Posts Tagged ‘Hermann von Helmholtz’

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

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)

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]