D25.1 Temperature Dependence of Gibbs Free Energy

Jia Zhou; John Moore; Etienne Garand

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 Δ_rG°. The 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). For most reactions the values of Δ_rH° and Δ_rS° change very little as temperature changes, so their signs do not change. Four scenarios are possible:

Both Δ_rH° and Δ_rS° are positive—This is an endothermic reaction with an increase in system entropy. Δ_rG° is negative if TΔ_rS° > Δ_rH°, and positive if TΔ_rS° < Δ_rH°. Such a reaction is product-favored at high temperatures and reactant-favored at low temperatures.
Both Δ_rH° and Δ_rS° are negative—This is an exothermic reaction with a decrease in system entropy. Δ_rG° is negative if |TΔ_rS°| < |Δ_rH°| and positive if |TΔ_rS°| > |Δ_rH°|. Such a reaction 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—This is an endothermic reaction with a decrease in system entropy. Δ_rG° is positive regardless of the temperature. Such a reaction is reactant-favored at all temperatures.
Δ_rH° is negative and Δ_rS° is positive—This is an exothermic reaction with an increase in system entropy. Δ_rG° 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 Δ_rH° and Δ_rS°**. There are four possible combinations of signs of Δ_rH° and Δ_rS°. 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 (Δ_rH° < 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 Δ_rS° > 0). Because the standard enthalpy change is negative and the standard entropy change is positive the reaction is product-favored (Δ_rG° < 0) at all temperatures.

The following 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.

In this course, unless specified otherwise, we will assume that Δ_rH° and Δ_rS° for a given reaction have the same values at all temperatures. 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 (scenario 1) and green (scenario 2) plots cross from product-favored to reactant-favored (cross Δ_rG° = 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 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 reaction is product-favored at “high” or “low” temperatures is simply indicating whether the temperature is above or below T_{Δ_rG°=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.