Entropy in Depth

The previous section introduced entropy as “disorder increases.” That’s a good starting point, but the MCAT sometimes pushes deeper. How do you actually calculate an entropy change? What connects entropy to Gibbs free energy (the spontaneity equation in chemistry)? And — most puzzling — how can highly ordered structures like proteins fold spontaneously if entropy always increases?

This section answers all three. The protein-folding question is especially MCAT-relevant because it bridges thermodynamics, biochemistry, and biology — and the resolution is one of the most elegant ideas in physical chemistry.

Calculating Entropy Change

For a reversible process at constant temperature, the entropy change is:

This formula explains why the same amount of heat has a bigger effect on entropy at low temperatures. Adding 100 J to a system at 200 K raises entropy by 0.5 J/K, but adding 100 J at 1000 K raises entropy by only 0.1 J/K.

The Universe Always Wins

For any spontaneous process:

Δ$S_{\text{universe}}$ = Δ$S_{\text{system}}$ + Δ$S_{\text{surroundings}}$ > 0

The system’s entropy can decrease - but only if the surroundings’ entropy increases by more. Look at specific examples:

Freezing water at -10 °C: Δ$S_{\text{system}}$ < 0 (liquid to solid = less disorder). But the heat released by freezing warms the surroundings, and because the surroundings are at a lower temperature, the entropy gain of the surroundings is large. Net result: Δ$S_{\text{universe}}$ > 0. The process is spontaneous.

Protein folding: A protein collapses from a random coil into a specific 3D shape - clearly lowering the protein’s own entropy. But the folding process releases water molecules that were ordered around hydrophobic residues, sharply raising the entropy of the surrounding water. The water’s entropy gain more than makes up for the protein’s entropy loss.

Connection to Gibbs Free Energy

Entropy is one half of the spontaneity equation. The full picture comes from Gibbs free energy, covered in General Chemistry - Thermochemistry:

ΔG = ΔH - TΔS

A process is spontaneous when ΔG < 0. Notice that high temperature amplifies the entropy term (TΔS), making entropy-driven processes more favorable at high temperatures.

Entropy and the Arrow of Time

Entropy gives time its direction. The laws of physics (Newton’s laws, Maxwell’s equations) work the same forward and backward in time. But the second law doesn’t - entropy increases toward the future, never toward the past. A video of an egg unscrambling itself looks wrong because it violates the second law, even though it would satisfy Newton’s laws perfectly.

The Third Law of Thermodynamics

At absolute zero (0 K), a perfect crystal has exactly one microstate - every atom is in its lowest energy position. Therefore, its entropy is exactly zero:

S = 0 at T = 0 K (for a perfect crystal)

This gives entropy an absolute reference point, unlike energy, which is always measured relative to some arbitrary zero. The MCAT may mention this but rarely tests it quantitatively.