The exact statement of the second law for any cycle — and the birth of entropy as a state function. Go around a closed loop and sum dQ/T: for a reversible cycle it is exactly zero, for a real one it is strictly negative. Because the reversible loop closes to zero, dS = dQ_rev/T is path-independent, so entropy exists. Rendered, not quoted.
source Clausius, R. (1865). Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Ann. Phys. 201(7), 353–400. doi:10.1002/andp.18652010702 — the paper that named “Entropie”.
An ideal gas (γ = 7/5) taken around a closed reversible cycle exchanges heat only at the reservoir temperatures. Along a reversible path dQ_rev = dU + dW = nCvdT + (nRT/V)dV, so
The integrand is exact, therefore its loop integral vanishes and its endpoint integral is a state function — call it S. The Carnot loop is the calibration: heat Qh in at Th, Qc out at Tc, with Qc/Qh = Tc/Th.
The second law written as an integral. Clausius 1865 gives ∮dQ/T ≤ 0 (= 0 reversible), the inequality that bounds the-carnot-cycle and every engine beneath it: no cycle between two reservoirs can beat 1 − Tc/Th, because doing so would force ∮dQ/T > 0.
the-carnot-cycle → the-clausius-inequality → entropy(state)
Live re-check of the closed loop ∮dQ/T for the reversible Carnot cycle. It must be 0 (and ≤ 0). If Window 6 flips a sign, this badge catches it and turns red.
witness idleReservoirs & heat
Th = 600 K · Tc = 300 K
Qh = 1000 J absorbed at Th
γ = 7/5 · n = 1 mol · R = 8.314…
Carnot (reversible) T–S loop is a rectangle; enclosed area = Wnet = (Th−Tc)·ΔS. The corner sums close the loop.
Reversible loop closes exactly: ∮dQ/T = 0.
Irreversible loop: ∮dQ/T < 0.
Therefore S exists and ΔSuniverse ≥ 0.
No. Reversal negates every dQ but also swaps which leg touches which reservoir; ∮dQ/T for the reversed reversible cycle is still exactly 0. Any real (finite-rate, dissipative) traverse adds internal entropy generation that can only make ∮dQ/T more negative. The wall is the second law itself: you cannot cross it in either direction.
“Entropy is disorder / messiness.”→ Entropy is the state function whose differential is dQ_rev/T. It is defined by the closing of a loop, not by a metaphor.
“∮dQ/T = 0 for every heat engine.”→ Only for reversible cycles. Every real engine gives ∮dQ/T < 0 (strict deficit).
“dQ/T is an exact differential, so heat is a state function.”→ Q is path-dependent; only dQ_rev/T (with the 1/T integrating factor) is exact. That is exactly Clausius's insight.
Flip the sign on the rejected-heat term so the Carnot loop integrates to a positive value — the second law “violated,” entropy not conserved on a reversible loop. The Witness (7) re-runs and catches it.