# 4.21: Temperature and Free Energy

Iron ore $$\left( \ce{Fe_2O_3} \right)$$ and coke (an impure form of carbon) are heated together to make iron and carbon dioxide. The reaction is non-spontaneous at room temperature, but becomes spontaneous at temperatures above $$842 \: \text{K}$$. The iron can then be treated with small amounts of other materials to make a variety of steel products.

## Temperature and Free Energy

Consider the reversible reaction in which calcium carbonate decomposes into calcium oxide and carbon dioxide gas. The production of $$\ce{CaO}$$ (called quicklime) has been an important reaction for centuries.

$\ce{CaCO_3} \left( s \right) \rightleftharpoons \ce{CaO} \left( s \right) + \ce{CO_2} \left( g \right)$

The $$\Delta H^\text{o}$$ for the reaction is $$177.8 \: \text{kJ/mol}$$, while the $$\Delta S^\text{o}$$ is $$160.5 \: \text{J/K} \cdot \text{mol}$$. The reaction is endothermic with an increase in entropy due to the production of a gas. We can first calculate the $$\Delta G^\text{o}$$ at $$25^\text{o} \text{C}$$ in order to determine if the reaction is spontaneous at room temperature.

$\Delta G^\text{o} = \Delta H^\text{o} - T \Delta S^\text{o} = 177.8 \: \text{kJ/mol} - 298 \: \text{K} \left( 0.1605 \: \text{kJ/K} \cdot \text{mol} \right) = 130.0 \: \text{kJ/mol}$

Since the $$\Delta G^\text{o}$$ is a large positive quantity, the reaction strongly favors the reactants and very little products would be formed. In order to determine a temperature at which $$\Delta G^\text{o}$$ will become negative, we can first solve the equation for the temperature when $$\Delta G^\text{o}$$ is equal to zero.

\begin{align} 0 &= \Delta H^\text{o} - T \Delta S^\text{o} \\ T &= \frac{\Delta H^\text{o}}{\Delta S^\text{o}} = \frac{177.8 \: \text{kJ/mol}}{0.1605 \: \text{kJ/K} \cdot \text{mol}} = 1108 \: \text{K} = 835^\text{o} \text{C} \end{align}

So at any temperature higher than $$835^\text{o} \text{C}$$, the value of $$\Delta G^\text{o}$$ will be negative and the decomposition reaction will be spontaneous. Figure $$\PageIndex{1}$$: This lime kiln in Cornwall was used to produce quicklime (calcium oxide), an important ingredient in mortar and cement.

Recall that the assumption that $$\Delta H^\text{o}$$ and $$\Delta S^\text{o}$$ are independent of temperature means that the temperature at which the sign of $$\Delta G^\text{o}$$ switches from being positive to negative $$\left( 835^\text{o} \text{C} \right)$$ is an approximation. It is also important to point out that one should not assume that absolutely no products are formed below $$835^\text{o} \text{C}$$ and that at that temperature, decomposition suddenly begins. Rather, at lower temperatures, the amount of products formed is simply not great enough to say that the products can be detected by monitoring the pressure of the $$\ce{CO_2}$$ gas that is produced. Above about $$700^\text{o} \text{C}$$, measurable amounts of $$\ce{CO_2}$$ are produced. The pressure of $$\ce{CO_2}$$ at equilibrium gradually increases with increasing temperature. Above $$835^\text{o}$$, the pressure of $$\ce{CO_2}$$ at equilibrium begins to exceed $$1 \: \text{atm}$$, the standard-state pressure. This is an indication that the products of the reaction are now favored above that temperature. When quicklime is manufactured, the $$\ce{CO_2}$$ is constantly removed from the reaction mixture as it is produced. This causes the reaction to be driven towards the products, according to Le Chatelier's principle.

## Summary

• The influence of temperature on free energy is described.