Theory: Maxwell–Boltzmann Speed Distribution
For an ideal gas in thermal equilibrium at temperature \(T\), molecular velocities follow the Maxwell–Boltzmann law.
The calculator plots a probability density and checks normalization numerically.
3D speed distribution (Maxwell)
\[
f(v)=4\pi v^2\left(\frac{m}{2\pi kT}\right)^{3/2}\exp\!\left(-\frac{mv^2}{2kT}\right),
\qquad v\ge 0,
\]
where \(m\) is the mass of one molecule, \(k\) is Boltzmann’s constant, and \(f(v)\,dv\) is the probability that a molecule’s speed lies in \([v,v+dv]\).
The density satisfies \(\int_0^\infty f(v)\,dv=1\).
\[
v_{\rm mp}=\sqrt{\frac{2kT}{m}},\qquad
\langle v\rangle=\sqrt{\frac{8kT}{\pi m}},\qquad
v_{\rm rms}=\sqrt{\frac{3kT}{m}}.
\]
Optional variants (difficulty scaling)
2D speed (Rayleigh)
\[
f_{2D}(v)=\left(\frac{m}{kT}\right)v\,\exp\!\left(-\frac{mv^2}{2kT}\right),\qquad v\ge 0.
\]
1D velocity component (Gaussian)
\[
f(v_x)=\sqrt{\frac{m}{2\pi kT}}\exp\!\left(-\frac{m v_x^2}{2kT}\right),\qquad v_x\in(-\infty,\infty).
\]
These appear in transport theory and component-wise analysis (e.g., \(v_x\) in wall-collision derivations).
How temperature changes the curve
Typical speeds scale like \(\sqrt{T}\), so increasing \(T\) shifts the distribution to the right and broadens it.
Lower \(T\) produces a taller, narrower peak (because the area must remain 1).
Effusion application (website tip)
In effusion through a small hole, faster molecules are more likely to exit. A common model uses a flux-weighted distribution
proportional to \(v\,f(v)\). The calculator optionally reports:
\[
P(v>v_0)=\int_{v_0}^{\infty} f(v)\,dv,\qquad
P_{\rm eff}(v>v_0)=\frac{1}{\langle v\rangle}\int_{v_0}^{\infty} v\,f(v)\,dv.
\]
