Visual walkthrough — Thermodynamic potentials — U (internal), H (enthalpy), F (Helmholtz), G (Gibbs)

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Step 1 — What is energy bookkeeping? (the piggy bank)

WHAT. A gas trapped in a container holds a certain amount of internal energy. Call it $U$ — just a number of joules, like the money inside a piggy bank. Two things can change that number: we can heat the gas (pour energy in through the walls) or the gas can push its container wall outward and do work (spend energy).

WHY start here. Every potential we build is $U$ rewritten. If you cannot picture $U$ changing, nothing later makes sense.

PICTURE. Look at the figure: the orange arrow into the box is heat added; the teal arrow out is work done by the gas pushing its piston. The bank level rises with heat, falls with work.

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Step 2 — Naming the "slivers": what is $TdS$ and what is $pdV$?

WHAT. We now put faces on $\delta Q$ and $\delta W$.

The work sliver is easy to see: if the piston of area $A$ moves out by a tiny distance, the gas pushes with pressure $p$ over that area, and the enclosed volume grows by $dV$. Force $\times$ distance $=(pA)\times (\text{distance})=pdV$. So $\delta W=pdV.$

• $p$ = pressure (how hard the gas pushes, per unit area).
• $dV$ = the tiny increase in volume as the piston slides out.

The heat sliver needs a new character. When we heat the gas reversibly (slowly, so it is never jolted), the heat added is $\delta Q=TdS.$

• $T$ = absolute temperature (how hot).
• $S$ = entropy — a measure of the gas's internal "spread-outness" or disorder (see Entropy and the Second Law). $dS$ is a tiny change in that disorder.

WHY these two products? Notice the pattern: each energy sliver is an intensive quantity (something you can measure with one probe: $T$, $p$) times a change in an extensive quantity (something that scales with system size: $S$, $V$). This "intensive × d(extensive)" shape is the heart of everything below.

WHY entropy at all? Because "amount of heat" is not a property of the gas — but $TdS$ is an exact bookkeeping entry. Entropy is the accountant nature invented so that heat can be tracked cleanly.

PICTURE. The figure shows the piston push (teal, giving $pdV$) and a thermometer + a scatter of dots getting more spread out (orange, giving $TdS$).

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Step 3 — Why this holds on any path (state function)

WHAT. We derived $dU=TdS-pdV$ assuming a slow, reversible process. Now we claim it holds always, even for a violent, irreversible change.

WHY it's allowed. $U$, $S$, and $V$ are all state functions: their values depend only on where the gas is (its current $T$, $p$, etc.), not on how it got there. Picture two points on a map — the difference in altitude between them is fixed no matter which trail you hike. Since $dU$, $dS$, $dV$ are differences of state functions, the relation between them is a fixed feature of the landscape, path-independent. We are allowed to compute it along the easy (reversible) trail and trust it everywhere.

PICTURE. Two paths between the same start and end dots on a $(S,V)$ map: a wiggly irreversible path (plum, dashed) and a smooth reversible one (teal). The net change in altitude $U$ is identical.

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Step 4 — The problem: we can't hold $S$ and $V$ fixed in the lab

WHAT. $U(S,V)$ is beautiful but impractical. In a real lab you rarely clamp entropy or volume. You clamp temperature (a thermostat / water bath) and pressure (leave the beaker open to the atmosphere).

WHY this matters. We want an energy function whose natural variables are the knobs we actually turn. So we must trade a variable we can't control ($S$ or $V$) for one we can ($T$ or $p$).

PICTURE. A lab bench: a sealed rigid box (controls $V$), a thermostat bath (controls $T$), an open beaker under the sky (controls $p$). We circle the knobs a human can grab.

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Step 5 — The Legendre move: subtract off a rectangle

WHAT. To swap $V$ (which we can't fix) for $p$ (which we can), we add $pV$ to $U$. To swap $S$ for $T$, we subtract $TS$. Why "add here, subtract there"? The geometry decides, not us.

WHY it works — the rectangle picture. Suppose $U$ is a curve rising with $V$ (at fixed $S$). Its slope is $-p$. We want a new function whose independent variable is the slope, not $V$. The trick: at each point, subtract the tangent-line's rectangle so that what's left changes with the slope instead of with $V$. Concretely, the differential of the product $pV$ is $d(pV)=pdV+Vdp.$

• The $pdV$ piece is exactly the annoying term in $dU$ we want to cancel.
• The $Vdp$ piece is the new term, carrying the slope $p$ as the variable.

PICTURE. The curve $U$ vs $V$, the tangent at a point, and the shaded rectangle of area $pV$. Sliding along the slope instead of along $V$ is the whole idea of the transform.

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Step 6 — Enthalpy $H$: swap $V\to p$

WHAT. Build the potential natural in $(S,p)$ by adding $pV$: $H\equiv U+pV.$

WHY add $pV$. The $-pdV$ in $dU$ is the term stopping $V$ from becoming the slope $p$. Adding $pV$ injects $+pdV+Vdp$; the $+pdV$ cancels the $-pdV$, leaving the fresh $Vdp$.

The differential, term by term: $dH=dU+pdV+Vdp=(\underset{\text{kept}}{\underbrace{TdS}}\underset{\text{cancels}}{\underbrace{-pdV}})+\underset{\text{cancels}}{\underbrace{pdV}}+\underset{\text{new}}{\underbrace{Vdp}}=TdS+Vdp$

So $H=H(S,p)$, and $T=(\partial H/\partial S{)}_p$, $V=(\partial H/\partial p{)}_S$.

PICTURE. A ledger-arrow diagram: start at $dU$, the $-pdV$ and $+pdV$ tokens annihilate (crossed out), $Vdp$ appears. The natural-variable label flips from $V$ to $p$.

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Step 7 — Helmholtz $F$ and Gibbs $G$: swap $S\to T$ (and both)

WHAT. Build the potential natural in $(T,V)$ by subtracting $TS$: $F\equiv U-TS,$ and the one natural in $(T,p)$ by doing both swaps: $G\equiv U-TS+pV=H-TS=F+pV.$

WHY subtract $TS$. The $+TdS$ term blocks $S$ from becoming the slope $T$. Subtracting $TS$ injects $-TdS-SdT$; the $-TdS$ cancels the $+TdS$, leaving the new $-SdT$.

The two differentials, term by term: $dF=dU-TdS-SdT=(\underset{\text{cancels}}{\underbrace{TdS}}-pdV)\underset{\text{cancels}}{\underbrace{-TdS}}\underset{\text{new}}{\underbrace{-SdT}}=-SdT-pdV$ $dG=dH-TdS-SdT=(\underset{\text{cancels}}{\underbrace{TdS}}+Vdp)\underset{\text{cancels}}{\underbrace{-TdS}}-SdT=-SdT+Vdp$

So $F=F(T,V)$ and $G=G(T,p)$.

PICTURE. A 2×2 map of the four potentials placed on axes "variable I chose for the first slot" (S or T) vs "second slot" (V or p). Arrows labelled $+pV$ and $-TS$ move you between the corners.

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Step 8 — Degenerate & edge cases (never leave a gap)

WHAT. Check the corners where a term dies.

• Isochoric ($dV=0$, rigid box): $dU=TdS$, and $dF=-SdT$. Heat now bumps $U$ directly; no work escapes.
• Isobaric ($dp=0$, open): $dH=TdS=\delta Q$ (Step 6). $H$ becomes the heat tracker.
• Isothermal ($dT=0$, bath): $dF=-pdV$ (all energy change is work) and $dG=Vdp$ (used in worked example (b): $\Delta G=nRT\ln (p/p_0)$, see Chemical potential and the grand potential).
• Adiabatic-reversible ($dS=0$, insulated & slow): $dU=-pdV$ and $dH=Vdp$. Energy changes are pure work.
• Zero-entropy limit ($T\to 0$): the $TS$ correction vanishes, so $F\to U$ and $G\to H$ — the "messiness tax" disappears near absolute zero (Third Law flavour).

WHY show all of them. Each real experiment clamps one variable; the reader must recognise which potential goes flat and which term survives, in every case.

PICTURE. A grid: rows = the four constraints, columns = which term of each $d\Phi$ survives vs dies. Dead terms greyed, live terms glowing.

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The one-picture summary

WHAT. One diagram compresses the whole derivation: start at $U(S,V)$, apply $+pV$ (right) and $-TS$ (down), read the differential in each corner. This is the Legendre transforms square, and its exact-differential corners generate every Maxwell relation and the $C_p - C_V$ result.

Recall Feynman retelling (plain words, cover and recall)

Start with a piggy bank of energy $U$. Two things move the level: heating it (that's $TdS$, temperature times a change in messiness) and letting it push its wall out (that's $pdV$, pressure times growth in size). That single sentence is $dU=TdS-pdV$. But in a real lab I don't get to dial "messiness" or "size" — I dial a thermostat (temperature) and I leave things open to the air (pressure). So I re-package the same energy so its handles are the ones I actually own. To swap "size" for "pressure," I bolt on $pV$; the swap cancels the size-term and hands me a pressure-term — that's enthalpy $H$. To swap "messiness" for "temperature," I peel off $TS$ (the tax I can't spend); that's Helmholtz $F$. Do both and I get Gibbs $G$, whose handles $T$ and $p$ are exactly a chemist's knobs. Every "add $pV$" or "subtract $TS$" is forced by the plus/minus sign already sitting in the first sentence — I never guess. And in special cases (rigid box, open beaker, insulated, near absolute zero), one term simply switches off, telling me which bank balance to watch.