Topic 2

Thermodynamics

The study of heat, work, and the inexorable flow of energy from concentrated to dispersed forms. Thermodynamics governs engines, stars, living cells, and the fate of the universe itself.

  1. Temperature & thermal equilibrium
  2. The ideal gas law
  3. Kinetic theory of gases
  4. The four laws of thermodynamics
  5. Entropy & the arrow of time
  6. Heat engines & efficiency

1. Temperature & Thermal Equilibrium

Temperature is a measure of how hot or cold something is, but more precisely it measures the average kinetic energy of the particles in a substance. When two objects touch, heat โ€” energy transfer due to temperature difference โ€” flows from the hotter to the colder until they reach the same temperature, thermal equilibrium.

$$T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15$$
Thermal equilibrium โ€” Two systems are in thermal equilibrium when, when placed in thermal contact, they no longer exchange heat. The zeroth law (transitivity of equilibrium) is what makes temperature a meaningful, measurable quantity.

2. The Ideal Gas Law

Gases are collections of many particles bouncing around. In the ideal limit (no volume, no interactions), pressure, volume, and temperature are linked:

$$\boxed{PV = nRT = Nk_B T}$$

where $n$ is moles, $R = 8.314\,\text{J/(molยทK)}$ is the gas constant, $N$ is the number of molecules, and k_B is Boltzmann's constant. The combined gas law $PV/T = \text{const}$ is useful for comparing states.

An ideal gas in a box. Increase temperature to speed up particles (more collisions = higher pressure); increase particle count to see density effects. Walls are elastic barriers.

3. Kinetic Theory of Gases

The ideal gas law becomes intuitive through kinetic theory. Pressure comes from particles slamming into the walls. Every translational degree of freedom carries average energy $\tfrac{1}{2}k_BT$:

$$\tfrac{1}{2}m\langle v^2\rangle = \tfrac{3}{2}k_B T,\qquad \langle v\rangle = \sqrt{\frac{8k_BT}{\pi m}},\qquad v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}}$$

The root-mean-square speed tells us how fast molecules move on average. At room temperature, an oxygen molecule ($m \approx 5\times10^{-26}$ kg) has $v_{\text{rms}} \approx 480$ m/s โ€” faster than the speed of sound!

Example. A helium atom ($m = 6.6\times10^{-27}$ kg) at $T=300$ K. Find $v_{\text{rms}}$.
$v_{\text{rms}} = \sqrt{3k_BT/m} = \sqrt{3(1.38\times10^{-23})(300)/(6.6\times10^{-27})} \approx 1360$ m/s.
Helium is the second-lightest element, so its atoms zip around nearly 3ร— faster than oxygen's.

4. The Four Laws of Thermodynamics

Zeroth Law โ€” If A is in thermal equilibrium with B, and B with C, then A is in equilibrium with C. (This is what makes thermometers work.)
First Law (Energy Conservation) โ€” Energy is conserved. Heat $Q$ added to a system minus work $W$ done by the system equals the change in internal energy: $\Delta U = Q - W$. For an ideal gas, $U = \tfrac{3}{2}Nk_BT$ (depends only on temperature).
$$\boxed{\Delta U = Q - W}$$
Second Law โ€” The total entropy of an isolated system never decreases; heat never flows spontaneously from cold to hot. Equivalently: no engine can convert heat completely into work without other effect.
Why the Second Law is special. The first law is time-symmetric โ€” you can reverse a movie of a billiard ball bouncing and the physics still looks fine. But reverse a movie of an egg splattering and the yolk spreading, and everyone sees the difference. The Second Law introduces an irreversibility โ€” the famous "arrow of time" โ€” that the underlying microscopic laws (which are themselves time-reversible) somehow produce in bulk.
Third Law โ€” As $T \to 0$ K, the entropy of a perfect crystal approaches zero. (You can get close to absolute zero, but never reach it.)
LawStatementKey implication
ZerothTransitivity of equilibriumTemperature is well-defined
First$\Delta U = Q - W$No free energy; perpetual motion impossible
Second$\Delta S \ge 0$ (isolated)Processes are irreversible
Third$S \to 0$ as $T \to 0$Absolute zero unreachable

5. Entropy & the Arrow of Time

Entropy $S$ counts the number of microscopic states (the multiplicity $\Omega$) that correspond to a macrostate:

$$S = k_B \ln \Omega$$

This is Boltzmann's tombstone equation. It connects the macroscopic, everyday notion of "disorder" to the microscopic reality of atoms jiggling. Consider a box of gas initially confined to one side; when the partition opens, the gas spreads to fill the whole box. The number of accessible positions (and hence $\Omega$) jumps enormously โ€” entropy increases, even though every molecule obeys reversible Newton's laws.

Before: low entropy particles crowded opens After: high entropy particles spread out
Entropy increases when a gas spreads from a confined region to fill its container โ€” the same thing that makes a hot cup of coffee cool down.

6. Heat Engines & Efficiency

A heat engine absorbs heat $Q_H$ from a hot reservoir, converts some into work $W$, and dumps waste $Q_C$ into a cold reservoir. The efficiency $\eta$ is how much of that heat becomes useful work.

$$\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$$
$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

The Carnot efficiency is the theoretical maximum for any engine operating between two temperatures (in Kelvin). Real engines fall far below it โ€” a car's internal combustion engine is typically ~25% efficient, limited by material temperatures and friction.

Refrigerators reverse the flow. A refrigerator does work to move heat from cold to hot, which is the opposite of natural flow. That's why your fridge's back coils feel warm โ€” it's dumping the extracted heat (plus the work done by the compressor) into your kitchen. The coefficient of performance $\text{COP} = Q_C / W$ measures how much cooling you get per unit of electrical work.

Key Takeaways

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