Section 02
The Idea, Step by Step
Start: heat only flows one way
Leave a hot cup of cocoa on the table and it always cools down to room temperature — it never spontaneously gets hotter while the room cools. Drop an ice cube into warm water and the two meet somewhere in the middle. Energy spreads out on its own, and that one-way flow is the whole story of thermodynamics: it tells us which way "downhill" is for heat.
Build: energy is just bookkeeping
To keep track, we name three things. The internal energy $U$ is the total jiggling energy of all the molecules in a gas. Heat $Q$ is energy that flows in because of a temperature difference. Work $W$ is energy the gas spends pushing a piston outward. The First Law simply says energy is never created or destroyed, only moved around:
First Law — energy accounting
$$\Delta U = Q - W$$
Suppose you add $Q = 1000\;\text{J}$ of heat to a gas and it pushes the piston out doing $W = 400\;\text{J}$ of work. Then its internal energy rises by $\Delta U = 1000 - 400 = 600\;\text{J}$, and the gas gets hotter. The pressure, volume and temperature that book-keep this are tied together by the ideal-gas law $PV = nRT$.
Deepen: why only one direction?
The First Law alone would happily allow the cocoa to heat itself back up — energy is conserved either way. The Second Law is what picks a direction. Define entropy $S$, a measure of how many microscopic arrangements $\Omega$ match the state we actually see: $S = k_B \ln \Omega$. An isolated system always drifts toward the overwhelmingly more numerous spread-out arrangements, so $\Delta S_{\text{universe}} \ge 0$. That is exactly why a heat engine running between a hot reservoir $T_H$ and a cold one $T_C$ can never convert all its heat into work — its best possible efficiency is the Carnot limit $\eta = 1 - T_C/T_H$. The sliders map straight onto these symbols: $P_0$ and $V_0$ set the gas state, while $T_H$ and $T_C$ set the two reservoirs that fix the ceiling on efficiency.
Try this in the sim above
Switch to Heat Engine mode and slide $T_C$ up toward $T_H$ — watch $\eta$ collapse toward zero, showing you need a big temperature gap to get useful work. Then drag $T_C$ far below $T_H$ and see the efficiency climb (but never reach 100%). Finally open Entropy mode and watch the particles always spread outward and never re-gather on their own — the arrow of time made visible.
Section 03
Equations & Derivation
1st Law — Energy Conservation
$$\Delta U = Q - W$$
Ideal Gas Law
$$PV = nRT,\quad n = \frac{PV}{RT},\quad R = 8.314\;\text{J mol}^{-1}\text{K}^{-1}$$
2nd Law — Entropy & Carnot Efficiency
$$\Delta S_{\text{universe}} \geq 0,\quad \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$
3rd Law
$$\lim_{T \to 0} S = 0 \quad \text{(perfect crystal)}$$
The Four Laws
1
Zeroth Law. Defines temperature: if A↔C and B↔C thermally, then A↔B. Temperature is transitive.
2
First Law. $\Delta U=Q-W$. Energy is conserved — heat in minus work done by system = change in internal energy. For a closed cycle: $\Delta U=0$ so $Q_{\text{net}}=W_{\text{net}}$.
3
Second Law. Entropy of an isolated system never decreases. No process is possible whose only result is transfer of heat from cold to hot, or complete conversion of heat to work.
4
Carnot Efficiency. Maximum efficiency of any heat engine between $T_H$ and $T_C$: $\eta_C=1-T_C/T_H$. Irreversible engines are always less efficient.
5
Third Law. Entropy approaches a constant (zero for perfect crystal) as $T\to0$. Absolute zero is unreachable in finite steps.
Ref: Halliday, Resnick & Walker 10th Ed., Ch.18–20; Callen — Thermodynamics (2nd Ed.); Fermi — Thermodynamics.
Section 05
Common Misconceptions
❌ Perpetual motion machines are theoretically possible with very low friction.
✅ The 1st Law forbids perpetual motion of the 1st kind (creates energy). The 2nd Law forbids perpetual motion of the 2nd kind (100% heat-to-work). Both are absolute — no amount of engineering can circumvent them.
📖 HRW 10th Ed., §18-1 & §20-1.
❌ Entropy always increases everywhere.
✅ Entropy increases in isolated systems. Open systems (engines, organisms, crystals) can decrease locally by increasing surroundings' entropy more. The total (system + surroundings) always increases or stays constant.
📖 HRW 10th Ed., §20-2.
❌ Temperature measures total energy.
✅ Temperature measures average kinetic energy per molecule, not total energy. Two objects at the same temperature can have very different internal energies (large cold block vs small hot block). Internal energy also includes potential energy of molecular interactions.
📖 Serway & Jewett 8th Ed., §17.3.
Misconception research: Arons — Guide to Introductory Physics Teaching; Chi et al. (1981), Cogn. Sci. 5.