

Since (∂G/∂T)_{P,ξ} = –S and (∂G/∂P)_{T,ξ} = V,

For the sake of brevity, I will introduce at this point a shortcut that Zemansky did not use, but which does not in any way alter the results of his reasoning.

For any extensive property X which varies according to the degree of advancement of a chemical reaction ξ at constant temperature and pressure, the slope of the curve has the following property

Applying this fact to the above equation, we find that in order to maintain the equilibrium condition ∂G/∂ξ=0, dT, dP and dξ must be such that

Setting dP=0 yields the result

When ΔG=0, the denominator is positive. At equilibrium therefore, (∂ξ/∂T)_{P} and ΔH have the same sign. So for an endothermic reaction (positive ΔH) the degree of reaction advancement at equilibrium increases as the temperature increases. This accords with Le Châtelier’s principle.

Setting dT=0 yields the result

When ΔG=0, the denominator is positive. At equilibrium therefore, (∂ξ/∂P)_{T} and ΔV have opposite signs. For a reaction resulting in a reduction of volume, the degree of reaction advancement at equilibrium increases as the pressure increases. This accords with Le Châtelier’s principle.

Zemansky thus demonstrates that deductions from a mathematical definition of conservative behavior for a thermodynamic system consisting of a reaction mixture in stable equilibrium result in equations which *“express in a rigorous form that part of Le Châtelier’s principle which concerns chemical reaction in heterogeneous systems”*.

Le Châtelier never got to see this deduction of his principle. He died in 1936, just a year before Zemansky’s book was published.

– – – –

P Mander February 2016