Boltzmann Factor Calculator

The Boltzmann factor calculator computes a relative probability of two states of a system at thermal equilibrium.

Boltzmann factor for 0.025 eV0.37171e^(−E/kT) — the relative probability of occupying a state that much higher in energy

RMS molecular speed502.445 m/s√(3RT/M) — 1,808.8 km/h, and 1.465× the speed of sound in air

Mean speed462.911 m/s√(8RT/πM) — slightly below the RMS value

Most probable speed410.244 m/s√(2RT/M) — the peak of the Maxwell–Boltzmann distribution, and the lowest of the three

Ordering of the threemode < mean < RMSalways, in that order — the distribution has a long high-speed tail that pulls the mean and RMS up

Average kinetic energy per molecule6.071e-21 J0.037893 eV — three halves kT, and independent of what the gas is

Average energy per mole3.6561 kJwhich is why monatomic gases all share the same molar heat capacity of 3R/2

Mass of one molecule4.81e-23 g28.9647 atomic mass units

Mean free path65.6732 nm177× the molecular diameter — a molecule travels that many of its own widths between collisions

Collision frequency7.0487 GHzbillions of collisions a second at ordinary pressure

Number density2.503e+25 per m³

Mean free path at 1% of this pressure6.5673 µminversely proportional to pressure — which is how a vacuum system reaches the free-molecular regime

kT at this temperature0.025262 eVabout 1/40 eV at room temperature, which is the figure worth remembering

Energy gap for a factor of 10000.174502 eV

The formula

v_rms = √(3RT/M); λ = kT ÷ (√2 πd²P)

Temperature is molecular kinetic energy

The average kinetic energy of a gas molecule is 3kT/2 and depends on nothing but the temperature — not on which gas it is. That is what temperature is, microscopically. Heavier molecules therefore move more slowly at the same temperature, in inverse proportion to the square root of their mass, which is why hydrogen escapes the atmosphere and nitrogen does not.

The speeds are large: nitrogen at room temperature averages around 500 m/s, faster than sound. Yet a smell crosses a room slowly, because each molecule collides billions of times a second and follows a random walk rather than a straight line. The mean free path at atmospheric pressure is only about 70 nanometres.

Three different averages of the same distribution appear here, and they are always ordered the same way: most probable below mean below RMS. The gap exists because the Maxwell–Boltzmann distribution has a long high-speed tail which drags the higher averages up. Which one to use depends on the question — RMS for energy, mean for collision rates.

kT at room temperature is about 1/40 of an electronvolt. That single number tells you at a glance whether thermal energy can drive a process: chemical bonds at several eV are safe from it, while a 0.025 eV gap is crossed constantly.