This presentation is about using R to show what a Maxwell - Boltzmann Distribution is, and how it is used in statistics and physics.

The Maxwell Boltzmann Distribution is a probability distribution used in statistical mechanics and physics to define the distribution of molecular speeds in an ideal gas, and how that speed varies by the increase/decrease in temperature.

The distribution aims to show that at some Temperature, particles (in this case an ideal gas) do move near a particular speed, this speed is often called the most probable speed.

The distribution when graphed, tells us how statistically likely speed a gas molecule moving at a fixed temperature.

The probability density function for a particle speeds is:

\[ f(v) = 4\pi\left(\frac{m}{2\pi kT}\right)^{3/2} v^2 e^{-\frac{mv^2}{2kT}} \begin{aligned} v &= \text{particle speed} \\ m &= \text{particle mass} \\ k &= \text{Boltzmann constant} \\ T &= \text{temperature} \end{aligned} \]

differentiating the probability function with respect to speed, and solving for speed by setting it to equal to zero gives you the following equation. This is the most probable speed that any molecule is the most likely to possess in the system.

\[ v_{mp} = \sqrt{\frac{2kT}{m}} \]

The mean speed

\[ v_{mean} = \sqrt{\frac{8kT}{\pi m}} \]

```r v <- seq(0,6,length=200)

T1 <- 1 T2 <- 2 T3 <- 4

f1 <- v^2 * exp(-v^2/T1) f2 <- v^2 * exp(-v^2/T2) f3 <- v^2 * exp(-v^2/T3)

df <- data.frame( velocity = rep(v,3), probability = c(f1,f2,f3), temperature = factor(rep(c(“T = 1”,“T = 2”,“T = 4”), each=length(v))) )

ggplot(df, aes(x=velocity, y=probability, color=temperature)) + geom_line(linewidth=1.2) + labs( title=“Maxwell–Boltzmann Distribution at Different Temperatures”, x=“Particle Speed”, y=“Probability Density”, color=“Temperature” ) + theme_minimal()