# Question #5da9f

##### 1 Answer

We use the normalised Boltzmann distribution in multiple dimensions to find the Maxwell-Boltzmann distribution.

#### Explanation:

The distribution of molecular speeds is based on the Boltzmann weight of translational motions and gives the probability of finding a particle with a certain velocity.

Under thermal equilibrium conditions, the probability

The normalised Boltzmann distribution is stated as following:

This function gives the probability of finding a particle with a certain velocity in a certain direction. Here I chose to find only the particles with velocity

Now we now that particles can move in three dimensions, not just in one. Therefore we need to find the distribution of *speeds* instead of the distribution of *velocity*.

Remember that the speed is the magnitude of the velocity vector.

The speed can, therefore, be calculated as:

Now using that the probability is proportional to the exponential of

Then we combine them and obtain the Maxwell-Boltzmann distribution of molecular speeds is therefore given by:

And as you can see in this equation we write

This equation gives the probability of finding a molecule with mass

If you plot this function you get

With the probability on the vertical axis and the speed of the particle on the horizontal axis.