D20.5 Temperature Dependence of Gibbs Free Energy

John Moore; Jia Zhou; Etienne Garand

D20.5 Temperature Dependence of Gibbs Free Energy

Whether a 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 Δ_rG°. This equation

Δ_rG° = Δ_rH° − TΔ_rS°

shows that the sign of Δ_rG° depends on the signs of Δ_rH° and Δ_rS°, and, in some cases, the absolute temperature (which can only have positive values). Four possibilities exist:

Both Δ_rH° and Δ_rS° are positive—an endothermic process with an increase in system entropy. Δ_rG° is negative if TΔ_rS° > Δ_rH°, and positive if TΔ_rS° < Δ_rH°. Such a process is product-favored at high temperatures and reactant-favored at low temperatures.
Both Δ_rH° and Δ_rS° are negative—an exothermic process with a decrease in system entropy. Δ_rG° is negative if |TΔ_rS°| < |Δ_rH°| and positive if |TΔ_rS°| > |Δ_rH°|. Such a process is product-favored at low temperatures and reactant-favored at high temperatures. (Remember that |TΔ_rS°| represents the magnitude of TΔ_rS°, ignoring mathematical sign.)
Δ_rH° is positive and Δ_rS° is negative—an endothermic process with a decrease in system entropy. Δ_rG° is positive regardless of the temperature. Such a process is reactant-favored at all temperatures.
Δ_rH° is negative and Δ_rS° is positive—an exothermic process with an increase in system entropy. Δ_rG° is negative regardless of the temperature. Such a process 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 Δ_rH° and Δ_rS°**. There are four possible combinations of signs of Δ_rH° and Δ_rS°. For two combinations, whether a process is reactant-favored or product-favored under standard-state conditions depends on temperature.

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

The next figure illustrates the four scenarios graphically, where Δ_rG° is plotted versus temperature:

Δ_rG°	=	− Δ_rS°(T)	+	Δ_rH°
y	=	m(x)	+	b

Figure: Graphic representation of variation in Δ_rG° with temperature. These plots show the variation in Δ_rG° with temperature for the four possible combinations of arithmetic signs of Δ_rH° and Δ_rS°. Click on each “+” for more information.

For most reactions, neither Δ_rH° nor Δ_rS° change significantly as temperature changes. Thus, in the preceding figure, the plots representing Δ_rG° are linear because the slope of each plot (−Δ_rS°) is the same at all temperatures. The orange and green plots (representing examples of scenario 1 and 2, respectively) cross from product-favored to reactant-favored (cross Δ_rG° = 0) at a temperature that is characteristic to each specific process. This temperature is represented by the x-intercept, the value of T for which Δ_rG° is zero:

Δ_rG° = 0 = Δ_rH° − TΔ_rS°

$T_{{\Delta} _{\text{r}}G^{\circ}=0} = \dfrac{{\Delta} _{\text{r}}H^{\circ}}{{\Delta}_{\text{r}}S^{\circ}}$

Hence, saying a process is product-favored at “high” or “low” temperatures is simply indicating whether the temperature is above or below T_{Δ_rG°=0} for that process. 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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Chemistry 109 Fall 2021 Copyright © by John Moore; Jia Zhou; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.