# Gibbs free energy in chemical reactions

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 5.2.

The Gibbs free energy is defined as

$\displaystyle G\equiv U-TS+PV=H-TS \ \ \ \ \ (1)$

The enthalpy ${H}$ is the total energy required to create the system from nothing, in which the environment at constant pressure ${P}$ must be pushed back to create the volume ${V}$ in which the new system is to be stored.

For a process occurring at constant temperature and pressure, the change in ${G}$ is

$\displaystyle \Delta G=\Delta H-T\Delta S \ \ \ \ \ (2)$

We can calculate changes in ${G}$ for a process such as a chemical reaction by considering the values for the reactants. Consider the reaction in which nitrogen and hydrogen combine to form ammonia:

$\displaystyle \mbox{N}_{2}+3\mbox{H}_{2}\rightarrow2\mbox{NH}_{3} \ \ \ \ \ (3)$

We can look up the relevant values in Schroeder’s book, where he gives values for 1 mole at ${T=298\mbox{ K}}$ and a pressure of 1 bar. The tabulated values are

 ${\Delta H\mbox{ (kJ)}}$ ${S\mbox{ (J K}^{-1}\mbox{)}}$ ${\mbox{N}_{2}}$ 0 191.61 ${\mbox{H}_{2}}$ 0 130.68 ${\mbox{NH}_{3}}$ ${-46.11}$ 192.45

For the reaction 3 we combine 1 mole of ${\mbox{N}_{2}}$ with 3 moles of ${\mbox{H}_{2}}$ to get 2 moles of ${\mbox{NH}_{3}}$, so we have

 $\displaystyle \Delta H$ $\displaystyle =$ $\displaystyle -92.22\times10^{3}\mbox{ J}\ \ \ \ \ (4)$ $\displaystyle \Delta S$ $\displaystyle =$ $\displaystyle 2\times192.45-3\times130.68-191.61\ \ \ \ \ (5)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -198.75\mbox{ J K}^{-1}\ \ \ \ \ (6)$ $\displaystyle \Delta G$ $\displaystyle =$ $\displaystyle \Delta H-T\Delta S\ \ \ \ \ (7)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -32.99\times10^{3}\mbox{ J} \ \ \ \ \ (8)$

This value is for 2 moles, so for one mole of ${\mbox{NH}_{3}}$ we have

$\displaystyle \Delta G=-16.5\mbox{ kJ} \ \ \ \ \ (9)$

which is close to the value of ${-16.45\mbox{ kJ}}$ given in Schroeder’s table. (I’m not sure if we’re supposed to be able to get closer with the given data.)