Foundations — Thermodynamic potentials — U, H, F, G (preview)

This page assumes nothing. We build every letter, every $d$, every $\partial$ from the ground up, in the order the parent note needs them. If you can read a thermometer and imagine a piston, you can finish this page.

0. The five characters in the story

Before any formula, meet the physical quantities. Two describe how much stuff and how spread out the energy is; two describe how hard the system is being pushed.

Why exactly these four? Because they come in two partner pairs:

• $(T,S)$ — the thermal pair. Multiply them and $TS$ has units of energy.
• $(p,V)$ — the mechanical pair. Multiply them and $pV$ also has units of energy.

Each pair is one "thing you can push on" and one "thing that responds." This pairing is the secret skeleton behind all four potentials, and we build the picture of it in §4.

1. What the little $d$ means: a tiny change

The parent note is full of things like $dU$, $dS$, $dT$. The symbol $d$ never means "multiply by d." It means "a tiny change in."

Why do physicists work with tiny changes instead of big ones? Because the rules ($TdS$, $pdV$) are only exactly true for a step so small that $T$ and $p$ don't change during the step. Take a tiny honest step, then add up millions of them (that's integration) to cover a big change. Look at the staircase in the figure: each tread is one $dX$; the whole climb is $\Delta X$.

2. Energy and the First Law: $dU=\delta Q-\delta W$

You can change that wallet in exactly two ways: pour heat in, or let the gas do work.

The First Law is just energy conservation written for these tiny bits: $dU=\delta Q-\delta W.$

3. Two substitutions that turn $\delta Q,\delta W$ into $TdS$ and $pdV$

The parent note jumps straight to $dU=TdS-pdV$. Here is where those pieces come from.

Work by a gas on a piston. Push a piston out by tiny volume $dV$ against pressure $p$: $\delta W=pdV.$ Why? Force = pressure × area, distance moved × area = volume swept, so force × distance = $pdV$. Picture: the piston slides out a sliver, the gas spent $pdV$ of energy shoving it.

Heat and entropy. For a reversible (idealised, done infinitely gently) step, the definition of entropy is exactly $\delta Q=TdS.$ Why can we say this? This is literally how entropy is defined — see Entropy and the Second Law. Read it backwards: entropy change $dS=\delta Q/T$ is "heat added, discounted by how hot you already are."

Substitute both into the First Law: $dU=\delta Q-\delta W=TdS-pdV.$

That single equation — connecting the two energy pairs from §0 — is the seed of the whole parent note. Everything ($H,F,G$, Maxwell relations, equilibrium) grows from it. See First Law of Thermodynamics.

4. The partner-pair square: why swapping is possible

Look again at the two pairs. Inside $dU=TdS-pdV$:

• $T$ sits in front of $dS$ → we say $T$ is the conjugate ("partner") of $S$.
• $p$ sits in front of $dV$ → $p$ is the conjugate of $V$.

Why does the parent note care? Because in a lab you can control one member of each pair easily and the other only clumsily:

So we will want to rewrite the energy so the easy-to-control partner becomes the "input variable." The tool that swaps a partner for its conjugate is the Legendre transform — introduced next, and covered fully in Legendre Transform.

5. The slope symbol ${\left(\frac{\partial U}{\partial S}\right)}_V=T$

The parent note reads off $T$ and $-p$ as partial derivatives. Here is that notation from zero.

Why does this matter for the topic? Because $dU=TdS-pdV$ can be pattern-matched against the generic rule for how a two-variable function changes: $dU={\left(\frac{\partial U}{\partial S}\right)}_{V}dS+{\left(\frac{\partial U}{\partial V}\right)}_{S}dV.$ Line up the coefficients of $dS$ and $dV$ on both sides: ${\left(\frac{\partial U}{\partial S}\right)}_V=T,{\left(\frac{\partial U}{\partial V}\right)}_S=-p.$ This "read the coefficient" move is used in every worked example of the parent note.

6. Building the other three wallets

Now every symbol is earned, the parent's definitions read cleanly:

• Add $pV$ → you're accounting for the room the system had to make against pressure → enthalpy $H$.
• Subtract $TS$ → you're setting aside the "entropy tax" the second law charges at temperature $T$ → Helmholtz free energy $F$.
• Do both → Gibbs free energy $G$ (the chemist's wallet — see Gibbs Free Energy and Chemical Equilibrium).

Why add/subtract exactly these products? Because each is one of the energy chunks from §0, and adding it is precisely the Legendre swap that promotes an easy lab variable to the driver's seat.

7. The prerequisite map

Read it top-down: the ideas of a tiny change and the four state variables feed the First Law; heat and work substitutions turn it into the master equation; conjugate pairs make the Legendre swap possible; that produces the four potentials; and partial derivatives let you extract everything else.

Equipment checklist

Test yourself — cover the right side and answer aloud.