In the 1800s James Clerk Maxwell and Ludwig Boltzmann independently investigated the distribution of velocities of atoms or molecules in an ideal gas at a given temperature.  The distribution of molecular speeds became known as the Maxwell-Boltzmann distribution.  In this lessons we will explore this distribution as a function of temperature and mass of the gaseous particles.       

The Maxwell-Boltzmann distribution is

$f\left(v\right) \= 4\cdot \mathrm{\π}\cdot {\left(\frac{M}{2\cdot \mathrm{\π}\cdot R\cdot T}\right)}^{\frac{3}{2}}\cdot v^2\cdot e^{}\mathrm{xp}\left(-\frac{M\cdot v^2}{2\cdot R\cdot T}\right)$

where v is the speed of the gas particles, M is the molar mass, T is the temperature, and R is the gas constant.  
   
From the distribution we can approximate the probability P(v) of finding a gas particle with a speed v between v-dv/2 and v+dv/2 as

$P\left(v\right) \approx f\left(v\right)\mathrm{dv}$

where dv is a small (differential) interval about the velocity v.

If the distribution is correctly defined, the total probability of finding the particle at some speed must be unity, that is

${\int }_0^{\infty }f\left(v\right)\mathrm{dv}\=1 \.$

Before we begin, we define the ideal gas constant in SI units

We can define the Maxwell-Boltzmann distribution with Maple

Let us check that the total probability of the particle having some speed is unity for a gas with a molar mass of 0.1 kg/mol at a temperature of 273 K

(a) Does the total probability equal the expected value of unity?

We can explore the Maxwell-Boltzmann distribution as a function of the temperature and molar mass of the gas.

Using the slide rules to change temperature T in K and molar mass M in kg/mol (SI units), answer the following questions:

(b) Approximately from the graph, what is the most probably velocity for water H₂O at 275 K?

(c) Approximately from the graph, what is the most probably velocity for methane CH₄ at 550 K?