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Concept 04 / 06Thermodynamics
Gibbs Energy, Spontaneity and Equilibrium
Gibbs energy G = H - TS as the criterion of spontaneity at constant temperature and pressure, the four combinations of ΔH and ΔS, crossover temperature, standard Gibbs energy of formation, ΔG° = -RT ln K, ΔG = ΔG° + RT ln Q, and the link to useful work and cell emf.
Figure 1
Total entropy is the fundamental criterion of spontaneity, but it needs information about the surroundings. At constant temperature and pressure, a function of the system alone does the job.
Gibbs energy. G=H−TS. At constant T and P:
ΔG=ΔH−TΔS.
Since ΔStotal=ΔSsys−TΔHsys=−TΔG, we get:
ΔG<0: spontaneous; ΔG>0: non-spontaneous (the reverse is spontaneous); ΔG=0: equilibrium.
−ΔG also equals the maximum useful (non-expansion) work obtainable: ΔG=wuseful, max. For an electrochemical cell, ΔG=−nFEcell.
Four cases (see figure):
ΔH<0, ΔS>0: ΔG<0 at all temperatures (combustion).
ΔH>0, ΔS<0: ΔG>0 at all temperatures; never spontaneous.
ΔH<0, ΔS<0: spontaneous at low T, below T=ΔSΔH (freezing of water, N2+3H2→2NH3).
ΔH>0, ΔS>0: spontaneous at high T, above T=ΔSΔH (melting, decomposition of CaCO3).
The crossover temperature T=ΔSΔH is where ΔG=0 (assuming ΔH and ΔS roughly independent of temperature).
Standard Gibbs energy. ΔrG∘=ΔrH∘−TΔrS∘, or ΔrG∘=∑ΔfG∘(products)−∑ΔfG∘(reactants), with ΔfG∘=0 for elements in their standard states.
Gibbs energy and equilibrium. Under non-standard conditions,
ΔG=ΔG∘+RTlnQ,
where Q is the reaction quotient. At equilibrium ΔG=0 and Q=K, so
ΔG∘=−RTlnK=−2.303RTlogK.
ΔG∘<0 means K>1 (products favoured); ΔG∘>0 means K<1. At 298K, each factor of 10 in K corresponds to 5.71kJmol−1 in ΔG∘.
If Q<K, ΔG<0 and the reaction proceeds forward; if Q>K, it proceeds backward.
Key Takeaways:
- ΔG=ΔH−TΔS; spontaneous if ΔG<0 (constant T, P).
- Crossover temperature T=ΔSΔH.
- ΔG∘=−RTlnK; ΔG=ΔG∘+RTlnQ.
- ΔG=−nFE; −ΔG is the maximum useful work.
- Pitfall: convert ΔS from JK−1 to kJK−1 before combining with ΔH in kilojoules.
- Pitfall: ΔG∘>0 does not mean no reaction at all; it means K<1 and the equilibrium lies towards reactants.
Brain-Teaser 1: For CaCO3(s)→CaO(s)+CO2(g), ΔH∘=+178.3kJmol−1 and ΔS∘=+160.4JK−1mol−1. Above what temperature is the decomposition spontaneous under standard conditions?
Solution: T=ΔSΔH=160.4178300=1112K (about 839∘C). Lime kilns operate above this temperature.
Brain-Teaser 2: A reaction has K=10 at 298K. Find ΔG∘.
Solution: ΔG∘=−2.303×8.314×298×log10=−5.71kJmol−1. A modest negative ΔG∘ already means products are favoured tenfold.
Brain-Teaser 3: For the freezing of water, ΔH=−6.01kJmol−1 and ΔS=−22.0JK−1mol−1. Find ΔG at 263K and at 283K, and interpret.
Solution: At 263K: ΔG=−6.01−263×(−0.0220)=−6.01+5.79=−0.22kJmol−1: water freezes spontaneously. At 283K: ΔG=−6.01+6.23=+0.22kJmol−1: freezing is not spontaneous (ice melts). The crossover 22.06010=273K is the melting point.
