Entropy — Clausius definition dS = dQ_rev - T: Foundations

This page builds every letter, ratio, and squiggle the parent note leans on — starting from what a "state" even is — so that when you meet $d S = d Q_{r e v} / T$ you already own every piece.

0. The stage: a "state" and a "system"

Before any symbol, picture the thing we study.

Figure — Entropy — Clausius definition dS = dQ_rev - T

Look at the figure. Each point on the diagram is one state. A curve joining two points is a process — a specific route the gas takes from state $A$ to state $B$ . The whole topic is one question: what quantities care only about the two dots, and what quantities care about the whole curve?

1. $V$ — Volume

Picture the piston in the figure sliding out: the box gets bigger, $V$ increases. Volume is a state quantity — you can point at the gas and say "it occupies $V$ right now" without knowing its history. We need it because gases do work by pushing their boundary outward, and that push changes $V$ .

2. $P$ — Pressure

Picture the gas molecules drumming on the piston face. More frequent, harder hits ⇒ higher $P$ . Pressure is also a state quantity. We need it because the work the gas does depends on how hard it pushes while the wall moves.

3. $T$ — Absolute temperature

Temperature is the "crowding" number in the parent note's water analogy: pour heat into a cold (small $T$ ) thing and $d Q / T$ is a big jump; into a hot (large $T$ ) thing, a small jump.

4. $U$ — Internal energy

For an ideal gas, $U$ depends only on $T$ (faster jiggle = more energy), a fact the parent note uses in Example 1.

5. $Q$ and $d Q$ — Heat (the slippery one)

This is the crux: $Q$ is a path quantity. Different routes between the same two dots deliver different total $Q$ . We need heat because it is one of the two ways (the other is work) energy enters the gas.

6. $W$ and $d W = P d V$ — Work

Why $d W = P d V$ ? Force on the piston is (pressure) × (area) $= P A$ . Push it out a tiny distance $d x$ and the work is force × distance $= P A d x$ . But $A d x$ is exactly the new sliver of volume $d V$ . So $d W = P d V$ . Look at the shaded strip in the figure: its area under the $P$ – $V$ curve is the work. Since the shape of the curve sets that area, work — like heat — is a path quantity.

7. $d$ vs $Δ$ vs $\int$ vs $\oint$ — the calculus notation

8. Exact vs inexact differentials — the mathematical heart

The figure shows two paths (violet, orange) between the same two dots. For an exact quantity both paths give the same total (altitude climbed on a hill: only endpoints matter). For an inexact quantity — heat $d Q$ , work $d W$ — the two totals differ (fuel burned on a hilly detour: the route matters).

9. Reversible — the word doing all the work

10. The Carnot ratio $\frac{Q _{H}}{T _{H}} = \frac{Q _{C}}{T _{C}}$

Why the parent note needs it: this single equation says the heat/temperature ratio balances around one loop. Tile any reversible loop with tiny Carnot cycles and the balance spreads to every loop, giving $\oint d Q_{r e v} / T = 0$ — the existence proof for $S$ . See Carnot Cycle and Efficiency and Clausius Inequality.

A state quantity vs a path quantity — give one of each. ::: State: $U, P, V, T, S$ (depend only on the dot). Path: $Q, W$ (depend on the curve).
Why must temperature be in kelvin for $d Q / T$ ? ::: Kelvin starts at absolute zero and is always positive, so $1/ T$ never divides by zero or flips sign.
Where does $d W = P d V$ come from? ::: Force $= P A$ , distance $= d x$ , work $= P A d x = P d V$ since $A d x = d V$ .
What does an inexact differential fail to satisfy? ::: Path-independence — $\int_{A}^{B}$ depends on the route and $\oint \neq = 0$ .
What is an integrating factor and which one works for heat? ::: A multiplier that turns an inexact differential exact; for $d Q_{r e v}$ it is $1/ T$ .
What does $\oint d X = 0$ tell you about $X$ ? ::: $X$ is a state function (returns to its value after any round trip).
Why is the word reversible required in $d S = d Q_{r e v} / T$ ? ::: Only then is $T$ well-defined and equal for system and surroundings, making $d Q / T$ exact.
What does the Carnot ratio $Q_{H} / T_{H} = Q_{C} / T_{C}$ let us conclude? ::: The heat/temperature ratio balances around a loop, extending to $\oint d Q_{r e v} / T = 0$ for any reversible cycle.

Connections

Parent: Clausius definition of entropy
First Law of Thermodynamics — supplies $d Q = d U + P d V$
Exact and Inexact Differentials — why $1/ T$ is an integrating factor
Reversible vs Irreversible Processes — the meaning of "rev"
Carnot Cycle and Efficiency — the ratio $Q_{H} / T_{H} = Q_{C} / T_{C}$
Clausius Inequality — where the loop integral becomes $\leq 0$
Second Law of Thermodynamics — where entropy is heading
Statistical Entropy — Boltzmann S = k ln W — the microscopic meaning of $S$

Foundations — Entropy — Clausius definition dS = dQ_rev - T