Research: Science & Education
The Temperature Dependence of ⌬G ⴗ and the Equilibrium Constant, Keq ; Is There a Paradox? Frances H. Chapple Department of Chemistry, Willamette University, Salem, Oregon 97301
Both reaction spontaneity, determined by ∆G°, and the position of equilibrium, defined by the equilibrium constant, Keq, are important concepts in introductory and physical chemistry. Knowledge of how these quantities change with temperature is important for the thermal control of both reaction spontaneity and the composition of the equilibrium mixture. An apparent paradox may arise in students’ minds when they learn that whereas the sign of ∆S° determines the temperature dependence of ∆G°, it is ∆H ° that is responsible for the shift in Keq with temperature. How can this be, when ∆G° and ln Keq are directly related? That it is the sign of ∆S° that determines whether ∆G° increases or decreases with temperature is clear both from ∆G° = ∆H° – T∆S° and from the differential of this equation, (∂∆G°/∂T ) p = (∂∆H°/∂T )p – ∆S° – T(∂∆S °/∂T)p = ∆Cp – ∆S° – T(∆Cp /T) = ᎑∆S° The apparent paradox arises when we consider the temperature dependence of the position of equilibrium, rather than of ∆G°. LeChâtelier’s principle predicts that it is the exothermicity or endothermicity of the reaction, rather than ∆S°, that is the determining factor. The solution to the apparent discrepancy lies in the fact that Keq is related to ∆G° by the equation ln Keq = ᎑∆G°/RT. It is thus the temperature dependence of ∆G°/T, and not of ∆G° alone, that determines whether Keq increases or decreases with temperature. The Gibbs–Helmholtz equation (∂(∆G°/T)/∂(1/T))p = ∆H°
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Table 1. Effect of Temperature on ⌬G ⴗ and K e q ∆H ° (k J)
∆S ° (J/K)
T = 273 K ∆G ° (k J)
K eq
T = 373 K ∆G ° (k J)
K eq
3
10
0.27
0.89
᎑0.73
1.27
᎑3
᎑10
᎑0.27
1.13
0.73
0.79
3
᎑10
5.73
0.08
6.73
0.11
᎑3
10
᎑5.73
12.49
᎑6.73
8.76
Note: Calculations of ∆ G° and K eq are based on the assumption that ∆ H° and ∆S ° are temperature independent.
and the derived equation (∂ ln Keq/∂(1/T))p = ᎑∆H°/R provide the thermodynamic foundation for Le Châtelier’s principle. Thus, for an exothermic reaction (∆H° < 0), Keq can favor reactants at lower temperatures (Keq < 1) and products at higher temperatures (Keq > 1). The situation is illustrated for a hypothetical reaction in Table 1. In summary, when considering the effect of temperature on equilibrium concentrations of products and reactants (Keq), it is the sign and magnitude of ∆H° that is the determining factor. When, however, the standard-state spontaneity of a reaction (∆G°) is of concern, it is the sign and magnitude of ∆S° that determines the effect of temperature.
Journal of Chemical Education • Vol. 75 No. 3 March 1998 • JChemEd.chem.wisc.edu
