Entropy (S) – created in 1865 by Rudolf Clausius
Entropy as a property of matter is admittedly difficult to comprehend, but this doesn’t mean it is difficult to use in classical thermodynamics. In fact it is precisely entropy that enabled the creation of the first fundamental thermodynamic relation — the equation from which all others were derived.
Chapter 5 established the two definitions of entropy and their connection. Boltzmann showed that entropy measures the number of accessible microstates: S = kBln W. Clausius showed that entropy changes can be calculated from measurable quantities: dS = δQrev/T. And Chapter 5 showed that these are the same thing seen from different levels of description — dividing δQrev by T gives precisely the fractional increase in accessible states caused by adding that thermal energy. The two definitions are not competing; they are complementary. One is microscopic, one is macroscopic, and both give the same answer for every substance ever measured.
Chapter 5 also established the Third Law and the absolute entropy integral:
S(T) = ∫₀ᵀ (Cp/T) dT + Σ (ΔHtransition / Ttransition)
The heat capacity Cp that appears here is not an abstraction — it is the physically meaningful property developed in Chapter 9, quantifying how many ways atoms and molecules can store energy. This integral is how absolute entropy values are obtained from calorimetric measurements in practice, and it is the source of the entropy values used in thermodynamic calculations throughout this book.
With that foundation in place from Chapter 5, we can now put entropy to work.
The fundamental equation
Combining Clausius’s entropy definition with the reversible work expression W = PdV and the First Law dU = Q − W, we arrive at:
dU = TdS − PdV
The power of this relation lies in what it eliminated. Heat Q and work W are path-dependent quantities — they depend on how a process is carried out, not just on the beginning and end states. By substituting TdS and PdV for Q and W respectively, Clausius expressed the change in internal energy entirely in terms of state properties. The equal sign now carries its full weight: this relation holds regardless of path. It applies to any process connecting two equilibrium states because all quantities in it — U, S, V, T, P — are properties of those states alone.
It was this equation that gave J. Willard Gibbs the foundation for his landmark 300-page publication “On the Equilibrium of Heterogeneous Substances” — from which enthalpy, Gibbs energy, Helmholtz energy, chemical potential, and the modern thermodynamics of mixtures, phase equilibria, and chemical reactions all descend.
Entropy in engineering: the isentropic process
Entropy proved especially valuable in the engineering analysis of expansion work. In a turbine, gas expands and does shaft work. If the expansion is both adiabatic (δQ = 0) and reversible — the best-case scenario — then dS = 0.
Both conditions are necessary. An adiabatic process alone is not sufficient — an adiabatic but irreversible expansion, such as a sudden unresisted expansion into a vacuum, still generates entropy internally even though no heat crosses the boundary. Only when the process is simultaneously adiabatic and reversible does entropy remain constant.
This isentropic condition represents the theoretical ideal for turbine performance — the maximum work extractable for a given pressure drop. Note that the second step in Sadi Carnot’s theoretical heat cycle is exactly this: adiabatic reversible expansion. It was Clausius who identified that entropy does not change in that step, giving the isentropic process its precise thermodynamic meaning.
Real turbines fall short of this ideal due to irreversibilities — friction, turbulence, heat losses — and entropy increases as a result. The gap between actual and isentropic performance defines turbine efficiency. Engineers use the Mollier diagram — a plot of enthalpy versus entropy — to visualize this directly: isentropic expansion appears as a vertical line, and the horizontal deviation from that line measures the entropy generated by irreversibilities. The more the actual path deviates from vertical, the less work was extracted.
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