Graphing the Maxwell-Boltzmann Energy Distribution curve [closed]

Question

I was trying to graph the Maxwell-Boltzmann kinetic energy distribution curve

$f(E)= \left(\frac{1}{\pi k_BT}\right)^{3/2} 2\pi \cdot E^{1/2} e^{-\frac{E}{k_B T}}$

but I kept on getting a straight line. I got the equation from this wiki and also found it here Kinetic Energy in Maxwell-Boltzmann distribution

It might not be a straight line and just be very very small. Could I be using the wrong constant? I've been using the Boltzmann constant, $k_B = 1,38 \cdot 10^{-23} \rm{\,J\, K^{-1}}$.

I am using this to determine the number of particles with sufficient kinetic energy to undergo a chemical reaction.

What could I be missing?

Answer 1

I think your scales are just too smooshed? 10^(-23) is too small to plot comfortably.

it is common to re-scale/non-dimensionalize your variables when dealing with numbers that are too small or too big.

if you apply change of variable $$x=E/kT$$ and demand that $$f(E)dE=f(x)dx$$

then $dE=kTdx$ and $E=kTx$ so the probability density function becomes:

$$f(E)dE=f(x)dx=(1/\pi)^{3/2}2\pi(1/kT)^{3/2}(kTx)^{1/2}e^{-x}(kT)dx$$ $$f(x)dx=2\pi(1/\pi)^{3/2} x^{1/2}e^{-x}dx$$

this is much nicer to plot: https://www.desmos.com/calculator/vga3av2hil

what your variable is measuring now is multiples of kT (your characteristic unit of energy), effectively removing any large numbers and units from the problem.