Posts Tagged ‘Max Planck’

Constantin Carathéodory (1873-1950)

Constantin Carathéodory (1873-1950)


This post has been translated into Greek by Giorgos Vachtanidis, and can be seen here.

Readers may also be interested to know that my supplementary blogpost “Carathéodory revisited” contains a proof of Carathéodory’s theorem: “If a differential dQ = ΣXidxi, possesses the property that in an arbitrarily close neighborhood of a point P defined by its coordinates (x1, x2,…, xn) there are points which cannot be connected to P along curves satisfying the equation dQ = 0, then dQ is integrable.” The supplementary post can be seen here

– – – –

Back in the days when I was a college student – the era when we wore our hair long, when elbow patches were commonplace and Woodstock was still fresh in our minds, the teaching of thermodynamics took place along two main routes.

The first was the classical route focused on heat and its convertibility into work, led philosophically by Carnot, Mayer and Joule, and developed mathematically by Clausius, Thomson (later Lord Kelvin), Helmholtz and Rankine. The second was the statistical route founded on a molecular model, and associated especially with the names of Boltzmann and Maxwell.

Nobody mentioned the third route. None of us were taught anything about the axiomatic approach to thermodynamics, published in 1909 in Mathematische Annalen under the title “Untersuchungen über die Grundlagen der Thermodynamik” [Examination of the foundations of thermodynamics] by a 36-year-old Greek mathematician called Constantin Carathéodory, who at the time was living in Hannover, Germany.


The title page of Carathéodory’s 1909 paper in Mathematische Annalen

It is clear from the outset of his paper that Carathéodory had studied Gibbs’ magnum opus “On the Equilibrium of Heterogeneous Substances (1875-1878)”. And just like Gibbs, Carathéodory uses the internal energy U and the entropy S (introduced together with the absolute temperature T) as the fundamental building blocks upon which he constructs his version of thermodynamics.

But whereas Gibbs introduces entropy via the classical route taken by Clausius, Carathéodory finds it through a mathematical approach based on the geometric behavior of a certain class of partial differential equations called Pfaffians, named for the German mathematician Johann Friedrich Pfaff (1765-1825) who first studied their properties.

Carathéodory’s investigations start by revisiting the first law and reformulating the second law of thermodynamics in the form of two axioms. The first axiom applies to a multiphase system change under adiabatic conditions:

Ufinal – Uinitial + W = 0

Nothing original here, since this is an axiom of classical thermodynamics due to Clausius (1850). It asserts the existence of a form of energy known as internal energy U – an intrinsic property of a system whose changes under adiabatic conditions are equal and opposite to the external work W performed (for a closed system not in motion).

In Carathéodory’s approach however, heat is regarded as a derived rather than a fundamental quantity that appears when the adiabatic restriction is removed, i.e. ΔU+W ≠ 0.

The second axiom is a different matter altogether, and constitutes the real novelty of Carathéodory’s approach:


This can be rendered in English as “In the neighborhood of any equilibrium state of a system (of any number of thermodynamic coordinates), there exist states that are inaccessible by reversible adiabatic processes.”

For a single substance, this postulate is obvious enough since reversible adiabatic processes are isenthalpic – a known result of classical thermodynamics. For such processes, all attainable states are represented by points on a curve for which entropy S = constant. There are other points which do not lie on this curve, and which represent states which cannot be reached by adiabatic transition.

But Carathéodory’s arguments go further, making this axiom applicable to a system of multiple bodies and multiple independent variables.

He shows that if in the neighborhood of any given point, corresponding to coordinates x1, x2,…, there are points not expressible by solutions of the Pfaffian equation X1dx1 + X2dx2 +… = 0, then for the expression X1dx1 + X2dx2 +… itself there exists an integrating factor.

The significance of this discovery is that via Carathéodory’s first axiom, the equation of adiabatic condition dQ = 0 admits an integrating factor, which when multiplying dQ renders the product an exact differential of a function whose value is therefore independent of the path between sets of coordinates.

The integrating factor (denominator) in this case is the absolute temperature T, and the path-independent integral ∫dQrev/T is the entropy change ΔS. This conjugate force-displacement pair, whose product is heat, arises directly from the geometric behavior and solutions of Pfaffians.

Using these partial differential expressions, Carathéodory obtains a formal thermodynamics without recourse to peculiar notions such as the flow of heat, or cumbrous conceptions such as imaginary heat engines and cycles of operation. In short, Carathéodory reduces the argument to a clean-cut consideration of lines and surfaces, together with a pair of axioms regarding the possibility of reaching certain states by adiabatic means.

It sounds very neat and tidy, as well as highly original, so how come it didn’t figure on our college curriculum? The answer to that question can be found in the reception afforded to Carathéodory’s masterwork by the scientific establishment of the day.

– – – –

Carathéodory’s thermodynamic theory got off to a rather inauspicious start in that it was ignored for the first 12 years of its existence. World War I came and went. Then in 1921, the German mathematician and physicist Max Born took note of Carathéodory’s work and published a set of articles on it entitled Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik [Critical considerations on the traditional representation of thermodynamics] in Physikalische Zeitschrift.

That got the ball rolling, but only as far as Max Planck, who besides being the towering authority in thermodynamics at the time, also turned out to be a severe critic of the new axiomatic method:

“nobody has up to now ever tried to reach, through adiabatic steps only, every neighborhood of any equilibrium state and to check if they are inaccessible,” Planck wrote, adding “this axiom gives us no hint which would allow us to differentiate the inaccessible from the accessible states.”

Others, impressed by the elegance of Carathéodory’s method, tried to render its formal austerity palatable to a wider audience. But these efforts met with no success, and when Lewis and Randall’s hugely influential and curriculum-setting textbook Thermodynamics and the Free Energy of Chemical Substances appeared in 1923, there was not a mention of Carathéodory or his theory.

Although there have been some notable attempts down the decades to champion Carathéodory’s cause, the axiomatic theory of 1909 has failed to achieve inclusion in mainstream academic teaching, and has been consigned to the catalogue of interesting curiosities. Planck’s enduring criticism of the theory’s failure to provide a compelling physical concept of entropy, together with the equally enduring difficulty of the math, seem to have played the deciding role. cc01a

Constantin Carathéodory (left) looking dapper in the company of his father, brother-in-law and sister. Carlsbad, Czechoslovakia 1898. Photo credit Wikimedia Commons

Suggested further reading:

1. The Structure of Physical Chemistry, C.N. Hinshelwood, Oxford University Press (1951)
Chapter III, Thermodynamic Principles, contains a concise introduction to Carathéodory’s theory, together with a discussion comparing its strengths and weaknesses with the classical approach. This book has been reissued as part of the Oxford Classic Texts series.

2. Constantin Carathéodory and the axiomatic thermodynamics, L. Pogliani and M. Berberan-Santos, Journal of Mathematical Chemistry Vol. 28, Nos. 1–3, 2000
This paper reviews the development of Carathéodory’s theory and explores some aspects of Pfaffians, the mathematical tools of axiomatic thermodynamics. A brief biography is also included.
PDF downloadable from

and if you’re feeling brave…

3. Examination of the foundations of thermodynamics – English translation of Carathéodory’s 1909 paper
PDF downloadable from

– – – –

P Mander January 2014

gibbs helmholtz cartoon


The Gibbs-Helmholtz equation was first deduced by the German physicist Hermann von Helmholtz in his groundbreaking 1882 paper “Die Thermodynamik chemischer Vorgänge” (On the Thermodynamics of Chemical Processes). In it, he introduced the concept of free energy (freie Energie) and used the equation to demonstrate that free energy – not heat production – was the driver of spontaneous change in isothermal chemical reactions, thereby overthrowing the famously incorrect Thomsen-Berthelot principle.

Although Gibbs was first to state the relations A = U – TS and G = U + PV – TS, he did not explicitly state the Gibbs-Helmholtz equation, nor did he explore its chemical significance. So the honors for this equation really belong more to Helmholtz than to Gibbs.

But from the larger historical perspective, both of these gentlemen can rightly be considered the founders of chemical thermodynamics – Gibbs for his hugely long and insanely difficult treatise “On the Equilibrium of Heterogeneous Substances” (1875-1878), and Helmholtz for his landmark paper referred to above. These works had a significant influence on the development of physical chemistry.

– – – –


The confusing thing about the Gibbs-Helmholtz equation is that it comes in three different versions, but most physical chemistry texts don’t say why. This is not helpful to students. Chemical thermodynamics is difficult enough already, so CarnotCycle will begin by giving the reason.

It just so happens that the form of calculus used in the Gibbs-Helmholtz equation has the following property:

For a thermodynamic state function (f) and its natural variables (x,y)

If we choose as our state function (f) the Gibbs Free Energy G and assign its natural variables, temperature T to (x) and pressure P to (y), we obtain:

Since (∂G/∂T)P = –S, and G ≡ H – TS

This is the Gibbs-Helmholtz equation. If we apply this equation to the initial and final states of a process occurring at constant temperature and pressure, and take the difference, we obtain:

where ΔH is the enthalpy change of a process taking place in a closed system capable of PV work; the three equivalent versions of the equation are determined by the properties of calculus:




A further useful relation can be derived from (2) and (3) using the equation ΔG° = –RTlnKp for a gas reaction where each of the reactants and products is in the standard state of 1 atm pressure.

Substituting –RlnKp for ΔG°/T in (2) and (3) yields



These are equivalent forms of the van ‘t Hoff equation, named after the Dutch physical chemist and first winner of the Nobel Prize in Chemistry, J.H. van ‘t Hoff (1852-1911). Approximate integration yields


– – – –

Applications of the Gibbs-Helmholtz equation

1. Calculate ΔHrxn from ΔG and its variation with temperature at constant pressure
This application of (1) is useful particularly in relation to reversible reactions in electrochemical cells, where ΔG identifies with the electrical work done –nFE. Scroll down to see the worked example GH1

2. Calculate ΔGrxn for a reaction at a temperature other than 298K
ΔH usually varies slowly with temperature, and can with reasonable accuracy be regarded as constant. Integration of (2) or (3) enables you to compute ΔGrxn for a constant-pressure process at a temperature T2 from a knowledge of ΔG and ΔH at temperature T1. Scroll down to see the worked example GH2

3. Calculate the effect of a temperature change on the equilibrium constant Kp
ΔH usually varies slowly with temperature, and can with reasonable accuracy be regarded as constant. The integrated van ‘t Hoff equation (6) allows the equilibrium constant Kp at T2 to be calculated with knowledge of Kp and ΔH° at T1. Scroll down to see the worked example GH3

– – – –

Insight: The Φ function and the meaning of –ΔG/T

By 1897, Hermann von Helmholtz was dead and Max Plank was professor of theoretical physics at the University of Berlin where he published Treatise on Thermodynamics, a popular textbook which ran to several editions. In it, Planck introduced the Φ function (originally deduced by François Massieu in 1869) as a measure of chemical stability:

planck function

The pencilled note in my German copy – found in a charity sale at a downtown church – correctly identifies Φ with –ξ/T, which in modern notation is –G/T

S is the entropy of the system, and (U+pV)/T is the enthalpy H of the system divided by its temperature T. Since G ≡ H – TS, we can immediately identify Φ with –G/T.

Planck’s formula indicates that Φ tends to be large when S is large and H is small, i.e. when the energy levels are closely spaced and the ground level is low – the criteria for chemical stability.

The Φ function gets even more interesting when one considers the meaning of ΔΦ.

ΔΦ = –ΔG/T = –ΔH/T + ΔS = ΔSsurroundings + ΔSsystem = ΔSuniverse

ΔΦ and –ΔG/T equate to the increase in the entropy of the universe: a measure of the ultimate driving force behind chemical reactions. The larger the value, the more strongly the reaction will want to go.

A direct association between –ΔG/T and the equilibrium constant K is thus implied, and this can be confirmed by rearranging the relation ΔG° = –RTlnKp to: –ΔG°/T = RlnKp

– – – –

Worked Example GH1

An electrochemical cell has the following half reactions:
Anode (oxidation):  Ag(s) + Cl → AgCl(s) + 1e
Cathode (reduction): ½Hg2Cl2(s) + 1e → Hg(l) + Cl

The EMF of the cell is +0.0455 volt at 298K and the temperature coefficient is +3.38 x 10-4 volt per kelvin. Calculate the enthalpy of the cell reaction, taking the faraday (F) as 96,500 coulombs.


Use version 1 of the Gibbs-Helmholtz equation

Substitute for ΔG using the relation ΔG = –nFE


[note that if the EMF is positive, the reaction proceeds spontaneously in the direction shown in the half reactions. If the EMF is negative, the reaction goes in the opposite direction]

The complete cell reaction for one 1 faraday is:
Ag(s) + ½Hg2Cl2(s) → AgCl(s) + Hg(l)
Each mole of silver transfers one mole of electrons (1e) to one mole of Cl ions. So n = 1.
F = 96,500 C
E = 0.0455 V
T = 298 K
(∂E/∂T)P = 3.38 x 10-4 VK-1

ΔHrxn = –1 x 96,500 (0.0455 – 298 (3.38 x 10-4)) joules
Dimensions check: remember that V= J/C, so C x V = J

ΔHrxn = 5329 J = 5.329 kJ

[note that this electrochemical cell makes use of a spontaneous endothermic reaction]

– – – –

Insight: The effect of temperature on EMF

From version 1 of the Gibbs-Helmholtz equation

and the relations ΔG = –nFE and (∂ΔG/∂T)P = –ΔS, it can be seen that

For many redox reactions that are used to power electrochemical cells, ΔSrxn is typically small (less than 50 JK-1). As a result (∂E/∂T)P is usually in the 10-4 to 10-5 range, and hence electrochemical cells are relatively insensitive to temperature.

winter text

– – – –

Worked Example GH2

The Haber Process for the production of ammonia is one of the most important industrial processes: N2(g) + 3H2(g) = 2NH3(g)
ΔG°(298K) = –33.3 kJ
ΔH°(298K) = –92.4kJ
Calculate ΔG° at 500K


Use version 3 of the Gibbs-Helmholtz equation

Making the assumption that ΔH° remains approximately constant, perform integration

Solve for ΔG°(T2)


ΔG°(T1) = –33.3 kJ
ΔH° = –92.4 kJ
T2 = 500K
T1 = 298K

Inserting these values into the integrated equation yields the result ΔG°(500K) ≈ 6.76 kJ. Compared with the negative value of ΔG° at 298K, the small positive value of ΔG° at 500K shows that the reaction has just become unfeasible at this temperature, pressure remaining constant at 1 atm.

– – – –

Insight: Reading the Haber process equation

N2(g) + 3H2(g) = 2NH3(g) ΔH°(298K) = –92.4 kJ

The Haber process is exothermic (negative ΔH) and results in a halving of volume (negative ΔS). Since ΔG = ΔH – TΔS, increasing the temperature will drive ΔG in a positive direction, leading to an upper temperature limit on reaction feasibility.

Le Châtelier’s principle shows that the Haber process is thermodynamically favored by low temperature and high pressure. In practice however a compromise has to be struck, since low temperature slows the rate at which equilibrium is achieved while high pressure increases the cost of equipment and maintenance.

fh text

– – – –

Worked Example GH3

The Haber Process for the production of ammonia is one of the most important industrial processes:
N2(g) + 3H2(g) = 2NH3(g) ΔH°(298K) = –92.4 kJ
The equilibrium constant KP at 298K is 6.73 x 105. Calculate KP at 400K.


Use the approximate integral of the van ‘t Hoff equation (6) to solve for KP(T2)


KP(T1) = 6.73 x 105, ln KP(T1) = 13.42
ΔH°(298K) = –92400 J (mol-1)
R = 8.314 J K-1 mol-1
T2 = 400K
T1 = 298K

Inserting these values into the integrated equation yields the approximate result ln KP(T2) ≈ 3.91, therefore KP(400K) ≈ 49.92. This is reasonably close to the measured value of KP(400K) = 48.91.

Compared to the large value of KP at 298K, the small positive value of KP at 400K shows that the reaction is approaching the point where it will shift to become reactant-favorable rather than product favorable.

– – – –

Insight: Legendre transformation and the Gibbs-Helmholtz equations

For an exact differential expression

the transforming function

can be written in terms of the natural variables of Y

This Legendre transformation is the means by which we obtain the Gibbs-Helmholtz equations. Taking Y=G(T,P) as an example, ℑ1 executes the transformation

while the transforming function

reverses the positions of the natural variables and executes the transformation

Setting Y=G(T,P) generates six Gibbs-Helmholtz equations, in each of which one of the two natural variables is held constant. Since there are four state functions – U, H, G and A – the total number of Gibbs-Helmholtz equations generated by this procedure is twenty-four. To this can be added a parallel set of twenty-four equations where U, H, G and A are replaced by ΔU, ΔH, ΔG and ΔA.

– – – –


P Mander updated August 2019