Entropy Changes and Gibbs Free Energy

Entropy, \(S\), is a measure of how much disorder or randomness exists in a system. Gases have more entropy than liquids, and liquids have more entropy than solids, because particles have more ways to arrange themselves as they become more free to move. The entropy change of a reaction, \(\Delta S\), compares the entropy of the products with the entropy of the reactants:

\(\Delta S = S_{products} - S_{reactants}\)

If a reaction produces more gas molecules than it starts with, \(\Delta S\) is usually positive. If the number of gas molecules decreases, or a reaction converts a liquid into a solid, \(\Delta S\) is usually negative. Entropy values are normally given in joules per mole per kelvin, \(J\,mol^{-1}K^{-1}\), which is important to remember when you combine entropy with enthalpy values measured in kilojoules per mole.

Whether a reaction happens on its own, without any continuous outside input of energy, depends on both its enthalpy change and its entropy change. These two quantities are combined into a single value called the Gibbs free energy change, \(\Delta G\), using the equation:

\(\Delta G = \Delta H - T\Delta S\)

Here \(\Delta H\) is the enthalpy change of the reaction (often obtained from calorimetry or from a Born-Haber cycle for ionic compounds), \(T\) is the absolute temperature in kelvin, and \(\Delta S\) is the entropy change of the system. Because \(\Delta H\) is usually quoted in kilojoules per mole and \(\Delta S\) in joules per mole per kelvin, you must divide \(\Delta S\) by 1000 (or convert \(\Delta H\) to joules) before substituting into the equation, otherwise the units will not match.

A reaction is thermodynamically spontaneous, meaning it is energetically favourable to occur, whenever \(\Delta G\) is negative. If \(\Delta G\) is positive the reaction is not spontaneous under those conditions, and if \(\Delta G\) equals zero the system is at equilibrium. Because \(T\) is always positive, the signs of \(\Delta H\) and \(\Delta S\) decide how spontaneity depends on temperature:

Consider a reaction with \(\Delta H = -92.4\ kJ\,mol^{-1}\) and \(\Delta S = -198.7\ J\,mol^{-1}K^{-1}\) at \(T = 298\ K\). First convert entropy into kilojoules: \(\Delta S = -0.1987\ kJ\,mol^{-1}K^{-1}\). Then substitute into the equation:

\(\Delta G = \Delta H - T\Delta S = -92.4 - (298 \times -0.1987)\)

\(\Delta G = -92.4 - (-59.2) = -33.2\ kJ\,mol^{-1}\)

Since \(\Delta G\) is negative, this reaction is spontaneous at 298 K, even though \(\Delta S\) is negative, because the large negative \(\Delta H\) dominates at this temperature. Enthalpy changes like this are often built up from lattice energies and other terms discussed in enthalpy, lattice energy and electron affinity.

Because \(\Delta G = \Delta H - T\Delta S\), and \(\Delta H\) and \(\Delta S\) change very little with temperature, \(\Delta G\) varies almost linearly with \(T\). For the reaction above, as temperature rises the term \(T\Delta S\) becomes more negative more slowly than the fixed \(\Delta H\) term stays negative, so \(\Delta G\) actually increases with temperature. At some temperature \(\Delta G\) reaches zero, and the reaction is no longer spontaneous beyond that point. Setting \(\Delta G = 0\) gives \(T = \Delta H \div \Delta S\), which for this example is approximately 465 K.

This kind of temperature dependence explains why some reactions, such as the thermal decomposition of certain solids, only become spontaneous once heated past a particular temperature, while others are spontaneous at every temperature you would realistically use in a laboratory.

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