# D23.5 Gibbs Free Energy and Work

When we talk about the thermodynamics of a reaction, we are concerned primarily with the difference in energy between reactants and products, but not with the mechanism or the rate by which reactants change into products (a topic for a later unit).

When the Gibbs free energy of the products is lower than that of the reactants, a reaction is said to be exergonic. Conversely, an endergonic reaction is one in which the products are higher in Gibbs free energy than the reactants.

When there is a decrease in Gibbs free energy as a reaction occurs, ΔrG corresponds to the maximum useful work that can be done by the reaction system, ΔrG = −wmax. (The negative sign indicates that wmax is positive when ΔrG is negative.) Conversely, if a reaction has positive ΔrG, work must be done on the system to force the reaction to occur, in other words, the work done by the reaction is negative. The minimum work that must be done to the system is given by ΔrG.

Exercise: Exergonic and Endergonic Reactions

When considering a reaction under standard-state conditions the relevant thermodynamic quantity is ΔrG°. If a reaction is exergonic under standard-state conditions, ΔrG° < 0.

Exercise: Gibbs Free Energy and Work

# Coupled Reactions

One way to allow a reactant-favored process to occur is to couple it with a reaction that is product-favored. For example, consider the recovery of aluminum from alumina (Al2O3) ore:

Al2O3(s) → 2 Al(s) + O2(g)          ΔrG° (298 K) = 1576.4 kJ/mol

At least 1576.4 kJ of work must be done to change 1 mol Al2O3(s) into 2 mol Al(s) and 1.5 mol O2(g) (at 1 bar). In a modern aluminum manufacturing plant, this work is supplied electrically and the electricity is often provided by burning coal. Assuming coal to be mainly carbon, the combustion reaction is:

C(s) + O2(g) → CO2(g)          ΔrG° (298 K) = −394.4 kJ/mol

Thus the ΔrG° values indicate that, under standard-state conditions and ideal 100% efficiency, at least four moles of carbon/coal must burn to process each mole of Al2O3 ore. (In practice the aluminum smelting process is only 17% efficient, so it is necessary to burn nearly 6 times the theoretical amount of coal.) Coupled reactions occur simultaneously and there is a means of exchanging energy between them. The energy exchange occurs via the electric power grid in this specific case.

In other words, a reaction that is endergonic under standard-state conditions can be coupled to a separate exergonic reaction that drives the endergonic reaction (the thermodynamically unfavorable one) to occur. The Δr values for the two coupled reactions are summed to yield the overall Δr. For the alumina example, multiply the carbon combustion reaction by 4, add the two reaction equations, and apply Hess’s Law gives:

Al2O3(s) + 4 C(s) + 4 O2(g) → 2 Al(s) + O2(g) + 4 CO2(g)          ΔrG°(298 K) = -1.2 kJ/mol

The overall reaction now has a negative ΔrG° and is product-favored.

Under nonstandard-state conditions, a reaction with ΔrG < 0 can drive a reaction with ΔrG > 0, provided energy can be transferred from one to the other.