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

This page assumes nothing. If the parent note wrote a symbol, we build it here from a picture first. Read top to bottom — each block uses only what came before.

0. The system: what are we even looking at?

Why we need this picture. Every symbol below is a number you could read off this one snapshot. Thermodynamics is the study of how these snapshot-numbers relate and how they change when you push the piston or heat the gas.

1. Volume $V$ — how much room

The picture: the length of the cylinder times its cross-section — the shaded region in the figure above. That's it. $V$ is the most concrete symbol in the whole subject: you can see it.

Why the topic needs it. $V$ is one thing you can physically clamp (lock the piston) or let float. Which one you clamp decides which potential is useful — so $V$ must be understood first.

2. Pressure $p$ — how hard the gas pushes

The picture: tiny gas molecules bouncing off the piston. Each bounce is a little shove. Add up all the shoves over the piston's area and divide by that area — that's $p$ (the orange arrows).

Why the topic needs it. $p$ is the other thing you can clamp (open beaker = fixed atmospheric pressure). $V$ and $p$ are the two "mechanical" handles.

3. Temperature $T$ — how vigorous the jiggling

The picture: the same bouncing molecules as before, but now the length of their motion arrows is what $T$ tracks — long arrows = hot, short arrows = cold.

Why the topic needs it. $T$ is the handle a thermostat controls. Most real experiments hold $T$ fixed (a beaker sitting in a room), which is exactly why potentials like $F$ and $G$ — built to use $T$ as a natural variable — exist.

4. Internal energy $U$ — the total energy inside

The picture: imagine summing the energy of every single molecule — that grand total is $U$.

Why the topic needs it. $U$ is the root potential. The other three ($H$, $F$, $G$) are all just $U$ with pieces added or subtracted.

5. Heat $\delta Q$ and work $\delta W$ — energy crossing the boundary

The picture: two arrows crossing the box wall in the figure of §0 — a red "heat in" arrow and a blue "work out" arrow.

Why the topic needs it. This is the equation the parent note starts from. Everything — all four potentials — is this law, rewritten.

6. Entropy $S$ — the "spread-out-ness" tax

The picture: a few marbles in a box. If there's only one tidy arrangement (all in a corner), $S$ is low. If they can be anywhere (many arrangements), $S$ is high. The figure shows "few ways" (left, low $S$) versus "many ways" (right, high $S$).

Why the topic needs it. Substituting $\delta Q_{\text{rev}}=TdS$ and $\delta W=pdV$ into the First Law turns the messy $\delta$-law into the clean master equation.

7. Assembling the master equation

Reading it as a picture. $dU$ is the total change in the piggy bank. $+TdS$ = energy deposited by heat. $-pdV$ = energy spent pushing the piston. Every potential in the parent note is built by algebraically rearranging this one line.

8. What a derivative ${\left(\frac{\partial U}{\partial S}\right)}_V$ means

The picture: stand on the energy-landscape $U(S,V)$. Walk one step in the $S$-direction only (never sideways in $V$). The steepness of that step is the partial derivative.

Why the topic needs it. From $dU=TdS-pdV$, matching term-by-term gives $T=(\partial U/\partial S{)}_V$ and $-p=(\partial U/\partial V{)}_S$. These are the "equations of state for free" the parent note promised — and they're just readings of slope.

9. Exact differentials → why mixed partials are equal

10. The prerequisite map

Equipment checklist

Cover the right side and test yourself — you're ready for the parent note when every line is instant.