Maxwell–Boltzmann Speeds
Compare three characteristic molecular speeds as temperature and molar mass change.
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Maxwell–Boltzmann Distribution
In a classical ideal gas, the speeds of molecules are distributed according to the Maxwell–Boltzmann distribution. This statistical distribution depends on both the macroscopic absolute temperature of the gas and the microscopic molar mass of the particles. Instead of all particles having a single average speed, there is a continuous range of speeds defined by a probability density function.
Worked example: N2 at 300 K
Let's calculate the three characteristic speeds and the density for nitrogen gas (N2, molar mass M = 0.028 kg/mol) at room temperature (T = 300 K), using the ideal gas constant R = 8.314462618 J/(mol K).
- Calculate the most probable speed (vmp): vmp = (2RT / M)1/2 = (2 × 8.314462618 × 300 / 0.028)1/2 = (178167.056)1/2 ≈ 422.098 m/s.
- Calculate the mean speed: vmean = (8RT / (πM))1/2 = (8 × 8.314462618 × 300 / (π × 0.028))1/2 ≈ 476.287 m/s.
- Calculate the root-mean-square (RMS) speed: vrms = (3RT / M)1/2 = (3 × 8.314462618 × 300 / 0.028)1/2 ≈ 516.963 m/s.
- Calculate the density at v = 500 m/s: The exponential term is exp(-Mv2 / (2RT)) = exp(-0.028 × 5002 / (2 × 8.314462618 × 300)) = exp(-1.403). The normalization factor is 4π(M / (2πRT))3/2 ≈ 3.018 × 10-8. Multiplying by v2 gives f(500) ≈ 0.001844 s/m.
Interactive Experiment
Open Maxwell–Boltzmann Speeds to visualize the distribution. The module allows you to change the temperature and molar mass to see how the speed distribution curve and the characteristic speeds change.
- Set the temperature to 300 K and molar mass to 0.028 kg/mol (Nitrogen). Verify that the characteristic speeds match our worked example. The most probable speed is at the peak of the density curve, with the mean and RMS speeds at progressively higher values.
- Increase the temperature to 600 K. The entire distribution broadens and shifts to higher speeds, reflecting the increased kinetic energy, while the peak density decreases because the total area must remain 1.
- Change the molar mass to 0.002 kg/mol (Hydrogen) at 300 K. The lighter molecules move significantly faster on average. The distribution is much wider and the speeds are higher than those of nitrogen at the same temperature.
- Use the speed probe to find the probability density at different speeds. Notice how the density drops significantly at extremely high speeds or approaches zero at 0 m/s.
Model Limitations
The Maxwell-Boltzmann distribution relies on several assumptions. It assumes a classical dilute ideal gas, meaning that the particles do not interact with each other except through perfectly elastic collisions and they do not occupy significant volume.
At very low temperatures or high densities, quantum effects become significant, and the Maxwell-Boltzmann statistics must be replaced by Fermi-Dirac or Bose-Einstein statistics. Additionally, the model is non-relativistic; it allows for speeds that could hypothetically exceed the speed of light if the temperature were high enough, which breaks down for extremely high-energy gases where relativistic corrections are required.
References
For an introduction to kinetic theory and the distribution of molecular speeds, see:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Maxwell–Boltzmann Speeds module.
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