Ideal gas law PV = nRT — derivation from kinetic theory: Foundations

This page assumes nothing. Before you touch the derivation, you need a small toolbox of symbols. We will earn each one — plain meaning, the picture it stands for, and why the topic can't proceed without it.

1 · The box: $L$ , $V$ , and "a wall"

Picture a cube-shaped container of gas. Every side has length $L$ (a length, measured in metres).

Figure — Ideal gas law PV = nRT — derivation from kinetic theory

Why the topic needs it. Two walls face each other a distance $L$ apart. A ball that bounces off one wall must travel across the box and back — a round trip of $2 L$ — before it can hit the same wall again. That distance $2 L$ is what later sets how often collisions happen. And $V = L^{3}$ is exactly the $V$ that appears in $P V = n R T$ , so the box's size is literally one of the four letters in the final law.

2 · Speed, velocity, and its components $v_{x}, v_{y}, v_{z}$

A ball doesn't just have a speed (how fast); it has a direction too. Together, speed + direction is called velocity.

The three components combine into the full speed $v$ by the 3-D Pythagoras rule: $v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$

Why the topic needs it. When a ball hits the right-hand wall, only its $x$ -motion matters — the wall faces the $x$ -direction. The sideways parts $v_{y}, v_{z}$ just skim along the wall and change nothing. So we must be able to isolate $v_{x}$ from the full speed. That splitting is the heart of the derivation.

3 · Mass $m$ and momentum $p = m v$

Why the topic needs it. A wall never feels "speed" — it feels a kick, and a kick is a change in momentum. When a ball hits and bounces back, its $x$ -momentum flips from $+ m v_{x}$ to $- m v_{x}$ , a change of $2 m v_{x}$ . That $2 m v_{x}$ is the size of the shove the wall receives. Momentum is the currency the whole derivation is paid in.

4 · Force $F$ and Newton's second law $F = Δ p /Δ t$

Why the topic needs it. Each bounce delivers momentum $2 m v_{x}$ . If those bounces come every $Δ t$ seconds, dividing the momentum by the time gives the average force on the wall. This is the bridge from "a ball bounced" to "the wall feels a steady push." See Pressure and Newton's Second Law for the deeper version.

5 · Pressure $P = F / A$

Why the topic needs it. $P$ is the very first letter of $P V = n R T$ . Everything above — components, momentum, force — exists to compute this one number.

6 · Averages: $⟨ ⟩$ and why $⟨ v_{x} ⟩ = 0$ but $⟨ v_{x}^{2} ⟩ \neq = 0$

We can't track billions of balls one by one. We track averages, written with angle brackets $⟨ \cdot ⟩$ .

Why the topic needs it. The push depends on how hard and how often balls hit — both grow with $v_{x}$ , so the force per ball scales with $v_{x}^{2}$ . Averaging over all balls turns the messy sum into a clean $N ⟨ v_{x}^{2} ⟩$ .

7 · Isotropy: $⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩$

Combine this with $v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$ , average both sides: $⟨ v^{2} ⟩ = 3 ⟨ v_{x}^{2} ⟩ \Rightarrow ⟨ v_{x}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩$

Why the topic needs it. This is where the famous factor $\frac{1}{3}$ in $P = \frac{1}{3} \frac{N m ⟨ v ^{2} ⟩}{V}$ is born. See Kinetic Theory of Gases for the full swarm picture and Root Mean Square Speed for what $⟨ v^{2} ⟩$ means.

8 · Counting the gas: $N$ , $n$ , $N_{A}$ , and the constants $k_{B}$ , $R$ , $T$

Why the topic needs it. These turn the mechanics result $P V = \frac{1}{3} N m ⟨ v^{2} ⟩$ into the thermodynamic $P V = n R T$ . The link "temperature = average kinetic energy" comes from the Equipartition Theorem and Temperature and Internal Energy; the constants themselves are unpacked in Boltzmann Constant and Gas Constant.

Prerequisite map

Equipment checklist

Hide the right side and test yourself — you are ready for the derivation only when every line is automatic.

What does

V = L^{3}

mean and why is

2 L

special?

V

is the cube's volume;

2 L

is the round-trip distance a ball travels before hitting the same wall again.

Why does only

v_{x}

matter when a ball hits the

x

-wall?

The wall faces the

x

-direction;

v_{y}, v_{z}

slide along it and change nothing.

How do

v_{x}, v_{y}, v_{z}

combine into speed?

v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

(Pythagoras done twice).

What is momentum and how much changes on an elastic bounce?

p = m v

; the

x

-momentum flips

+ m v_{x} \to - m v_{x}

, a change of size

2 m v_{x}

State Newton's second law in momentum form.

F = Δ p /Δ t

— force is the rate momentum changes.

Define pressure and give the wall area for the cube.

P = F / A

; the wall area is

A = L^{2}

Why is

⟨ v_{x} ⟩ = 0

but

⟨ v_{x}^{2} ⟩ \neq = 0

Signed velocities cancel; squares are all positive so they don't.

What does isotropy give you?

⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩

Relate

N

n

N_{A}

N = n N_{A}

, with

N_{A} \approx 6.022 \times 1 0^{23}

per mole.

How are

R

and

k_{B}

related, and why must

T

be in kelvin?

R = N_{A} k_{B}

;

T

measures energy so it must start from absolute zero.