Question

From the expressions we derived for the kinetic theory of gases, it can be shown that the fraction of gas molecules having speeds in excess of a particular speed c is

f(c) = 1 + [2*alpha/sqrt(pi)] exp(-alpha²) - erf(alpha)

where alpha = c/c_m , c_m is the most probable speed and erf() is the error function. Consider now 1000 H₂ atoms in thermal equilibrium at 0^oC. Consider the range of speeds from 0 to 3,000 m/s divided into 100 m/s intervals.

(a) Usint the equation above, plot the number of molecules in each increment

(b) Also, on your plot, show c_m, c bar (average speed), and c_rms (rms speed)

Answer 1

I used matlab to solve this problem:

Matlab Code:

clc;
clear;
format long     %for more accuracy
R=8314;   %universal gas constant    
M=1;      %molecular mass of hydrogen atom
c=0:100:3000;     %velocity of atoms 0<c<3000
T=273;    %temperature (K)
cm=sqrt(2*R*T/M)   %most probable velocity
c_bar=sqrt(8*R*T/(pi*M)) %average velocity
c_rms=sqrt(3*R*T/M) %rms velocity
alpha=c./cm;     %veloicty/most probable velocity
N=1000;   %number of atoms
t=length(c); %number of velocity ranges
old_fr=0; %old fraction
for i=t:-1:1   %loop is goinf from c=3000m/s to c=0m/s
    fr = 1+(2*alpha(i)/sqrt(pi))*exp(-alpha(i)^2)-erf(alpha(i)); %finding fraction
    new_fr=fr-old_fr; %actual fraction of atoms in given range= calculated fraction of atoms - fraction of atom in higher velocuty range
    num_atom(i)=new_fr*N; %number of atom= fraction*total number of atoms
    old_fr=fr;  
end
%plotting
plot (c,num_atom,'-*','color','r');
hold on
plot(cm,0,'o','color','g')
hold on
plot(c_bar,0,'+','color','b')
hold on
plot(c_rms,0,'<','color','y')
xlabel('Velocity of atom(m/s)','FontSize', 16); ylabel('Number of atom','FontSize', 16);
title('Number of atoms in each velocity ranges ','FontSize', 20);
legend('number of atoms','most probable velocity','average velocity','rms velocity');
grid on

PLot obtained:

Please comment if any doubt or errors are coming, i will help you ..