Maxwell Distribution of Speeds
In this simulation of the Maxwell distributioin of speeds for an idea gas, you can vary both the temperature and the molecular mass of the gas and oberseve the changes in the probability density function. script tp_LE_Mawell.mlx
For an ideal gas, the meaning of the temperature measured in kelvin is that it gives the average translational kinetic energy of the gas molecules
Temperature is a direct measure of the molecular translational kinetic energy of a gas. When two ideal gases with molecules of different mass are at the same temperature, then they will have the same average molecular translational kinetic energy.
However, we would not expect all the molecules in a gas to be travelling at the same speed. The way in which the speed of the molecules vary was first proposed by James Clerk Maxwell in 1860. At this time, no experimental evidence was possible because of the nonexistence of proper vacuum equipment. It was not until 1926 that Otto Stern was able to partially confirm Maxwell’s predictions.
Maxwell predicted that the speeds of all the gas molecules would be distributed according to his Maxwell distribution of speeds function. The prediction gives a probability density function which is called the Maxwell-Boltzmann Distriution. It gives the fraction of molecules with speed v in the range 
.
.
where
v = speed of a gas molecule
f(v) dv = fraction of molecules with speeds in the range
T = temperature [K]
k = Boltzmann constant
R = Universal gas constant
M = molecular mass of the gas
m = mass of a gas molecule
We can express the Maxwell-Boltzmann Distribution in terms of a probability density function 
where 
gives the probability of the gas molecules of having a speed in the range .
where gives the probability of the gas molecules of having a speed in the range .
where N is the normalizing constant such that
And the probability to find a gas molecule with speed in the range is
is
The average speed of the molecules can be expressed in a number of different ways;
Most probable speed 
Average speed 
Root-mean-square speed 
Average translational kinetic energy of a molecule 
close all
clc
clear
Input variables: temperature T [K] and molecular mass M [g]
T = 300;
M = 8;
% velocity limits: probability(v1 < v < v2)
v1 = 1000;
v2 = 2000;
Setup and Calculations
R = 8.3145; % Universal Gas constant
k = 1.38e-23; % Boltzmann constant
vMin = 0; % velocity range [m/s]
vMax = 3000;
nV = 1001;
v = linspace(vMin,vMax,nV);
indexv1 = find(v>v1,1);
indexv2 = find(v>v2,1);
indexv = indexv1:indexv2;
if mod(length(indexv),2) == 0; indexv = indexv1:indexv2+1; end
M = M*1e-3; % molecular mass [kg]
% Maxwell-Boltzmann probability density funtion
fn = (M/T)^(1.5) .*v.^2 .* exp(-(M.*v.^2) ./ (2*R*T));
N = simpson1d(fn,vMin,vMax); % normalization function
pD = fn./N; % probability density
% Velocities and average translational kinetic energy
v_P = sqrt(2*R*T/M); index1 = find(v>=v_P,1);
v_avg = sqrt(8*R*T/(pi*M)); index2 = find(v>=v_avg,1);
v_rms = sqrt(3*R*T/M); index3 = find(v>=v_rms,1);
KE_avg = 1.5*k*T;
prob12 = simpson1d(pD(indexv),v1,v2);
Graphics and output variables
figure(1)
Ylim = 4e-3;
xP = v(indexv1:indexv2); yP = pD(indexv1:indexv2);
Harea = area(xP, yP);
set(Harea,'FaceColor',[0.9 0.9 1],'EdgeColor',[0.9 0.9 1])
hold on
xP = v; yP = pD;
plot(xP,yP,'b','linewidth',2)
xlabel('velocity v [ m.s^{-1} ]')
ylabel('Prob. Density p_D [ s.m{-1} ]')
xP = [v_P v_P]; yP = [0 pD(index1)];
plot(xP,yP,'r','linewidth',2)
xP = [v_avg v_avg]; yP = [0 pD(index2)];
plot(xP,yP,'m','linewidth',2)
xP = [v_rms v_rms]; yP = [0 pD(index3)];
plot(xP,yP,'k','linewidth',2)
grid on
set(gca,'fontsize',12)
ylim([0 Ylim])
%legend('p_D','v_P','v_{avg}','v_{rms}')
tm1 = 'T = '; tm2 = num2str(T,'%4.0f '); tm3 = ' K';
tm = [tm1 tm2 tm3];
Htext = text(100,0.95*Ylim,tm);
set(Htext,'fontsize',14,'color','b')
tm1 = 'M = '; tm2 = num2str(M*1000,'%4.0f '); tm3 = ' g';
tm = [tm1 tm2 tm3];
Htext = text(1300,0.95*Ylim,tm);
set(Htext,'fontsize',14,'color','b')
tm1 = 'v_P = '; tm2 = num2str(v_P,'%4.0f '); tm3 = ' m.s^{-1}';
tm = [tm1 tm2 tm3];
Htext = text(100,0.84 *Ylim,tm);
set(Htext,'fontsize',14,'color','r')
tm1 = 'v_{avg} = '; tm2 = num2str(v_avg,'%4.0f '); tm3 = ' m.s^{-1}';
tm = [tm1 tm2 tm3];
Htext = text(1300,0.84 *Ylim,tm);
set(Htext,'fontsize',14,'color','m')
tm1 = 'v_{rms} = '; tm2 = num2str(v_rms,'%4.0f '); tm3 = ' m.s^{-1}';
tm = [tm1 tm2 tm3];
Htext = text(100,0.73 *Ylim,tm);
set(Htext,'fontsize',14,'color','k')
tm1 = 'KE_{avg} = '; tm2 = num2str(KE_avg,'%2.2e '); tm3 = ' J';
tm = [tm1 tm2 tm3];
Htext = text(1300,0.73 *Ylim,tm);
set(Htext,'fontsize',14,'color','m')
tm1 = 'Probability(v_1 < v < v_2) = '; tm2 = num2str(prob12,'%2.2f '); tm3 = ' ';
tm = [tm1 tm2 tm3];
Htext = text(1100,0.62 *Ylim,tm);
set(Htext,'fontsize',14,'color','b')
DOING PHYSICS WITH MATLAB
MATLAB DOWNLOAD DIRECTORY
Need to download the function simpson1d.m as well as the script tp_LEMaxwell.mlx