12.2 Entropy, Free Energy & Spontaneity
Key Takeaways
- The Second Law of Thermodynamics requires that total entropy of the universe increase for any spontaneous process (ΔS_universe > 0), even though a system's own entropy can decrease as long as the surroundings' entropy increases by more.
- Entropy increases in the order crystalline solid < liquid < gas for a given substance, because gas particles have access to the most possible positions and momenta.
- Gibbs free energy combines enthalpy and entropy as ΔG = ΔH − TΔS; a process is spontaneous when ΔG < 0, non-spontaneous when ΔG > 0, and at equilibrium when ΔG = 0.
- The relationship ΔG° = −RT ln Keq means Keq > 1 gives a negative ΔG° (products favored at equilibrium), Keq < 1 gives a positive ΔG° (reactants favored), and Keq = 1 gives ΔG° = 0.
- At room temperature (298 K), each tenfold change in Keq corresponds to roughly a 5.7 kJ/mol change in ΔG°, a useful no-calculator estimation shortcut.
Where Section 12.1 asked how much energy a reaction absorbs or releases, this section asks a different question entirely: will the reaction happen on its own? Answering that requires two more state functions — entropy and free energy — that combine energy changes with matter's natural tendency toward greater dispersal of energy. In living systems the same framework decides whether ATP hydrolysis can drive an endergonic biosynthetic step, whether ice melts at body temperature, and whether a metabolic pathway can run forward under cellular concentrations.
The Second Law of Thermodynamics
The Second Law of Thermodynamics states that for any spontaneous process, the total entropy of the universe (system plus surroundings) increases: ΔS_universe > 0. Entropy (S) is a state function commonly described as a measure of disorder, or more precisely, the number of energetically equivalent ways a system's particles and energy can be arranged — its microstates. Boltzmann's relation S = k ln W captures this idea: more accessible arrangements (larger W) means higher entropy. Unlike energy, entropy is not conserved: every real, spontaneous process increases the total entropy of the universe, even when a system's own entropy decreases locally.
A refrigerator, for example, lowers entropy inside the fridge but increases entropy even more in the surroundings through the heat and work it exhausts out the back. Living organisms are open systems that maintain low internal entropy (highly ordered macromolecules, ion gradients, and membranes) by exporting entropy to the surroundings through heat release and waste products. Local order in a cell never violates the Second Law; the universe's total entropy still rises.
This distinction is one the MCAT tests directly: a system's own entropy (ΔS of the system alone) can decrease during a spontaneous process, provided the surroundings' entropy increases by a larger amount. The Second Law constrains the universe's total entropy, not any single system considered in isolation.
Relative Entropy: Gas > Liquid > Crystal
Because entropy tracks the number of accessible microstates, the physical state of matter predicts relative entropy directly. Gas particles move freely through a large volume with many possible positions and momenta, so gases have the highest entropy. Liquid particles sit close together but can still slide past one another, giving intermediate entropy. A crystalline solid locks particles into a fixed, highly ordered lattice with the fewest accessible arrangements, giving the lowest entropy. For a given substance:
S(gas) > S(liquid) > S(crystalline solid)
This ranking predicts the sign of ΔS for many reactions without any calculation at all. A reaction that increases the number of gas moles, or converts a solid or liquid into a gas, has ΔS > 0. A reaction that decreases the number of gas moles, or forms a solid from dissolved ions (precipitation, freezing), has ΔS < 0. For example, 2 KClO3(s) → 2 KCl(s) + 3 O2(g) has ΔS > 0 because three moles of gas appear from an entirely solid starting point. Dissolving a solid into solution generally increases entropy as well, since the ions gain the freedom of the entire solution volume instead of a single fixed lattice position.
Biological examples of entropy change:
- Hydrolysis of a peptide into many free amino acids increases the number of independent particles → ΔS of the system is typically positive.
- Protein folding decreases the conformational entropy of the chain itself (fewer microstates for the backbone), but hydrophobic burial and solvent release often increase solvent entropy enough that the net ΔS_universe is still positive under folding conditions.
- Condensation of water vapor in the lungs (and evaporation from skin) are entropy-driven phase changes that also couple to heat transfer covered in Section 12.1.
Which of the following processes is expected to have a positive ΔS (an increase in entropy) for the system?
Gibbs Free Energy and Spontaneity
Gibbs free energy (G) combines a system's enthalpy and entropy into a single state function that predicts spontaneity directly, without separately tracking the surroundings:
ΔG = ΔH − TΔS
where T is the absolute temperature in kelvin. At constant temperature and pressure, a process is spontaneous (thermodynamically favorable, exergonic) when ΔG < 0, non-spontaneous (endergonic) when ΔG > 0 (the reverse process is favorable instead), and at equilibrium when ΔG = 0.
The standard free energy change (ΔG°) applies this same logic under standard conditions (1 atm, 1 M solutes, a specified temperature, usually 298 K). A negative ΔG° means the reaction as written is favorable under those standard conditions. Biochemistry often reports ΔG°′, the standard free energy change under biochemical standard conditions (pH 7, and often [Mg2+] fixed), which is the form used for ATP hydrolysis tables — but the algebraic relationship ΔG = ΔH − TΔS and the comparison of ΔG to zero remain the same.
Critical distinction the MCAT loves to test: spontaneous does not mean fast. ΔG describes whether a reaction is thermodynamically favorable; it says nothing about the rate at which the reaction proceeds, which is governed instead by kinetics and activation energy, covered later in this chapter. Diamond converting to graphite has ΔG° < 0 at room temperature, yet the process is so slow it is imperceptible on any human timescale. Peptide bond hydrolysis is also highly favorable thermodynamically but kinetically slow without a protease — a recurring MCAT theme linking this section to enzyme catalysis.
Predicting Spontaneity: The Four ΔH/ΔS Cases
Because ΔG = ΔH − TΔS, the signs of ΔH and ΔS alone sort every reaction into one of four categories:
| ΔH | ΔS | Resulting ΔG | Spontaneity |
|---|---|---|---|
| Negative | Positive | Always negative | Spontaneous at all temperatures |
| Negative | Negative | Negative only at low T | Spontaneous at low temperatures only |
| Positive | Positive | Negative only at high T | Spontaneous at high temperatures only |
| Positive | Negative | Always positive | Never spontaneous (as written) |
Worked example. A reaction has ΔH = +40 kJ/mol and ΔS = +100 J/(mol·K) — note the unit mismatch, which must always be resolved (convert kJ to J, or J to kJ) before combining the two terms. This reaction falls into the "positive ΔH, positive ΔS" row, so it is spontaneous only above some threshold temperature. Find that temperature by setting ΔG = 0:
0 = ΔH − TΔS, so T = ΔH ÷ ΔS = 40,000 J/mol ÷ 100 J/(mol·K) = 400 K
Below 400 K, ΔG is positive (non-spontaneous, since the unfavorable +40 kJ/mol enthalpy term dominates); above 400 K, ΔG is negative (spontaneous, since the −TΔS term grows large enough to outweigh ΔH). This is exactly the pattern behind melting: ice melting has ΔH > 0 and ΔS > 0, is non-spontaneous below 0°C, and becomes spontaneous above it, with the melting point itself being the crossover temperature where ΔG = 0.
At 298 K, a reaction has an equilibrium constant Keq = 0.01. What is the approximate sign and magnitude of ΔG° for this reaction?
Free Energy and the Equilibrium Constant
Standard free energy connects directly to the position of equilibrium through:
ΔG° = −RT ln Keq
where R is the gas constant, 8.314 J/(mol·K), and T is absolute temperature. This equation explains sign relationships the MCAT tests directly: when Keq > 1 (products favored at equilibrium), ln Keq is positive, making ΔG° negative. When Keq < 1 (reactants favored), ln Keq is negative, making ΔG° positive. When Keq = 1, ln Keq = 0 and ΔG° = 0.
A useful estimation shortcut at room temperature (298 K): RT ln 10 ≈ (8.314 J/(mol·K))(298 K)(2.303) ≈ 5,700 J/mol, or about 5.7 kJ/mol. That means every factor of 10 change in Keq corresponds to roughly a 5.7 kJ/mol swing in ΔG° at room temperature — a handy no-calculator checkpoint whenever a passage gives a Keq value and asks for a rough ΔG°, or the reverse.
Note the distinction: ΔG° (calculated from Keq) describes the reaction specifically under standard conditions. The more general relationship,
ΔG = ΔG° + RT ln Q
describes the free energy change at any actual, non-standard concentrations, where Q is the reaction quotient. Q equals Keq only once the system reaches equilibrium, at which point ΔG (not ΔG°) equals zero. This is why a reaction can still be spontaneous (ΔG < 0) in a given direction even when its ΔG° is positive, provided the actual concentrations are far enough from equilibrium — a common situation in cells, where product is continuously removed by a subsequent enzyme-catalyzed step, keeping Q small and ΔG negative.
Coupling Endergonic Reactions (Preview of Bioenergetics)
Many biosynthetic reactions have positive ΔG° values. Cells drive them by coupling them to strongly exergonic reactions (most famously ATP hydrolysis) so that the sum of the free-energy changes is negative. Coupling works only if the shared intermediate is shared chemically — the reactions are not merely "near each other" in space. That principle is developed fully in the bioenergetics section of the enzymes chapter; here the key thermo fact is that free energies add for sequential reactions, just as Hess's Law adds enthalpies.
Third Law Snapshot
The Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero is zero (a single microstate). Absolute entropies tabulated for substances are therefore always positive at T > 0 K. The MCAT rarely requires Third Law calculations, but knowing that absolute S values exist explains why ΔS°rxn can be computed as Σ S°(products) − Σ S°(reactants) in the same style as formation enthalpies.
A reaction has ΔH = −50 kJ/mol and ΔS = −60 J/(mol·K). Which statement correctly describes its spontaneity?
A metabolic reaction has ΔG° > 0 under standard conditions, yet proceeds spontaneously in a cell. Which explanation is most consistent with thermodynamic principles?
Common MCAT Traps
- Confusing ΔG and ΔG°. ΔG = 0 defines actual equilibrium; ΔG° = 0 only means Keq = 1. A reaction with a positive ΔG° can still be spontaneous (ΔG < 0) at concentrations far from equilibrium.
- Forgetting the unit mismatch between ΔH and ΔS. ΔH is typically given in kJ/mol while ΔS is given in J/(mol·K) — plugging both directly into ΔG = ΔH − TΔS without converting units first is one of the most common numeric errors on this topic.
- Equating "spontaneous" with "fast." Spontaneity (governed by ΔG) and rate (governed by kinetics and activation energy) are independent quantities; a strongly favorable reaction can still be kinetically trapped and proceed at an imperceptible rate.
- Misapplying the entropy ranking across different substances. Gas > liquid > crystal compares phases of the same substance; it does not automatically mean every gas has higher absolute entropy than every solid of a different, more complex substance.
- Assuming living systems violate thermodynamics. Ordered cells export entropy; local decreases in system entropy are allowed when surroundings entropy increases by more.