Ideal Gas Law and Maxwell-Boltzmann Speed Distribution — 3D Lab Simulation
Gas particles in a piston box: drag temperature, volume and amount to watch PV = nRT and the speed distribution emerge
Theoretical Foundation & Historical Context
In 1834 Clapeyron combined the laws of Boyle, Charles and Avogadro into PV = nRT. In 1860 Maxwell showed that molecular speeds in a gas are not equal but follow a definite probability distribution, which Boltzmann later grounded in statistical mechanics. This simulation lets non-interacting N2 particles collide elastically with the walls and a piston: the pressure measured from the momentum delivered to the walls matches nRT/V, and the speed histogram follows the Maxwell-Boltzmann curve.
Law: Ideal gas law PV = nRT and the Maxwell-Boltzmann distribution | Scientist: Émile Clapeyron, James Clerk Maxwell & Ludwig Boltzmann | Year: 1860
Key Learning Objectives & Competencies
- ✓ Relate pressure, volume and temperature through PV = nRT and predict how each slider changes the others.
- ✓ Explain pressure as the average momentum delivered by particle-wall collisions.
- ✓ Read v_p, mean and rms speed from the Maxwell-Boltzmann curve and see how they scale with the square root of T.
Mathematical Formulations & Differential Equations
P·V = n·R·T = N·k_B·T
R = 8.314 J/(mol·K); with V in litres and T in kelvin, P = nRT/V comes out in kPa
f(v) = 4π·(m / 2πk_B·T)^(3/2) · v² · exp(−m·v² / 2k_B·T)
Area under the curve is 1; the peak shifts right and flattens as T rises
v_p = √(2k_B·T/m), ⟨v⟩ = √(8k_B·T/πm), v_rms = √(3k_B·T/m)
v_p < ⟨v⟩ < v_rms; mean kinetic energy ⟨E⟩ = (3/2)·k_B·T
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| T | Absolute temperature (K) | — | — |
| V | Box volume (L) | — | — |
| n | Amount of gas (mol) | — | — |
| m | Mass of one N2 molecule = 28.0134 u | — | — |
| k_B | Boltzmann constant 1.380649·10⁻²³ J/K | — | — |
Real-World & Industrial Engineering Applications
Designing gas cylinders, pistons and pressure vessels.
Gas-phase reaction rates: only the fast tail of the distribution clears the activation energy.
Planetary atmospheres: light gases such as hydrogen escape when their tail exceeds escape velocity.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
Why does pressure rise with temperature at constant volume?
Molecules move faster, so they hit the walls more often and harder, raising the momentum transferred per unit area and time.
Do the particles collide with each other here?
No. This is an ideal gas: particles do not interact, so the Maxwell-Boltzmann distribution comes from the initial velocities and is preserved by the wall reflections.
Why is the histogram a little noisy?
Only tens to a few hundred particles are drawn, so statistical fluctuations are large; with 10²³ real molecules the curve would be essentially smooth.
Related Simulations & Experiments
Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.