7.6 Gibbs Free Energy

Learning Objectives

• To gain an understanding of Gibbs free energy.
• To understand the relationship between the sign of Gibbs free energy change and the spontaneity of a process.
• To be able to determine Gibbs free energy using standard free energies of formation.

J. Willard Gibbs (1839–1903) proposed a single state function to determine spontaneity:

$G=H-TS$

where H is the enthalpy of the system, S is the entropy of the system, and G is Gibbs free energy.

The change in Gibbs free energy, ΔG, is the maximum amount of free energy available to do useful work. For an isothermal process, it can be expressed as:

$\Delta G=\Delta H-T\Delta S$

or at standard conditions:

$\Delta G^{\circ}=\Delta H^{\circ}-T\Delta S^{\circ}$

This single term, Gibbs free energy (G), allows us to avoid calculating the entropy of the surroundings. It is really just a simplification of our previous method of estimating spontaneity:

\begin{align*} \Delta S_{\text{universe}}&=\Delta S_{\text{sys}}+\Delta S_{\text{surr}} \\ \\ \Delta S_{\text{universe}}&=\Delta S_{\text{sys}}+\dfrac{-\Delta H_{\text{sys}}}{T} \end{align*}

Multiply both sides of the equation by −T:

\begin{align*} -T\left(\Delta S_{\text{universe}}\right)&=\left(\Delta S_{\text{sys}}+\dfrac{-\Delta H_{\text{sys}}}{T}\right)-T \\ \\ -T\Delta S_{\text{universe}}&=\Delta H_{\text{sys}}-T\Delta S_{\text{sys}} \end{align*}

Therefore, $\Delta G=-T\Delta S_{\text{universe}}$.

As a result of this relationship, the sign of Gibbs free energy provides information on the spontaneity of a given reaction:

\begin{align*} \text{If }\Delta G>0,&\text{ the reaction is nonspontaneous in the direction written.} \\ \text{If }\Delta G=0,&\text{ the reaction is in a state of equilibrium.} \\ \text{If }\Delta G<0,&\text{ the reaction is spontaneous in the direction written.} \end{align*}

The significance of the sign of a change in Gibbs free energy parallels the relationship of terms from the equilibrium chapter: the reaction quotient, Q, and the equilibrium constant, K.

If Q > K, the reaction is nonspontaneous in the direction written.If Q = K, the reaction is in a state of equilibrium.If Q < K, the reaction is spontaneous in the direction written.

Example 7.10

Calculate ΔG° for a reaction where ΔH° is equal to 36.2 kJ and ΔS° is equal to 123 J/K at 298 K. Is this a spontaneous reaction?

Solution

\begin{align*} \Delta G^{\circ}&=\Delta H^{\circ}-T\Delta S^{\circ} \\ &=36.2\text{ kJ}-(298\text{ K}\times 123\text{ J/K}) \\ &=-0.4\text{ kJ} \end{align*}

Therefore the reaction is spontaneous because ΔG° is negative.

Determining ΔG° from Standard Free Energy of Formation

The standard Gibbs free energy change, ΔG°, for a reaction can be calculated from the standard free energies of formation, ΔG°f .

$\Delta G^{\circ}{}_{\text{f}}=\underset{\text{products}}{\sum n\Delta G^{\circ}{}_{\text{f}}}-\underset{\text{reactants}}{\sum m\Delta G^{\circ}{}_{\text{f}}}$

where n and m are the coefficients in the balanced chemical equation of the reaction.

Standard free energies of formation values are listed in the appendix, “Standard Thermodynamic Quantities for Chemical Substances at 25°C.”

Example 7.11

Calculate the standard free energy change for the following reaction, using standard free energies of formation:

5C(s) + 2SO2(g) → CS2(g) + 4CO(g)

Is this a spontaneous reaction?

Solution

\begin{align*} \Delta G^{\circ}&=\underset{\text{products}}{\sum n\Delta G^{\circ}{}_{\text{f}}}-\underset{\text{reactants}}{\sum m\Delta G^{\circ}{}_{\text{f}}} \\ &=[(4\times -137.2\text{ kJ/mol})+(67.1\text{ kJ/mol})] \\ &\phantom{=}-[(5\times 0\text{ kJ/mol})+(2\times -300.1\text{ kJ/mol})] \\ &=(-481.7\text{ kJ/mol})-(-600.2\text{ kJ/mol}) \\ &=118.5\text{ kJ/mol} \\ \end{align*}

ΔG° has a positive value, so this is not a spontaneous process.

In the Gibbs free energy change equation, the only part we as scientists can control is the temperature. We have seen how we can calculate the standard change in Gibbs free energy, ΔG°, but not all reactions we are interested in occur at exactly 298 K. The temperature plays an important role in determining the Gibbs free energy and spontaneity of a reaction.

$\Delta G=\Delta H-T\Delta S$

If we examine the Gibbs free energy change equation, we can cluster the components to create two general terms, an enthalpy term, ΔH, and an entropy term, –TΔS. Depending on the sign and magnitude of each, the sum of these terms determines the sign of ΔG and therefore the spontaneity (Table 7.5 “Spontaneity and the Signs of Enthalpy and Entropy Terms”).

Table 7.5 Spontaneity and the Signs of Enthalpy and Entropy Terms
ΔH ΔS TΔS ΔG Spontaneity
+ + + Nonspontaneous
+ Spontaneous
+ + or −
• Low temp: Spontaneous
• High temp: Nonspontaneous
+ + + or −
• Low temp: Nonspontaneous
• High temp: Spontaneous

Since all temperature values are positive in the Kelvin scale, the temperature affects the magnitude of the entropy term. As shown in Table 18.2 “Spontaneity and the Signs of Enthalpy and Entropy Terms,” the temperature can be the deciding factor in spontaneity when the enthalpy and entropy terms have opposite signs. If ΔH is negative, and –TΔS positive, the reaction will be spontaneous at low temperatures (decreasing the magnitude of the entropy term). If ΔH is positive, and –TΔS negative, the reaction will be spontaneous at high temperatures (increasing the magnitude of the entropy term).

Key Takeaways

• The change in Gibbs free energy (ΔG) is the maximum amount of free energy available to do useful work.
• If Δ> 0, the reaction is nonspontaneous in the direction written. If Δ= 0, the reaction is in a state of equilibrium. If Δ< 0, the reaction is spontaneous in the direction written.
• The standard Gibbs free energy change, ΔG°, for a reaction can be calculated from the standard free energies of formation, ΔG°f.
• Temperature can impact whether a reaction is spontaneous or not.