The Maxwell-Boltzmann Distribution

Probability in Physics/Chemistry!

The Maxwell-Boltzmann (MB) distribution describes how particles change in velocity, with respect to temperature. It illustrates that as the temperature increases, a larger proportion of the particles will travel at higher speeds. Let’s observe the following MB probability distribution:

graph

On the horizontal axis, we have speed, x, and on the vertical axis, we have a probability density function, P. In this particular graph, the blue distribution represents the lowest of the temperatures, the green distribution represents the highest of the temperatures, and the red distribution represents an intermediate temperature. We see that as the temperature increases, the mean speed for the particles also increases. One way this can be understood is through observing the shape of each of the distributions. If you notice, all of these distributions are skewed towards particles with higher speeds. To elaborate, if you examine the proportion of particles within different speed ranges in a given temperature, you will see that there will be a much smaller proportion of particles travelling at relatively high speeds, compared to the proportion of particles travelling at relatively low speeds. This indicates that these distributions are not symmetric - therefore these distributions are not normal.

“But, how do you actually know that as you increase the temperature, the mean speed of the particles also increases?” - you may ask. To answer this question, we must first analyze the skewed distribution. Let’s take a look at the following distribution, which is skewed towards higher x-values, as seen in the MB distribution:

formula

We see that the mode (the most occurring value) occurs “x-location” of the peak, the mean is situated closest to the skewed area, and the median is located in between the mode and the mean. Coming back to the MB distribution, as the temperature increases, the peak of the distribution increases. So, through logical reasoning, the mean, median, and mode of the speed also increase.

Now for the physics/chemistry description of the MB distribution. When you have a gas at a low temperature, there is a relatively high proportion of particles travelling at a slow speed, than at a higher speed. According to kinetic molecular theory, when you increase the temperature of a gas, the average kinetic energy of the particles also increases. So, there is now a higher proportion of particles travelling at faster speeds, compared to the lower temperature scenario. Going back to the shape of the probability distributions, we initially see distributions with high skew, but as the temperature increases, the distributions appear to be more symmetric. This means that the vast majority of particles are travelling at higher speeds.