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D25.1 Temperature Dependence of Gibbs Free Energy

Whether a chemical or physical reaction is product-favored, that is, whether the reactants are converted to products under standard-state conditions, is reflected in the arithmetic sign of its ΔrG°. The equation

ΔrG° = ΔrH° − TΔrS°

shows that the sign of ΔrG° depends on the signs of ΔrH° and ΔrS°, and, in some cases, the absolute temperature (which can only have positive values). For most reactions the values of ΔrH° and ΔrS° change very little as temperature changes, so their signs do not change. Four scenarios are possible:

  1. Both ΔrH° and ΔrS° are positive—This is an endothermic reaction with an increase in system entropy. ΔrG° is negative if TΔrS° > ΔrH°, and positive if TΔrS° < ΔrH°. Such a reaction is product-favored at high temperatures and reactant-favored at low temperatures.
  2. Both ΔrH° and ΔrS° are negative—This is an exothermic reaction with a decrease in system entropy. ΔrG° is negative if |TΔrS°| < |ΔrH°| and positive if |TΔrS°| > |ΔrH°|. Such a reaction is product-favored at low temperatures and reactant-favored at high temperatures. (Remember that |TΔrS°| represents the magnitude of TΔrS°, ignoring mathematical sign.)
  3. ΔrH° is positive and ΔrS° is negative—This is an endothermic reaction with a decrease in system entropy. ΔrG° is positive regardless of the temperature. Such a reaction is reactant-favored at all temperatures.
  4. ΔrH° is negative and ΔrS° is positive—This is an exothermic reaction with an increase in system entropy. ΔrG° is negative regardless of the temperature. Such a reaction is product-favored at all temperatures.

These four scenarios are summarized in this figure:

A table with three columns and four rows is shown. The first column has the phrase, “Delta sub r capital S standard greater than zero ( entropy increase ),” in the third row and the phrase, “Delta sub r capital S standard less than zero ( entropy decrease ),” in the fourth row. The second and third columns have the phrase, “Summary of the Four possibilities for Standard Reaction Enthalpy and Entropy Changes,” written above them. The second column has, “Delta sub r capital H standard greater than zero ( endothermic ),” in the second row, “Delta sub r capital G less than zero at high temperature, delta G greater than zero at low temperature, Process is product-favored at high temperature,” in the third row, and “Delta sub r capital G standard greater than zero at low temperature, Delta sub r capital G standard greater than zero at high temperature, Process is reactant-favored at all temperatures,” in the fourth row. The third column has, “delta sub r capital H standard less than zero ( exothermic ),” in the second row, “delta sub r capital G standard less than zero at low temperature, delta sub r capital G standard less than zero at high temperature, Process is product-favored at all temperatures,” in the third row, and “delta sub r capital G standard less than zero at low temperature, delta sub r capital G standard greater than zero at high temperature, Process is product-favored at low temperature.
Figure: Four combinations of signs of ΔrH° and ΔrS°. There are four possible combinations of signs of ΔrH° and ΔrS°. For two combinations, whether a reaction is reactant-favored or product-favored under standard-state conditions depends on temperature.

Activity: Temperature and Product-favored or Reactant-favored Reactions

The incomplete combustion of carbon is described by this equation:

2C(s) + O2(g)  ⇌  2 CO(g)

Without doing any calculations, write in your notebook an explanation of whether this process is product-favored at low temperatures, high temperatures, all temperatures, or no temperature.

Write in your notebook, then left-click here for an explanation.
Combustion heats things so combustion processes are exothermic (ΔrH° < 0). This particular reaction involves an increase in entropy due to the accompanying increase in the amount of gaseous species (net gain of one mole of gas means ΔrS° > 0). Because the standard enthalpy change is negative and the standard entropy change is positive the reaction is product-favored (ΔrG° < 0) at all temperatures.

The following figure illustrates the four scenarios graphically, where ΔrG° is plotted versus temperature:

ΔrG° = − ΔrS°(T) + ΔrH°
y = m(x) + b
Figure: Graphic representation of variation in ΔrG° with temperature. These plots show the variation in ΔrG° with temperature for the four possible combinations of arithmetic signs of ΔrH° and ΔrS°. Click on each “+” for more information.

In this course, unless specified otherwise, we will assume that ΔrH° and ΔrS° for a given reaction have the same values at all temperatures. Thus, in the preceding figure, the plots representing ΔrG° are linear because the slope of each plot (−ΔrS°) is the same at all temperatures. The orange (scenario 1) and green (scenario 2) plots cross from product-favored to reactant-favored (cross ΔrG° = 0) at a temperature that is characteristic to each specific reaction. This temperature is represented by the x-intercept, which is the temperature, in Kelvin, for which ΔrG° is zero:

ΔrG° = 0 = ΔrH° − TΔrS°
T_{{\Delta} _{\text{r}}G^{\circ}=0} = \dfrac{{\Delta} _{\text{r}}H^{\circ}}{{\Delta}_{\text{r}}S^{\circ}}

Hence, saying a reaction is product-favored at “high” or “low” temperatures is simply indicating whether the temperature is above or below TΔrG°=0 for that reaction. These relative terms are reaction-specific, that is, what is a “high” temperature for one reaction may very well be a “low” temperature for another reaction.

Exercise: Estimating Boiling Point

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Chem 109 Fall 2024 Copyright © by Jia Zhou; John Moore; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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