# 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_{r}*G°*. This equation

_{r}

*G° =*Δ

_{r}

*H° − T*Δ

_{r}

*S°*

shows that the sign of Δ_{r}*G°* depends on the signs of Δ_{r}*H°* and Δ_{r}*S°*, and, in some cases, the absolute temperature (which can only have positive values). Four possibilities exist:

**Both Δ**—an endothermic process with an increase in system entropy. Δ_{r}*H°*and Δ_{r}*S°*are positive_{r}*G°*is negative if*T*Δ_{r}*S° >*Δ_{r}*H°*, and positive if*T*Δ_{r}*S°*< Δ_{r}*H°*. Such a process is*product-favored at high temperatures and reactant-favored at low temperatures.***Both Δ**—an exothermic process with a decrease in system entropy. Δand Δ_{r}H°_{r}*S°*are negative_{r}*G°*is negative if |*T*Δ_{r}*S*°| < |Δ_{r}*H*°| and positive if |*T*Δ_{r}*S*°| > |Δ_{r}*H*°|. Such a process is*product-favored at low temperatures and reactant-favored at high temperatures.*(Remember that |*T*Δ_{r}*S*°| represents the magnitude of*T*Δ_{r}*S*°, ignoring mathematical sign.)**Δ**—an endothermic process with a decrease in system entropy. Δ_{r}*H°*is positive and Δ_{r}*S°*is negative_{r}*G°*is positive regardless of the temperature. Such a process is*reactant-favored at all temperatures.***Δ**—an exothermic process with an increase in system entropy. Δ_{r}*H°*is negative and Δis positive_{r}S°_{r}*G°*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 Δ* _{r}G°* is plotted versus temperature:

Δ_{r}G° |
= | − Δ_{r}S°(T) |
+ | Δ_{r}H° |

y | = | m(x) | + | b |

For most reactions, neither Δ_{r}*H*° nor Δ* _{r}S*° change significantly as temperature changes. Thus, in the preceding figure, the plots representing Δ

_{r}

*G°*are linear because the slope of each plot (−Δ

_{r}S°) 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

_{r}

*G*° = 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

_{r}

*G°*is zero:

_{r}

*G°*= 0 = Δ

_{r}

*H°*

*− T*Δ

_{r}

*S°*

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