# 4.22: Changes of State and Free Energy

Energy in a body of water can be gained or lost, dependent on conditions. When water is heated above a certain temperature, steam is generated. The increase in heat energy creates a higher level of disorder in the water molecules as they boil off and leave the liquid.

## Changes of State and Free Energy

At the temperature at which a change of state occurs, the two states are in equilibrium with one another. For an ice-water system, equilibrium takes place at $$0^\text{o} \text{C}$$, so $$\Delta G^\text{o}$$ is equal to 0 at that temperature. The heat of fusion of water is known to be equal to $$6.01 \: \text{kJ/mol}$$, and so the Gibbs free energy equation can be solved for the entropy change that occurs during the melting of ice. The symbol $$\Delta S_\text{fus}$$ represents the entropy change during the melting process, while $$T_\text{f}$$ is the freezing point of water.

\begin{align} \Delta G &= 0 = \Delta H - T \Delta S \\ \Delta S_\text{fus} &= \frac{\Delta H_\text{fus}}{T_\text{f}} = \frac{6.01 \: \text{kJ/mol}}{273 \: \text{K}} = 0.0220 \: \text{kJ/K} \cdot \text{mol} = 22.0 \: \text{J/K} \cdot \text{mol} \end{align}

The entropy change is positive as the solid state changes into the liquid state. If the transition went from the liquid to the solid state, the numerical value for $$\Delta S$$ would be the same, but the sign would be reversed, since the phase changes indicates going from a less ordered to a more ordered situation.

A similar calculation can be performed for the vaporization of liquid to gas. In this case, we use the molar heat of vaporization. This value is $$40.79 \: \text{kJ/mol}$$. The $$\Delta S_\text{vap}$$ is as follows:

$\Delta S = \frac{40.79 \: \text{kJ/mol}}{373 \: \text{K}} = 0.1094 \: \text{kJ/K} \cdot \text{mol} = 109.4 \: \text{J/K} \cdot \text{mol}$

The value is positive, reflecting the increase in disorder going from liquid to vapor. Condensation from vapor to liquid would give a negative value for $$\Delta S$$.

## Summary

• Calculations are shown for determining entropy changes at transition temperatures (ice $$\rightarrow$$ water or water $$\rightarrow$$ vapor and reverse).