Entropy and the Principle of Increase
Entropy derived from Carnot rather than asserted — and one bar that is never allowed to fall.
Skip to the animationEntropy is the property whose change is δQ_rev/T, discovered by noticing that Q/T returns to its starting value round a reversible cycle — and the entropy of an isolated system can never decrease, which is the second law in a form you can compute with.
Where it comes from
Carnot gives two expressions for the same efficiency: 1 − Q_C/Q_H and 1 − T_C/T_H. Equating them gives Q_H/T_H = Q_C/T_C.
So Q/T returns to its starting value round a reversible cycle. Anything whose cyclic integral vanishes is a property — by exactly the argument that made internal energy one. Entropy is not asserted; it is discovered.
The definition, and how to use it
dS = δQ_rev/T. Because S is a property, ΔS between two states is fixed regardless of the actual route — so for a real irreversible process you invent any convenient reversible path between the same end states and integrate along that.
The subscript "rev" is on the *definition*, not on where the result may be applied. Without this, entropy would be uncomputable for every process that actually happens.
The principle of increase
- 1Reversible process. Heat crosses at zero ΔT, so the system gains exactly what the surroundings lose, at the same T.
ΔS_universe = 0. - 2Heat across a finite ΔT. The same Q leaves at 600 K and arrives at 300 K. The cold body gains
Q/300, the hot losesQ/600, and the first is larger. - 3So
ΔS_universe > 0— with energy perfectly conserved throughout.
For an isolated system, ΔS ≥ 0, with equality only if reversible. A *system's* entropy may certainly fall — a refrigerator lowers its contents' entropy every second — but the total cannot, and the compressor work is what guarantees the surroundings rise by more.
Making it calculable
The Clausius inequality ∮δQ/T ≤ 0 is the second law in computable form, with equality only for a reversible cycle. The shortfall is the entropy generated, S_gen = ΔS_universe ≥ 0, which measures how irreversible the process actually was.
Multiplying by the ambient temperature gives the lost work: T₀·S_gen, in joules — the work you could have had and did not. This is what turns "that process is wasteful" into a number you can rank designs by.
What entropy is not
"Disorder" is a rough analogy: serviceable for gases, actively misleading for a tidy room or an ecosystem. The precise statement is Boltzmann's, S = k·ln W, where W counts the microstates consistent with the same macrostate.
This also answers the perennial objection about life. A growing organism lowers its own entropy while raising the universe's by very much more — the only entropy required to increase is that of an isolated system, never that of a chosen one.
The numbers you will be asked for
- Definition
dS = δQ_rev / T
Integrate along any reversible path between the same end states.
- Clausius inequality
∮ δQ/T ≤ 0
Equality only for a reversible cycle.
- Increase principle
ΔS_universe ≥ 0
Zero only if reversible.
- Ideal gas
Δs = c_v·ln(T₂/T₁) + R·ln(v₂/v₁)
Both terms are needed unless one variable is held fixed.
- Lost work
W_lost = T₀ · S_gen
Turns irreversibility into joules.
- Boltzmann
S = k · ln W
The statistical meaning, and the exact one.
Advantages and disadvantages
Advantages
- Converts the second law from a prohibition into a computable quantity.
- Being a property, it can be tabulated and looked up.
- Entropy generated measures irreversibility directly.
- It gives lost work in joules, which makes designs comparable.
Disadvantages
- Defined by a reversible path, which requires care when the real process is not.
- The disorder analogy misleads more often than it helps.
- Absolute entropy needs the third law as a reference point.
- It says nothing about rate — a permitted process may still be far too slow to use.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.