The Gibbs-Helmholtz equation

History

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.

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Derivation

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:

  (1)

(2)

  (3)

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

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

  (4)

  (5)

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

  (6)

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Applications of the Gibbs-Helmholtz equation

1. Calculate ΔH_rxn 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 ΔG_rxn 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 ΔG_rxn 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 K_p
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 K_p at T₂ to be calculated with knowledge of K_p and ΔH° at T₁. Scroll down to see the worked example GH3

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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:

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 = ΔS_surroundings + ΔS_system = ΔS_universe

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° = –RTlnK_p to: –ΔG°/T = RlnK_p

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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.

Strategy

Use version 1 of the Gibbs-Helmholtz equation

Substitute for ΔG using the relation ΔG = –nFE

Calculation

[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

ΔH_rxn = –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

ΔH_rxn = 5329 J = 5.329 kJ

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

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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, ΔS_rxn 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.

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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

Strategy

Use version 3 of the Gibbs-Helmholtz equation

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

Solve for ΔG°_(T2)

Calculation

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.

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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.

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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 K_P at 298K is 6.73 x 10⁵. Calculate K_P at 400K.

Strategy

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

Calculation

K_P(T₁) = 6.73 x 10⁵, ln K_P(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 K_P(T₂) ≈ 3.91, therefore K_P(400K) ≈ 49.92. This is reasonably close to the measured value of K_P(400K) = 48.91.

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

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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.

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P Mander updated August 2019