Theoretical Physics VI - Thermodynamics Sheet 9

<< Graphics`Graphics`

Problem 9.1

b)

Ta = 0.5; Tb = 1; cM = 0.06; M0 = 6.5;

DisplayTogether[Plot[Ta M, {M, 0, 10}], Plot[Tb M, {M, 0, 10}], 
        Plot[Tb M, {M, 0, 10}], 
        Plot[Tb M Exp[cM (M^2 - M0^2)/2], {M, 0, 10}], 
        Plot[Ta M Exp[cM (M^2 - M0^2)/2], {M, 0, 10}], 
        ImageSize -> 550, AxesLabel -> {"M", "H"}];

Problem 9.2

a)

mol = 6.022 10^23; a = 55.729 10^(-10)/mol; 
    b = 3.1 10^(-8); R = 8.314472; Tc = 8 a/ (27 b R); 
    T1 = Tc; T2 = 0.9 10^-26; T3 = 1.5 10^-26;

VdW = 2  a/v^3 - R T/(v - b)^2;

pcritVdW = a/v^2 - 2 a b/v^3;

p1VdW = R T1/(v - b) - a/v^2; p2VdW = R T2/(v - b) - a/v^2; 
    p3VdW = R T3/(v - b) - a/v^2;

DisplayTogether[Plot[p1VdW, {v, b + 0.3  b, 10^1 b}], 
        Plot[p2VdW, {v, b + 0.3  b, 10^1 b}], 
        Plot[p3VdW, {v, b + 0.3  b, 10^1 b}], 
        Plot[pcritVdW, {v, 2 b, 10^1 b}, PlotStyle -> Hue[1]], 
        ImageSize -> 550, AxesLabel -> {"V", "p"}, 
        AxesOrigin -> {0, 0}];

DisplayTogether[Plot[p1VdW, {v, b + 0.3  b, 10^2 b}], 
        Plot[p2VdW, {v, b + 0.3  b, 10^2 b}], 
        Plot[p3VdW, {v, b + 0.3  b, 10^2 b}], 
        Plot[pcritVdW, {v, 2 b, 10^2 b}, PlotStyle -> Hue[1]], 
        ImageSize -> 550, AxesLabel -> {"V", "p"}, 
        AxesOrigin -> {0, 0}];

d)

Reduce[(p2 + a2/v2^2) (v2 - b2) == R T2, v2]

Plot[p2VdW, {v, b + 0.3  b, 10^2 b}, ImageSize -> 550, 
        AxesLabel -> {"V", "p"}, 
        AxesOrigin -> {0, 0}];

f)

c0 = 0 ; TsmallerTc = Tc - 0.18 Tc; 
    pTsmallerTcVdW = R TsmallerTc/(v - b) - a/v^2;

T<T_c

DisplayTogether[
        Plot[c0 - R TsmallerTc Log[v - b] -  a/v, {v, 
            b + 10^(-15) b, 7 b}, PlotStyle -> Hue[1]], 
        ImageSize -> 550, 
        AxesLabel -> {"V", "p"}];

c0 = 0 ; TsmallerTc = Tc - 0.99 Tc;

T<<T_c

DisplayTogether[
        Plot[c0 - R TsmallerTc Log[v - b] -  a/v, {v, 
            b + 10^(-15) b, 7 b}, PlotStyle -> Hue[1]], 
        ImageSize -> 550, 
        AxesLabel -> {"V", "p"}];