Section 02
The Idea, Step by Step
Blow up a balloon and squeeze it: it pushes back. Nothing solid is in there — just air. So what is doing the pushing? Picture the inside as a room full of tiny, invisible balls flying around in every direction, bouncing off the walls billions of times a second. Each bounce is a tiny tap. Add up all those taps over the whole wall and you feel a steady push. That push is pressure. Now warm the gas up and the balls fly faster, tap harder, and the balloon swells. That speeding-up is what we call temperature.
To turn this picture into numbers we name three things: the temperature $T$ (in kelvin), the speed of a molecule $v$, and the gas pressure $P$. The single most important rule is that temperature is a direct measure of the average energy of motion of one molecule:
The core idea
$$\langle KE \rangle = \tfrac{3}{2}k_BT$$
Put in room temperature, $T=300$ K, and one molecule carries about $\tfrac{3}{2}(1.38\times10^{-23})(300)\approx 6.2\times10^{-21}$ J — a minuscule amount, but there are roughly $10^{25}$ of them in a breath, so it adds up.
From one molecule to a whole gas
Heavier molecules carry the same energy more slowly, so speed depends on the molar mass $M$. The root-mean-square speed captures this:
Typical molecular speed
$$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$$
For ordinary air ($M=0.029$ kg/mol) at 300 K this gives about $508$ m/s — faster than a passenger jet, right there in the room with you. Molecules don't all share this one speed, though; they spread out into the bell-shaped Maxwell-Boltzmann distribution $f(v)$, with most-probable, average, and rms speeds in the fixed ratio $1:1.13:1.22$. Counting up the wall taps from this swarm reproduces the ideal gas law exactly: $P=Nmv_{\text{rms}}^2/3V=nRT/V$. The sliders map straight onto these symbols — $T$, $M$, $V$ and $n$ each move a variable in those equations.
Try this in the sim above
In Maxwell-Boltzmann mode, drag $T$ from 300 up to 1200 K and watch the curve flatten and slide right — quadrupling $T$ only doubles $v_{\text{rms}}$, because of the square root. Next, hold $T$ fixed and drag the molar mass $M$ from hydrogen-light to xenon-heavy: the peak slides left as heavy molecules lumber along. Finally switch to Diffusion mode and compare a light gas with a heavy one to see Graham's law, rate $\propto 1/\sqrt{M}$, in action.
Section 03
Equations & Derivation
Maxwell-Boltzmann Speed Statistics
$$v_{\text{rms}} = \sqrt{\frac{3RT}{M}},\quad v_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}},\quad v_{\text{mp}} = \sqrt{\frac{2RT}{M}}$$
Average Kinetic Energy
$$\langle KE \rangle = \frac{3}{2}k_BT,\quad k_B = 1.38\times10^{-23}\;\text{J K}^{-1}$$
Pressure from Kinetic Theory
$$P = \frac{Nmv_{\text{rms}}^2}{3V} = \frac{nRT}{V}$$
Maxwell-Boltzmann Distribution
$$f(v) = 4\pi\!\left(\!\frac{M}{2\pi RT}\!\right)^{3/2}\!v^2 \exp\!\left(\!-\frac{Mv^2}{2RT}\!\right)$$
Symbol Definitions
| Symbol | Quantity | SI Unit |
|---|
| $v_{\text{rms}}$ | Root-mean-square speed | m s⁻¹ |
| $v_{\text{avg}}$ | Mean speed | m s⁻¹ |
| $v_{\text{mp}}$ | Most probable speed | m s⁻¹ |
| $k_B$ | Boltzmann constant = 1.38×10⁻²³ | J K⁻¹ |
| $M$ | Molar mass | kg mol⁻¹ |
1
Microscopic pressure. Gas molecules collide with walls, transferring momentum $\Delta p=2mv_x$ per collision. Averaging over all molecules: $P=Nm\langle v_x^2\rangle/V=Nmv_\text{rms}^2/(3V)=nRT/V$ — the ideal gas law derived from Newton's Laws.
2
Three characteristic speeds. Most probable $v_\text{mp}=\sqrt{2RT/M}$ is the peak of the distribution. Mean $v_\text{avg}=\sqrt{8RT/(\pi M)}$ is the average speed. RMS $v_\text{rms}=\sqrt{3RT/M}$ determines kinetic energy. All three $\propto\sqrt{T/M}$.
3
Equipartition theorem. Each translational degree of freedom contributes $\frac{1}{2}k_BT$ to average energy. Monatomic: $U=\frac{3}{2}nRT$. Diatomic (room temp): $U=\frac{5}{2}nRT$ (3 translational + 2 rotational).
4
Graham's Law. Effusion rate $\propto v_\text{rms}\propto 1/\sqrt{M}$. Lighter gases effuse faster. Uranium enrichment exploits the tiny difference between $^{235}$UF₆ and $^{238}$UF₆ diffusion rates.
Ref: Halliday, Resnick & Walker 10th Ed., §19-1 to §19-6; Serway & Jewett 8th Ed., Ch. 21; Reif — Fundamentals of Statistical and Thermal Physics.
Section 05
Common Misconceptions
❌ All molecules in a gas at the same temperature have the same speed.
✅ At any temperature, molecules have a broad Maxwell-Boltzmann distribution of speeds — from near zero to several times $v_\text{rms}$. Temperature determines the average KE, not individual speeds. This distribution explains evaporation, chemical reaction rates, and atmospheric escape.
📖 HRW 10th Ed., §19-5.
❌ Temperature and heat are the same quantity.
✅ Temperature (K) measures average molecular KE per molecule — an intensive property, independent of amount. Heat (J) is energy transferred due to temperature difference — an extensive property. Adding the same heat to a large cold object raises its temperature less than a small hot object.
📖 HRW 10th Ed., §18-3.
❌ The RMS speed equals the average speed.
✅ For any non-uniform distribution, RMS > mean: $v_\text{rms}=\sqrt{\langle v^2\rangle}>\langle v\rangle=v_\text{avg}$. The ratio is $v_\text{rms}/v_\text{avg}=\sqrt{3\pi/8}\approx1.085$. Using the wrong speed formula gives significant errors in pressure calculations.
📖 HRW 10th Ed., §19-5; Serway 8th Ed., §21.3.
Misconception research: Arons — Guide to Introductory Physics Teaching.