Force between two neutral atoms

Van der Waals equation

Sir James Dewar FRS (1842 – 1923)Heike Kamerlingh Onnes (1853 – 1926)Johannes Diderik van der Waals (1837 – 1923)

Isotherms for a van der waals gas MATLAB code clear all; v = 19:1:200; t=340; b=14; a=78435; for i=1:20 t = t-10; r = 8.314; p = r*t./(v-b)-(a./v.^2); plot(v,p) axis([ ]); xlabel('v'); ylabel('p'); hold on; end Set a = b = 0 and you get the isotherms for an ideal gas a and b were chosen to give a Tc of 200 K

At a particular temperature a vdw gas develops a point of inflexion. The gas can be liquefied below this temperature. To find the point of inflexion set: and

Which gives: Define dimensionless reduced variables

Which gives the reduced form of the vdw equation Can be applied to any gas (to some extent) since its independent of a and b (law of corresponding states)

Isotherms for the reduced form of the van der waals equation of state MATLAB code clear all; vr = 0.5:0.01:10; tr=2; for i=1:15 tr = tr-0.1; pr = (8*tr./(vr.*3-1))-(3./vr.^2); plot(vr,pr) axis([ ]); xlabel('v_r'); ylabel('p_r'); hold on; end

Below the critical temperature Tc, the vdw equation has a few problems Negative compressibility, unphysical Multivaluedness, unphysical

Free energy considerations can remove these problems (1) (2) (1) and (2) From (1)

Plotting the gibbs function using matlab… The system will choose the minimum Gibb’s free energy

B2 Maxwell construction

clc; clear all; close all Vr = linspace(0.5,3,100); figure(1); ylim([0 2]) xlim([0.25 3]) xlabel('V_r') ylabel('P_r') Tr = 0.9; Prfunc 8*Tr./(3*Vr - 1) - 3./(Vr.^2); Pr = Prfunc(Vr); plot(Vr,Pr) Clear command, clears the screen; clears all objects; closes all figures etc. Creates values of Vr from 0.5 to 3 in steps of (3-0.5)/100 Opens a figure window Y axis limits X axis limits Labels x axis Labels y axis Defines the van der Waals equation as a function of Tr and Vr Calculate Pr using the vdW equation Plots Pr vs. Vr

if Tr < 1 Pr_b = 1.0; vdW_Pr_b = [1 -1/3*(1+8*Tr/Pr_b) 3/Pr_b -1/Pr_b];

v = sort(roots(vdW_Pr_b)); A1 = (v(2)-v(1))*Pr_b - integral(Prfunc,v(1),v(2)); A2 = integral(Prfunc,v(2),v(3)) - (v(3)-v(2))*Pr_b; Finds the roots of the polynomial and sorts them in increasing order: v(1), v(2), and v(3) Calculates the area enclosed by the line and the vdW curve between v(1) and v(2) and then from v(2) and v(3). (clearer figure later).

Z = abs(A1-A2); while Z > vdW_Pr_b = [1 -1/3*(1+8*Tr/Pr_b) 3/Pr_b -1/Pr_b]; v = sort(roots(vdW_Pr_b)); Prfunc 8*Tr./(3*Vr - 1) - 3./(Vr.^2); A1 = (v(2)-v(1))*Pr_b - integral(Prfunc,v(1),v(2)); A2 = integral(Prfunc,v(2),v(3)) - (v(3)-v(2))*Pr_b; Z = abs(A1 - A2); Pr_b = Pr_b ; figure(1); hold off; plot(Vr,Pr) figure(1); hold on; plot([0.5 3],[Pr_b Pr_b],'k--') hold off; end Calculates the difference between the two areas If z > then this loop starts Defines the polynomial in the loop Calculates and sorts the roots Defines vdW equation in the loop Calculates the areas again Calculates the difference Lowers the test values Pr_b by Chooses figure 1 window; new plot will appear, old graph is removed Plots Pr vs. Vr Chooses figure 1window; new plot will appear keeping the old graph Plots the line for Pr_b vs. Vr in a black dashed line Next time a new graph is plotted the old plots will be removed Starts the loop again and checks if the difference between A1 and A2 (Z) > for the new value of Pr_b. Loop stops if Z < i.e A1  A2

Pr_b v(1) v(2) v(3) (v(2)-v(1))*Pr_b(v(3)-v(2))*Pr_b

Pr_b v(1) v(2) v(3) integral(Prfunc,v(1),v(2)) integral(Prfunc,v(2),v(3)) Calculates the area under the vdW curve from v(1) to v(2) and from v(2) to v(3).

Pr_b v(1) v(2) v(3) Calculates the relevant area for the Maxwell constructions (v(2)-v(1))*Pr_b - integral(Prfunc,v(1),v(2)) integral(Prfunc,v(2),v(3)) - (v(3)-v(2))*Pr_b

end A1 A2 Pr_b Ends the if loop Prints A1 and A2 Prints the value of pressure Pr_b which satisfies Maxwell’s condition.

P v T Liquefying a gas by applying pressure

P v T

P v T

P v T

P v T

P v T

P v T

Two phase region in the vdW gas

P (bar)

sPTOA