Foundations — Kinetic molecular theory — derivation of P = (1 - 3)ρv²_rms

Before you can follow the derivation of $P=\frac{1}{3}\rho v_{rms}^2$, every letter in that formula — and every idea hiding behind it — must mean something concrete to you. This page builds each one from nothing, in the order that they depend on each other. Nothing here is assumed; if the parent note used it, we define it here first.

1. A molecule and its mass $m$

Why the topic needs it: heavier balls carry more "oomph" (momentum) when they hit a wall. Every collision's kick scales with $m$, so $m$ must appear in the final pressure.

Do not confuse it with $M$ — that comes later and is a thousand-molecule-bundle mass, not one molecule.

2. Position, and the box: $L$ and $V$

Why the topic needs it: the molecule must travel across the box (distance related to $L$) between hits, and the wall it strikes has area $L^2$. Both facts feed directly into "how often" and "spread over how much area".

Look at the figure: the blue dot is a molecule, the cube edge is $L$, and the shaded yellow face is the wall of area $L^2$ we will keep our eye on throughout.

3. Velocity and its components $v_x,v_y,v_z$

A molecule doesn't just have a speed — it moves in a definite direction. To handle direction, we split the motion into three separate movements: how fast it moves left-right, forward-back, and up-down.

In the figure the true velocity (yellow arrow) is broken into three coloured arrows. Only the red $v_x$ arrow points into the yellow wall — that is the only one that will produce pressure on it.

4. Speed squared and the Pythagoras link

We need one more fact about those three arrows: how they combine back into the full speed.

Why the topic needs it: the derivation measures pressure using $v_x^2$ (one direction), but the physically meaningful quantity is the full speed $v$. This equation is the bridge that later lets us swap $\overline{v_x^2}$ for a piece of $\overline{v^2}$.

Why squared and not the plain speed? Squaring removes the sign — a molecule moving left ($v_x$ negative) contributes the same as one moving right. Pressure doesn't care about direction of travel, only about how energetically things hit, and energy depends on speed squared.

5. Momentum $p$ and its change $\Delta p$

Why the topic needs it: when a molecule hits the wall it comes in with $+m{v}_x$ and leaves with $-m{v}_x$. The change is $\Delta p=(-m{v}_x)-(+m{v}_x)=-2m{v}_x,$ so the wall receives $+2m{v}_x$. That factor of 2 (the ball reverses, it doesn't stop) is the heart of the whole derivation — see Newton's Laws — Impulse and Momentum.

The figure shows the incoming red arrow $+v_x$ and the reflected red arrow $-v_x$. The total swing between them is $2v_x$, not $v_x$ — this is the most-missed step in the topic.

6. Force $f$, and Force = rate of momentum change

Why the topic needs it: one collision is a sudden kick, but if kicks arrive fast enough they blur into a steady force. That's exactly how billions of collisions per second become a smooth force on the wall. See Newton's Laws — Impulse and Momentum.

7. Time between collisions $\Delta t$

Why the topic needs it: dividing "momentum per hit" ($2m{v}_x$) by "time per hit" ($2L/v_x$) gives the force one molecule exerts — and produces the crucial $v_x^2$ (fast molecules hit both harder and more often).

8. Pressure $P$ and area

Why the topic needs it: the whole goal is $P$. Once we have the total force $F$ on the wall of area $L^2$, pressure is just $F/L^2$. This is the quantity that links to Ideal Gas Equation PV = nRT.

9. Counting many molecules: $N$ and the bar $\overline{}$

Why the topic needs it: $F_x=\frac{mN}{L}\overline{v_x^2}$ — the total force is $N$ molecules times the average single-molecule effect.

10. Isotropy: the $\frac{1}{3}$

Why the topic needs it: this is where the famous $\frac{1}{3}$ comes from — pure geometry. We only measured the $x$-wall, but the total speed splits equally among 3 directions, so any one wall feels a third of the motion.

11. $v_{rms}$ — the right kind of "typical speed"

12. Density $\rho$

Why the topic needs it: the compact result $P=\frac{1}{3}\rho v_{rms}^2$ is written using $\rho$ because $\rho$ bundles $N$, $m$, and $V$ into one measurable quantity you can look up or weigh.

13. The molar quantities: $M$, $R$, $T$, $k_B$

These appear at the end of the topic when it connects speed to temperature.

Why the topic needs them: the payoff $v_{rms}=\sqrt{\frac{3RT}{M}}$ lets you compute a speed from just temperature and molar mass — the bridge to Average Kinetic Energy and Temperature and Graham's Law of Diffusion.

Prerequisite map

Read the arrows as "is needed for". Notice how components feed the $\frac{1}{3}$, while momentum and timing feed the force — two streams that meet at $P$.

Equipment checklist

Cover the answers and test yourself — you are ready for the derivation only if every line clicks.

Connections

• Parent: 2.4.05 Kinetic molecular theory — derivation of P = (1 - 3)ρv²_rms (Hinglish)
• Newton's Laws — Impulse and Momentum — where $\Delta p=2m{v}_x$ and force-as-rate come from.
• Ideal Gas Equation PV = nRT — the equation we compare our result against.
• Average Kinetic Energy and Temperature — turns $\overline{v^2}$ into $T$.
• Maxwell-Boltzmann Speed Distribution — the full family of speeds ($v_{mp},\overline{v},v_{rms}$).
• Graham's Law of Diffusion — a direct consequence of $v_{rms}\propto 1/\sqrt{M}$.