Entropy and the Second Law of Thermodynamics
The first law says energy is conserved; the second law says something the first law alone doesn't capture — that some perfectly energy-conserving processes never happen in reverse. A dropped egg conserves energy whether it shatters or (hypothetically) reassembles itself, but only one of those is ever observed. Entropy, ΔS=Q/T for a reversible process, is the quantity that formalizes this one-way arrow: total entropy of an isolated system never decreases, which is really a statement about which macroscopic outcomes are statistically overwhelmingly more likely than others.
What you'll learn
- Calculate the entropy change of a system for a reversible heat transfer.
- State the second law of thermodynamics and explain what it means for the total entropy of an isolated system.
- Calculate the total entropy change when heat flows spontaneously between two reservoirs at different temperatures.
- Explain why spontaneous heat flow from hot to cold (never cold to hot) is consistent with the second law.
- Distinguish reversible from irreversible processes in terms of total entropy change.
- Connect entropy to the statistical likelihood of macroscopic states, at a qualitative level.
1. Entropy and the Second Law
For a reversible heat transfer at a well-defined temperature T, entropy change is simply ΔS=Q/T. The second law of thermodynamics states that the total entropy of an isolated system never decreases — it stays the same for a reversible process and strictly increases for any real, irreversible one.
This is what breaks the forward/backward symmetry the first law alone allows: energy conservation permits heat to flow either from hot to cold or cold to hot equally well, but only the hot-to-cold direction increases total entropy, which is exactly why it's the only direction ever observed to happen spontaneously.
2. Reversible vs. Irreversible Processes
A reversible process proceeds through a continuous sequence of equilibrium states, with no finite imbalance (temperature difference, pressure difference, friction) driving it — an idealization real processes can only approach, never perfectly achieve, since any process happening at a finite rate necessarily involves some departure from equilibrium.
For irreversible processes, ΔS=Q/T using the *actual* process's heat and temperature doesn't apply directly — since entropy is a state function (depending only on initial and final states, not the path taken), its change must instead be computed along some reversible path connecting the same two states, even if the real process that actually occurred was nothing like that path.
Key equations
- ΔS = Q/T — Entropy change for a reversible heat transfer at (essentially) constant temperature T.
- ΔS_total = ΔS_hot + ΔS_cold ≥ 0 — Total entropy change when heat Q flows spontaneously from a hot reservoir to a cold one — always positive for any real (spontaneous) heat flow, consistent with the second law.