# 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 Δr. This equation

ΔrG° = ΔrH° − TΔr

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

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

These four scenarios are summarized in this figure:

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

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

 ΔrG° = − ΔrS°(T) + ΔrH° y = m(x) + b
Figure: Graphic representation of variation in Δr 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.

For most reactions, neither ΔrH° nor ΔrS° change significantly as temperature changes. Thus, in the preceding figure, the plots representing Δr are linear because the slope of each plot (−ΔrS°) 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 ΔrG° = 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 Δr is zero:

Δr = 0 = Δr − TΔr

Hence, saying a process is product-favored at “high” or “low” temperatures is simply indicating whether the temperature is above or below TΔrG°=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