Foundations — Internal energy of ideal gas U = (f - 2)nRT

Before you can trust the parent formula, you must be able to read every letter in it without hesitation. This page builds each symbol from absolute zero, in an order where each one leans only on the ones before it. Nothing is assumed.

0 — The mental picture we keep returning to

Picture a sealed box full of tiny hard balls, zooming in every direction, bouncing off the walls and each other. That is our gas. Everything below is a way of counting or measuring something about this swarm. Keep this picture in your head — every symbol points back to it.

1 — Counting the particles: $N$, $n$, $N_A$

That number is enormous (trillions of trillions), so counting one-by-one is hopeless. Instead we count in bundles called moles.

2 — Measuring the swarm's state: $T$, $P$, $V$

These three are what you could measure from outside the box, without seeing individual molecules.

3 — The universal glue: the ideal gas law and its constants $R$, $k_B$

Now connect the outside measurements ($P,V,T$) to the counts ($N,n$). This link is the Ideal gas law.

Two constants appear. They are the same idea at two counting scales.

4 — The energy we actually want: kinetic energy and $⟨v^2⟩$

Different molecules move at different speeds, so we need an average.

This is why the kinetic-theory result the parent quotes, $PV=\frac{1}{3}Nm⟨v^2⟩,$ puts $⟨v^2⟩$ (not $⟨v⟩$) on the right: it is really a statement about total kinetic energy dressed up as a pressure. See Kinetic theory of gases.

5 — The counting of "ways to move": degrees of freedom $f$

A single point-like atom can move along three independent directions — left/right, forward/back, up/down. That is 3 translational degrees of freedom. A dumbbell-shaped diatomic molecule can also tumble end-over-end about two axes, adding 2 rotational degrees of freedom, giving $f=5$. See Degrees of freedom for the full count.

6 — The half: where $\frac{1}{2}$ comes from

The $\frac{f}{2}$ in the formula is really $f\times \frac{1}{2}$. The $\frac{1}{2}$ is the energy nature puts into each channel per molecule: $\frac{1}{2}{k}_{B}T$. Multiply by $f$ channels → $\frac{f}{2}{k}_{B}T$ per molecule → multiply by $N$ molecules and swap $N{k}_B\to nR$ → $U=\frac{f}{2}nRT$. Every piece is now a symbol you have met.

7 — Symbols that ride along: $U$, $\Delta$, $Q$, $W$

Prerequisite map

Equipment checklist

Cover the right side and test yourself before moving to the derivation page.

Connections

• Ideal gas law — the $PV=nRT$ glue and the $N{k}_B=nR$ bridge
• Kinetic theory of gases — source of $⟨v^2⟩$ and $PV=\frac{1}{3}Nm⟨v^2⟩$
• Degrees of freedom — where $f$ is counted
• Equipartition theorem — the $\frac{1}{2}{k}_{B}T$-per-channel rule
• First law of thermodynamics — $\Delta U=Q-W$
• Molar specific heats Cv and Cp — what these symbols build toward next
• Parent topic →