---
title: "Energy and Matter: university physics revision - Kerfox"
description: "Revise university thermal physics: kinetic theory and the ideal gas, the Boltzmann factor, transport, the laws of thermodynamics, entropy, the thermodynamic potentials and phase changes."
image: "https://www.kerfox.app/og.webp"
url: "https://www.kerfox.app/university/physics/energy-and-matter"
---
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# Energy and Matter

Kinetic theory, the four laws, entropy, phases.

## What the course covers

Thermal physics, attacked from both ends. Kinetic theory builds a gas out of molecules obeying Newton and gets $P = \frac{1}{3}nm\overline{v^2}$, temperature as molecular kinetic energy, and estimates for diffusion, conduction and viscosity from one argument. One exponential, the Boltzmann factor $e^{-E/kT}$, then governs everything from reaction rates to the spread of molecular speeds. Classical thermodynamics starts instead from four measured laws, and its second law yields both an absolute temperature scale and entropy. The potentials $H$, $F$ and $G$ follow, then phase equilibria and the van der Waals gas.

### The ideal gas

1. Two ways to know a gas. Two theories, neither needing the other.
2. Six assumptions. Know which ones are lies.
3. Pressure from bouncing. A steady push, built from hail.
4. Why the average is a square. Zero average, and still 100 kPa.
5. Internal energy is pressure. One fact, stated twice.
6. The equation of state. Three quantities, only two choices.
7. Temperature is kinetic energy. What a thermometer really reads.
8. The ideal gas review. One bounce to 22.4 litres.

### Boltzmann and equipartition

1. Pressure falls with height. What holds the air up.
2. The exponential atmosphere. Density that decays as you climb.
3. The Boltzmann factor. The most reused exponential in physics.
4. Activation energies. Why one spark changes everything.
5. The distribution of speeds. Feynman gets it from the sky.
6. Three different speeds. One spread, three right answers.
7. Energy, not velocity. One exponential for every kind.
8. Equipartition. Half of kT, everywhere it fits.
9. Counting degrees of freedom. Molecules spin and stretch too.
10. Heat capacities, measured. The theory meets the table.
11. Why equipartition fails. Degrees of freedom freeze solid.
12. The first crack in classical physics. Maxwell knew, and said so.
13. Einstein, Debye and the real answer. Two models, one right curve.
14. The Boltzmann review. One exponential, and its limits.

### Collisions and transport

1. The mean free path. How far before you hit something.
2. How far without a collision. Survival, and its one law.
3. The random walk. Flying far, arriving nowhere.
4. Brownian motion. The random walk you can watch.
5. The transport model. Two assumptions, three laws.
6. Diffusion. Molecules carrying themselves.
7. Thermal conduction. Molecules carrying energy.
8. Viscosity. Molecules carrying momentum.
9. Weighing a molecule. Loschmidt counts atoms, 1865.
10. Independent of pressure. Pump the air out, lose nothing.
11. The sharpest prediction. A ratio that should not move.
12. What kinetic theory cannot do. The hole where the forces go.
13. The transport review. One argument, three coefficients.

### The first law

1. Drawing the boundary. Fussy words, for a reason.
2. Joule’s paddle wheel. A brewer settles what heat is.
3. Functions of state. What a system owns, and what it does not.
4. Heat. Not a substance, and never was.
5. The first law. Two routes in, one quantity changed.
6. You can only break even. Why the patents keep being refused.
7. Heat capacity needs a caveat. An incomplete question, until now.
8. Calorimetry. A bucket, a block, a thermometer.
9. Three ways to expand a gas. Same ending, three different accounts.
10. What Joule’s expansion proved. A measurement of nothing, carefully.
11. The first law review. What a system owns.

### Reversibility and work

1. Run the film backwards. The test you already know how to apply.
2. The infinitesimal test. One criterion, and three verdicts.
3. Work as an integral. When the force will not hold still.
4. Work on a gas. The one formula to remember.
5. Where the extra R goes. Lifting the world costs something.
6. Isothermal work. A logarithm, and a lower bound.
7. PV to the gamma. The steeper curve, and why.
8. Newton got the sound wrong. And the error runs backwards.
9. Why mountain air is cold. An assumption, quietly repaid.
10. Indicator diagrams. Where work becomes visible.
11. Every kind of work. All of them are Y dX.
12. The reversibility review. The idealisation that pays.

### The second law

1. Allowed but never seen. Energy is conserved either way.
2. Two statements. Wordy on purpose, both of them.
3. Work into heat is free. The asymmetry is the whole law.
4. The heat engine. Carnot abstracts every engine at once.
5. Efficiency. Miles per gallon, for an engine.
6. The two statements are one. Forbid either and you forbid both.
7. Run it backwards. A fridge and a heat pump are one machine.
8. The ratchet and pawl. The machine that ought to work.
9. The ratchet as an engine. Two temperatures, or nothing.
10. Maxwell’s demon. The same machine, wearing a hat.
11. Reversible laws, irreversible world. Newton runs both ways. The world does not.
12. Carnot’s theorem. No engine beats a reversible one.
13. The Carnot cycle. One example is enough.
14. The second law review. Direction, at last.

### Temperature

1. Temperature needs a law. Why two thermometers should agree.
2. The ideal gas scale. The one family that agrees.
3. Temperature from engines. A scale with no substance in it.
4. Carnot efficiency. The ceiling, as a number at last.
5. Goodbye to T sub I. Two scales that were always one.
6. The temperature review. What a thermometer was measuring.

### Entropy

1. Clausius’ inequality. The second law, as arithmetic.
2. Entropy exists. A vanishing loop, and a new quantity.
3. The arrow of time. The direction the first law could not give.
4. The Third Law. Absolute zero, and why nothing gets there.
5. Calculating entropy changes. Invent whatever path is convenient.
6. Irreversibility manufactures entropy. It is made, not moved.
7. Counting microstates. A probability with 10²⁵ in the exponent.
8. S equals k log omega. The integral becomes a count.
9. Mixing, and a paradox. The same sum, twice, one answer wrong.
10. What entropy is not. Disorder is the wrong word.
11. The entropy review. From an inequality to a count.

### Thermodynamic potentials

1. The central equation. Both laws, on one line.
2. The entropy of an ideal gas. One integration, and it is done.
3. Why we need new functions. Nothing is ever really isolated.
4. Enthalpy. Heat, when the vessel can expand.
5. Joule-Thomson, the fourth expansion. A gas that cools with nothing to push against.
6. Helmholtz free energy. Will it go, at fixed volume and temperature?
7. Gibbs free energy. The same argument, with the lid off.
8. Why “free” energy. The part you can actually take out.
9. Natural variables. Which pair each potential wants.
10. Maxwell relations. Not examinable, and worth seeing once.
11. Diamond is not forever. Graphite wins, and slowness saves the ring.
12. The potentials review. Four functions, four sets of constraints.

### Phases and real gases

1. When two phases coexist. One number decides where the boundary sits.
2. Clausius-Clapeyron. How steep a coexistence line is.
3. Squeezing ice into water. The one substance that gets it backwards.
4. Why skating works. The standard explanation fails its own arithmetic.
5. Boiling on Everest. The same equation, for a liquid and its vapour.
6. The pressure inside a drop. Surface tension squeezes small things hard.
7. Why rain needs dust. Condensation cannot start from nothing.
8. Atoms push and pull. The one thing the ideal gas leaves out.
9. The van der Waals equation. Two corrections, one equation.
10. Mean field. The same equation, from the free energy.
11. The kink and the Maxwell construction. A stretch of curve that cannot happen.
12. The critical point. Where the distinction stops existing.
13. Reduced variables. Every gas on the same curve.
14. Phase diagrams. The whole part on one plot.
15. The phases review. From one equality to the critical point.

## One lesson in full

**Three different speeds.** One spread, three right answers.

Ask how fast air molecules move and three different numbers come back, all correct: 415, 468 and 508 metres per second at the same temperature. Those are the most probable speed, the mean and the rms of the very same air.

### Where the peak sits

Every velocity is a point in velocity space, and the ones of speed $v$ fill a shell of area $4\pi v^2$. So the count at $v$ is the one-component law cubed, times that area: $F(v) = 4\pi v^2 (m/2\pi kT)^{3/2} e^{-mv^2/2kT}$. A rising shell against a falling exponential, which is why the peak is not at zero.

### Three ways to say average

The most probable speed is where the curve is tallest. The mean is the plain average of every speed. The root mean square squares first, then averages, then takes the root, which counts the fast molecules for far more.

The three, written out

Differentiate $4\pi v^2e^{-mv^2/2kT}$ and set it to zero for the peak; integrate $vF(v)$ for the mean; integrate $v^2F(v)$ and take the root for the last. Out come $v_p = \sqrt{2kT/m}$, $\bar v = \sqrt{8kT/\pi m}$ and $v_{\mathrm{rms}} = \sqrt{3kT/m}$. In units of $\sqrt{kT/m}$ those are 1.414, 1.596 and 1.732, so the three stand in the fixed ratio $1 : 1.128 : 1.225$ for every gas at every temperature. All three go as $\sqrt{T/m}$, so the order never changes and a heavier gas is a slower gas. The ratio $\bar v/v_{\mathrm{rms}} = \sqrt{8/3\pi} = 0.921$ is fixed too, so a quoted pair that fails it cannot be one gas at one temperature.

### So they come in order

Squaring rewards the fast tail, so the root mean square is always the largest of the three and the most probable the smallest. Their ratios are fixed at $1 : 1.128 : 1.225$ for every gas at every temperature. For air at 300 K, 415, 468 and 508 m/s.

### Which one to reach for

The physics picks the average, never the gas: pressure took $\overline{v^2}$, wall arrivals take $\bar v$. The power of $v$ decides, and that is already a hint about energy.

## A question to try

Helium sits at 300 K and xenon at 600 K. Which gas has the higher rms molecular speed?

1. Helium, because it is far lighter
2. Xenon, because it is hotter
3. It cannot be decided without the pressures
4. They are equal, since xenon is twice as hot

**The answer, and whyThe answer: Helium, because it is far lighter.**

The rms speed is √(3kT/m). Helium gives √(300/4) and xenon √(600/131), so helium comes out roughly four times faster. This is also why helium leaks out of a balloon overnight and why Earth cannot hold onto it at all.

- Xenon, because it is hotter. This is the trap: temperature fixes the average kinetic energy, not the speed. Xenon is about thirty times heavier than helium, and doubling its temperature nowhere near makes up for that.
- It cannot be decided without the pressures. Pressure does not enter. The rms speed of an ideal gas depends only on its temperature and the mass of its molecules.
- They are equal, since xenon is twice as hot. That would need the masses to be equal too. The rms speed goes as the square root of T/m, and here the mass ratio dominates the temperature ratio completely.

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