Thermodynamics½mv² = (3/2)k_BT

Kinetic Theory Simulator

Watch gas molecules bounce around in a box. Adjust temperature and see how average kinetic energy, pressure, and speed distribution change.

— vₚ— v̄— vᵣₘₛ

Parameters

Temperature T

Molecules N

Mass m

amu

T500 K

vᵣₘₛ0.0 m/s

v̄0.0 m/s

vₚ0.0 m/s

⟨KE⟩0.000 J

P (rel)0.0000

About the Kinetic Theory of Gases

The kinetic theory of gases explains macroscopic properties like pressure and temperature in terms of the microscopic motion of individual molecules. An ideal gas consists of point-like particles moving randomly with no intermolecular forces except during perfectly elastic collisions.

ℹTemperature is a direct measure of the average translational kinetic energy of molecules: ⟨KE⟩ = (3/2) k_B T. Doubling temperature doubles the average kinetic energy.

Key Variables

Symbol	Name	Unit	Description
T	Temperature	K	Absolute temperature of the gas
m	Molecule mass	kg	Mass of a single gas molecule
k_B	Boltzmann constant	J/K	1.381 × 10⁻²³ J/K
v_rms	RMS speed	m/s	√(3k_BT/m) — root-mean-square speed
v_mean	Mean speed	m/s	√(8k_BT/πm) — average speed
v_p	Most probable speed	m/s	√(2k_BT/m) — peak of distribution
P	Pressure	Pa	Force per unit area on container walls
⟨KE⟩	Avg kinetic energy	J	(3/2)k_BT per molecule

Maxwell-Boltzmann Speed Distribution

Not all molecules move at the same speed. The Maxwell-Boltzmann distribution gives the probability of finding a molecule with speed v:

f (v) = 4 π (\frac{m}{2 π k _{B} T})^{3/2} v^{2} exp (- \frac{m v ^{2}}{2 k _{B} T})

This distribution is asymmetric: it starts at zero, rises to a peak at v_p, then falls off with an exponential tail. The three characteristic speeds satisfy v_p < v_mean < v_rms.

Worked Example

Nitrogen gas (N₂, m = 28 × 1.66 × 10⁻²⁷ kg = 4.65 × 10⁻²⁶ kg) at T = 300 K:

v_{rms} = \frac{3 k _{B} T}{m} = \frac{3 \times 1.381 \times 1 0 ^{- 23} \times 300}{4.65 \times 1 0 ^{- 26}} \approx 517 m/s

⟨ K E ⟩ = \frac{3}{2} k_{B} T = \frac{3}{2} \times 1.381 \times 1 0^{- 23} \times 300 \approx 6.21 \times 1 0^{- 21} J

Pressure from Molecular Collisions

Pressure arises from momentum transfer when molecules collide with container walls. For N molecules in volume V:

P = \frac{N m v _{rms}^{2}}{3 V} = \frac{N k _{B} T}{V}

💡In this simulator, the displayed pressure is relative (proportional to momentum transferred per frame), not absolute Pa. Increase temperature or molecule count to see pressure rise.

Kinetic Theory Formulas

Characteristic Speeds

v_{p} = \frac{2 k _{B} T}{m}, v_{mean} = \frac{8 k _{B} T}{π m}, v_{rms} = \frac{3 k _{B} T}{m}

Speed	Formula	Ratio to v_p
v_p (most probable)	√(2k_BT/m)	1.000
v_mean (mean)	√(8k_BT/πm)	1.128
v_rms (root mean square)	√(3k_BT/m)	1.225

Energy and Pressure

⟨ K E ⟩ = \frac{3}{2} k_{B} T, P = \frac{n R T}{V} = \frac{N k _{B} T}{V}

Maxwell-Boltzmann Distribution

f (v) = 4 π (\frac{m}{2 π k _{B} T})^{3/2} v^{2} e^{- m v^{2} / (2 k_{B} T)}

ℹAll characteristic speeds scale as √T: doubling temperature increases all speeds by a factor of √2 ≈ 1.41.

Frequently Asked Questions

Why do molecules in the simulator not all move at the same speed?

Molecules continuously exchange energy through collisions, creating a statistical spread of speeds described by the Maxwell-Boltzmann distribution. Even at a fixed temperature, individual molecular speeds range from near zero to several times the mean speed.

What happens to pressure when temperature doubles?

From P = Nk_BT/V, pressure doubles when temperature doubles (at constant volume and number of molecules). Equivalently, since v_rms ∝ √T, the momentum transferred per collision increases, and so does the collision frequency, both contributing to the pressure increase.

Why is v_rms greater than v_mean?

The root-mean-square operation weights faster molecules more heavily (squaring amplifies large values). Because the Maxwell-Boltzmann distribution has a high-speed tail, v_rms ends up about 8.5% larger than v_mean.

What does molecule mass affect?

Heavier molecules move more slowly at the same temperature. All characteristic speeds scale as 1/√m, so quadrupling the mass halves all speeds. Pressure per molecule also changes, but total pressure in an ideal gas depends only on N, T, and V.

Is this an ideal gas simulation?

Yes. Molecules are treated as point particles with elastic wall collisions and no intermolecular forces. This is the ideal gas approximation, valid for low densities and temperatures well above condensation.