Why does the pressure jump up and down with one molecule but stay steady with 400?

Pressure is just the total push on the wall per second. One molecule pushes only at the instant it hits, roughly once a second, so the 0.5 s average is zero between hits and a spike at each hit. Hundreds of molecules hit the piston hundreds of times a second, and the kicks blur into a steady force — that is what a pressure gauge on a bicycle tyre feels.

Why does doubling the temperature double the pressure, not multiply it by four?

Doubling T doubles the kinetic energy, so speeds grow by √2. Each hit then delivers √2 times more impulse, and the molecule crosses the box √2 times faster so it hits √2 times more often. The two factors together give √2 × √2 = 2.

Why does pushing the piston in raise the pressure?

With the box half as wide, each molecule reaches the piston twice as often, so the number of hits per second doubles while the impulse of each hit is unchanged. Halving V doubles P at constant N and T — Boyle's law, which the readout PV/(NT) ≈ 1 confirms.

Why does changing the mass not change the pressure?

At the same temperature a heavy molecule moves more slowly: m is 4× larger, so v is 2× smaller. The impulse per hit 2mv is doubled, but the hit rate is halved, and the product is the same. Pressure depends only on N, T and V — that is why air and helium at the same N, T, V have the same pressure.

Gas Pressure: Molecules Hitting the Walls

Start with 1 molecule and watch each wall hit; add molecules, move the piston, heat up — PV/(NT) stays constant.

Gas

Number of molecules N1

Temperature T300 K

Box width (piston position)1 × max

Molecule mass

View

Slow motion (×0.15)Show impulse flashes on the walls

Start with one molecule: every flash on the piston is one hit, and the gauge jumps with each hit. Add molecules and the hits blur into a steady pressure. Pressure averages reset whenever you change N, T, the piston or the mass. Slow motion does not change the measured pressure — it is computed in simulation time.

Горячие клавиши

Space / Enter — play / pause
R — reset (new random molecules)
Drag the piston on the canvas to change the volume

Измеренные величины

Hits on piston per second0.01/s

Pressure on piston (0.5 s window)0.00rel.

Pressure on piston (average since reset)0.00rel.

Pressure, all walls (average)0.00rel.

Kinetic theory P = NkT/V1.00rel.

PV/(N·T) (normalised, should stay ≈ 1)—

T300K

Volume (relative)1.00× max

Mean speed0.00box/s

v_rms = √(2kT/m)2.40box/s

Averaging time0.0s

О модели

Up to 400 hard-sphere molecules move in a two-dimensional box whose right wall is a piston that can be dragged or set with the slider (0.4–1.0 of the full width). Molecules bounce elastically off the walls and off each other, so the total kinetic energy is conserved; temperature sets the speed scale through ½m⟨v²⟩ = kT (the two-dimensional form), and switching from light to heavy molecules (4×) halves the speeds at the same T. Every wall hit transfers an impulse 2m|v⊥| to that wall and is drawn as a flash. Pressure on the piston is the sum of those impulses over the last 0.5 s divided by the time window and the wall length — with one molecule it is a sequence of isolated kicks, with hundreds of molecules a steady value. Pressure is reported in relative units where 1 means one molecule at 300 K in the full box, so kinetic theory predicts P = N·(T/300 K)/V_rel, and the readout PV/(N·T) should stay near 1 whatever you change. The averages restart whenever N, T, the piston or the mass changes. A lone molecule never thermalises, so it is launched at about 45° to make ⟨v_x²⟩ = ⟨v_y²⟩; with two or more, collisions randomise the directions. The model omits gravity, attraction between molecules, rotation and vibration, and a moving piston does no work on the gas (its position changes instantly, with speeds unchanged). Because the spheres have a finite size, the measured PV/(NT) sits a few per cent above 1 at 400 molecules — a first hint of the van der Waals excluded volume. For the macroscopic gas-law view see the existing thermodynamics/ideal-gas and thermodynamics/gas-laws simulators.

Для кого: Grades 7–9 physics and chemistry, first meeting the particle model of gases and the idea that pressure is caused by molecular collisions.

Ключевые понятия

Gas pressure
Kinetic theory
Impulse 2m|v⊥|
Elastic collision
Piston / volume
Temperature and molecular speed
Ideal gas law PV = NkT

Как это работает

Hard-sphere molecules fly around a 2D box and bounce elastically off the walls. Each time a molecule hits a wall it gives the wall a push (impulse 2m|v⊥|); pressure is the total push per second divided by the wall length. With one molecule the pressure is a series of isolated kicks; with hundreds of molecules the kicks merge into a steady value equal to NkT/V. Doubling N, halving the volume with the piston, or doubling T each double the pressure, so PV/(NT) stays constant — the ideal-gas law emerging from collisions.

Основные формулы

one hit on a wall: Δp = 2 m |v⊥|

P = (Σ Δp over time τ) / (τ · wall length)

½ m ⟨v²⟩ = k T (2D) → P V = N k T

Часто задаваемые вопросы

Why does the pressure jump up and down with one molecule but stay steady with 400?: Pressure is just the total push on the wall per second. One molecule pushes only at the instant it hits, roughly once a second, so the 0.5 s average is zero between hits and a spike at each hit. Hundreds of molecules hit the piston hundreds of times a second, and the kicks blur into a steady force — that is what a pressure gauge on a bicycle tyre feels.
Why does doubling the temperature double the pressure, not multiply it by four?: Doubling T doubles the kinetic energy, so speeds grow by √2. Each hit then delivers √2 times more impulse, and the molecule crosses the box √2 times faster so it hits √2 times more often. The two factors together give √2 × √2 = 2.
Why does pushing the piston in raise the pressure?: With the box half as wide, each molecule reaches the piston twice as often, so the number of hits per second doubles while the impulse of each hit is unchanged. Halving V doubles P at constant N and T — Boyle's law, which the readout PV/(NT) ≈ 1 confirms.
Why does changing the mass not change the pressure?: At the same temperature a heavy molecule moves more slowly: m is 4× larger, so v is 2× smaller. The impulse per hit 2mv is doubled, but the hit rate is halved, and the product is the same. Pressure depends only on N, T and V — that is why air and helium at the same N, T, V have the same pressure.

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