SID: 440617329 ..
Question 1
Kb = 1.38064852e-23; %Kb = 8.6173303e-5 % ev Z = @(E,T) sum(exp(-E./(Kb*T))); P = @(Q,E,T) (1/Z(E,T))*exp(-E(Q)/(Kb*T)); f = 4.8e13; h = 6.62607004e-34; E = (0.5*f*h)*[1:0.1:101];
1a)
T = 300 P(1,E,T) %First state P(2,E,T) %Second state P(3,E,T) %Third state
1b)
T = 700 P(1,E,T) %First state P(2,E,T) %Second state P(3,E,T) %Third state
T1 = 200 T2 = 500 T3 = 800 states = 1:5; Prob1 = []; Prob2 = []; Prob3 = []; for s = states Prob1 = [Prob1, P(s,E,T1)]; Prob2 = [Prob2, P(s,E,T2)]; Prob3 = [Prob3, P(s,E,T3)]; end plot(states, Prob1); hold on; plot(states, Prob2); plot(states, Prob3);
Question 2
2a)
Please see attachment
2b)
Full working in attachment under 2a)
D(v)=(2πkTm)2πve−mv2/2kT ⟹ D(x)dx=2xe−x2dx, where x = uv, u = (m2kT)1/2
%% 2D case maxwell2d = @(x) 2*x.*exp(-x.^2); %% 3D case maxwell3d = @(x) 4/sqrt(pi)*(x.^2).*exp(-x.^2);
figure(1) ezplot(maxwell2d, [0,4]) hold on; ezplot(maxwell3d, [0,4]) title('Maxwell speed distrobution 2D vs 3D') legend('2D', '3D') y = 0:0.1:4; TwoDCharSpeed = 1/sqrt(2); % show v = u/sqrt(2) when x = 1/sqrt(2) ThreeDCharSpeed = 1; % show v = u at x = 1 plot(0*y + TwoDCharSpeed, y, 'b') plot(0*y + ThreeDCharSpeed, y, 'r') quad(maxwell2d, 0, inf)
2c)
We see that the 2D distrobution has a lower most likely speed at v = u/sqrt(2) compared to v = u for the 3D case.
Both have the same most likely velocity vector of 0.
Both distrobution amplitudes increase as we increase x, indicating the larger multiplicites of vectors available at higher speeds.
However, after the peak likely speed, the probability amplitudes begin to fall (with skewed right tails) as the increasing multiplicity of available vector states for a given speed is compeonsated by fact that the probability of obtaining any such invididual vector decays exponentially with increasing speed.
2d)
Please see attachment
2e)
Please see attachment
Question 3
Let x = kT/ep
Noting that 0 is parity even
3a)
% define a vectorized version of the Boltzmann factor J = [0:2:1000]; %even boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); figure(1) ezplot(@(x) aveJ(x),[0,7]); title('Average rotation level vs temperature for parahydrogen') xlabel('kT/ep'); ylabel('ave j'); hold on; figure(2) ezplot(@(x) heatC(x),[0,7]); title('Low-temperature heat capacity for parahydrogen') xlabel('kT/ep'); ylabel('C/k');
3b)
% define a vectorized version of the Boltzmann factor% J = [1:2:1001]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); figure(1) ezplot(@(x) aveJ(x),[0,7]); title('Average rotation level vs temperature for orthohydrogen') xlabel('kT/ep'); ylabel('ave j'); hold on; figure(2) ezplot(@(x) heatC(x),[0,7]); title('Low-temperature heat capacity for orthohydrogen') xlabel('kT/ep'); ylabel('C/k');
3c)
J = [0:2:250]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); J = [1:2:251]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC2 = @(x) sigma_E_sq(x)./(x.^2); figure(1) ezplot(@(x) (1/4)*heatC(x)+(3/4)*heatC2(x),[0,7]); title('Low-temperature heat capacity for normal hydrogen') xlabel('kT/ep'); ylabel('C/k'); ylim([0 1.5]) hold on; x = 0:7; y = 0:1; plot(x, 0*x+1) plot(x, 0*x+0.5) plot(0*y + 1.65, y)
%% Guess the x is around 1.6 Kb = 1.38064852e-23; ep = 0.0076; epj = ep*1.602e-19; T = (1.65*epj)/Kb %Kelvin
3d)
jodd = [1:2:1001]; jeve = [0:2:1000]; J = [jodd, jodd, jodd, jeve]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); figure(1) ezplot(@(x) aveJ(x),[0,7]); title('Average rotation level vs temperature for normal hydrogen - all terms') xlabel('kT/ep'); ylabel('ave j'); hold on; figure(2) ezplot(@(x) heatC(x),[0,7]); title('Low-temperature heat capacity for normal hydrogen - all terms') xlabel('kT/ep'); ylabel('C/k'); ylim([0 2.1])
Just for a nice figure
J = [0:2:1000]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); ezplot(@(x) heatC(x),[0,7]); hold on; J = [1:2:1001]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); ezplot(@(x) heatC(x),[0,7]); J = [0:2:250]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); J = [1:2:251]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC2 = @(x) sigma_E_sq(x)./(x.^2); ezplot(@(x) (1/4)*heatC(x)+(3/4)*heatC2(x),[0,7]); jodd = [1:2:1001]; jeve = [0:2:1000]; J = [jodd, jodd, jodd, jeve]; boltz = @(x) exp( -(J.*(J+1))'./x ); % matrix / vector division Z = @(x) (2*J+1)*boltz(x); % partition function p = @(x) boltz(x)./Z(x); % probability of a *fixed* J, matrix / vector division aveJ = @(x) (J.*(2*J+1))*p(x); % average J, including degeneracy term aveE = @(x) (J.*(J+1).*(2*J+1))*p(x); aveE_sq = @(x) (((J.*(J+1)).^2).*(2*J+1))*p(x); sigma_E_sq = @(x) aveE_sq(x)-aveE(x).^2; heatC = @(x) sigma_E_sq(x)./(x.^2); ezplot(@(x) heatC(x),[0,7]); title('Low-temperature heat capacity for hydrogen') xlabel('kT/ep'); ylabel('C/k'); legend('Parahydrogen', 'Orthohydrogen', 'Normal hydrogen', 'Equilibrium') ylim([0 2.1])
Question 4
Please see attachment