Maxwell-Boltzmann speed distribution

Gas molecules are constantly moving and colliding, each with the same mass and different speed and has a different kinetic energy. The statistical distribution of the velocities of the random motion of a gas particle is very well described by the 'Maxwell-Boltzmann distribution. Probability density (notes under the line) the size distribution of the ideal gas molecule has the form:


 * $$f(v) = \sqrt{\left(\frac{m}{2 \pi kT}\right)^3}\, 4\pi v^2 \mathrm{e}^{\left(- \frac{mv^2}{2kT}\right)}$$ m is molecular mass, k is Boltzmann constant (1,38.10-23 J.K-1) and T absolute temperature.

The most important characteristic is the root mean square value (because it is used to express the mean kinetic energy of the molecules):


 * $$ v_{k} = \sqrt{\frac{3kT}{m}} $$

An important parameter is also the maximum, i.e. the most likely speed (in the language of statistics, itlly the mode):


 * $$ v_{p} = \sqrt{\frac{2kT}{m}} $$

It is easily found by deriving the density.

The Maxwell-Boltzmann distribution is not symmetric, but positively skewed, i.e. it is not at the same time its mean value. The physical interpretation is that it is the speed with whthe most particles in the system move, ' but not  the average speed of individual particles. This can be calculated by integration: (it is solved by the substitution u=v2 and further by the per partes method)


 * $$\bar v = \int_{0}^{\infty}v f(v) dv=  \int_{0}^{\infty} \sqrt{\left(\frac{m}{2 \pi kT}\right)^3}\, 4\pi v^3 \mathrm{e}^{\left(- \frac{mv^2}{2kT}\right)} dv =\sqrt{\frac{8kT}{\pi m}}$$

The behavior of gases, which can be described using the Maxwell-Boltzmann distribution, depends on temperature. The higher the temperature, the more the maximum speed is shifted towards higher values ​​and the curve itself is flatter.

Notes under the line

 * 1) ↑ As a reminder or explanation, a probability density is a function that describes probabilistic behavior. For example, if we are interested in the probability that the velocity of a randomly selected particle lies in the interval v1 to v2, then the solution is the integral of the probability density with integration limits from v1 to v2.