Gibbs–Helmholtz equation

The Gibbs–Helmholtz equation is a thermodynamic equation used for calculating changes in the Gibbs energy of a system as a function of temperature. It is named after Josiah Willard Gibbs and Hermann von Helmholtz.

Equation

The equation is:^[1]

$\left( \frac{\partial (G/T) } {\partial T} \right)_p = - \frac {H} {T^2},$

where H is the enthalpy, T the absolute temperature and G the Gibbs free energy of the system, all at constant pressure p. The equation states that the change in the G/T ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor H/T².

Chemical reactions

The typical applications are to chemical reactions. The equation reads:^[2]

$\left( \frac{\partial ( \Delta G^\ominus/T ) } {\partial T} \right)_p = - \frac {\Delta H} {T^2}$

with ΔG as the change in Gibbs energy and ΔH as the enthalpy change (considered independent of temperature). The o denotes standard pressure (1 bar).

Integrating with respect to T (again p is constant) it becomes:

$\frac{\Delta G^\ominus(T_2)}{T_2} - \frac{\Delta G^\ominus(T_1)}{T_1} = \Delta H^\ominus(p)\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature T₂ with knowledge of just the Standard Gibbs free energy change of formation and the Standard enthalpy change of formation for the individual components.

Also, using the reaction isotherm equation,^[3] that is

$\frac{\Delta G^\ominus}{T} = -R \ln K$

which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.^[4]

Derivation

Background

The definition of the Gibbs function is

$H= G + ST \,\!$

where H is the enthalpy defined by:

$H= U + pV \,\!$

Taking differentials of each definition to find dH and dG, then using the fundamental thermodynamic relation (always true for reversible or irreversible processes):

$dU= T\,dS - p\,dV \,\!$

where S is the entropy, V is volume, (minus sign due to reversibility, in which dU = 0: work other than pressure-volume may be done and is equal to −pV) leads to the "reversed" form of the initial fundamental relation into a new master equation:

$dG= - S\,dT + V\,dp \,\!$

This is the Gibbs free energy for a closed system. The Gibbs–Helmholtz equation can be derived by this second master equation, and the chain rule for partial derivatives.^[5]

Derivation

Starting from the equation $dG = - S\,dT + V\,dp = \frac{\partial G}{\partial T}\,dT + \frac{\partial G}{\partial p}\,dp \,\!$ at constant pressure meaning dp = 0. Then dG reduces to $dG_p = - S\,dT = \left(\frac{\partial G}{\partial T}\right)_p \,dT \quad \rightarrow \quad -S = \left(\frac{\partial G}{\partial T}\right)_p. \,\!$ The dependence of the ratio G/T on T is found by the product rule of differentiation: $\left( \frac{\partial(G/T)} {\partial T} \right)_p = \frac{1}{T} \left(\frac{\partial G}{\partial T}\right)_p + G\frac{\partial (T^{-1})}{\partial T} = \frac{T \left( \frac{\partial G}{\partial T} \right)_p- G}{T^2} = \frac{-ST - G}{T^2} = -\frac{H}{T^2} \,\!$

Sources

1. Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7
2. Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3
3. Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, ISBN 0-19-855148-7
4. Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3
5. physical chemistry, p.W. Atkins, Oxford University press, 1978, ISBN 0-19-855148-7

External links

• Link - Gibbs–Helmholtz equation
• Link - Gibbs–Helmholtz equation