restart;vawgas.mwsSet up classical phase function for a VW gas. Use n as the function to make Phase function homogeneous degree 1.use new coefficient c for interaction length. Format VWGAS interms of c and values at critical point.Draw Critical Isothermwith(linalg):setup(U,S,V,n);Theta:=evalm(n*(exp(S/(n*Cv))*(V/n-b)^(-R/Cv)-a/(V/n+c*b)-U/n));HH:=factor(innerprod([n,S,U,V],grad(Theta,[n,S,U,V])));HOMOGDEG1:=factor(HH-Theta);T:=diff(Theta,S);P:=diff(Theta,V);UU:=diff(Theta,U);muu:=diff(Theta,n);factor((n*muu+S*T+P*V+U*UU)-Theta);Theta is homogeneous degree 1Thetan:=evalm(n*(exp(s/(Cv))*(v-b)^(-R/Cv)-a/(v+c*b)-u));Tn:=diff(Thetan,s);T;Pn:=diff(Thetan,v);mu:=diff(Thetan,n)+diff(Thetan,s)*(-s/n)+diff(Thetan,v)*(-v/n)+diff(Thetan,u)*(-u/n);Un:=diff(Thetan,u);factor((s*Tn+v*Pn+u*Un+n*mu)-Thetan);HOMOGENEOUS OF DEGREE 1 when all terms are taken into account.Tnn:=Tn;T0:=T;Pnn:=Pn;P0:=P;UU:=simplify((solve(Theta=0,U)));PTHETA:=diff(Theta,V);P:=-((diff(UU,V)));factor(-exp(S/n/Cv)*(-(-V+b*n)/n)^(-R/Cv)*V-exp(S/n/Cv)*(-(-V+b*n)/n)^(-R/Cv)*n*c*b+a*n);factor(P+PTHETA);Ta:=simplify(diff(UU,S));factor(T-diff(Theta,s));mu:=simplify((diff(Theta,n)));factor(-u-P*v+T*s+n*mu-Theta);restart;with(plots):Theta:=evalm(n*(exp(S/(n*Cv))*(V/n-b)^(-R/Cv)-a/(V/n+c*b)-U/n));PP0:=-diff(Theta,V);TT0:=diff(Theta,S);PP:=R*T/(V/n-b)-a/(V/n+c*b)^2;dPP:=(diff(PP,V));ddPP:=((diff(dPP,V)));TC:=factor(solve(dPP,T));ZZ:=subs(T=TC,ddPP);VCC:=(solve(ZZ,V));TCC:=factor(subs(V=VCC,TC));PCC:=factor(subs(T=TCC,V=VCC,PP));RATIO:=factor(n*R*TCC/(VCC*PCC));subs(c=3/2,RATIO);PH:=subs(T=TTH*TCC,V=VVH*VCC,PP)-(PPH*PCC);factor(simplify(subs(c=c,VVH=1/rho,PH)));Press:=collect((72*TTH*rho+27*rho^3-81*rho^2+9*PPH*rho+6*PPH*rho^2*c+32*TTH*rho*c^3+128*TTH*rho*c^2+32*TTH*rho^2*c^3+80*TTH*rho^2*c^2+8*TTH*rho^3*c^3+48*TTH*rho^2*c+8*TTH*rho^3*c^2+168*TTH*rho*c-8*PPH*c^3*rho-20*PPH*c^2*rho-2*PPH*c^3*rho^2-6*PPH*c*rho+PPH*c^2*rho^2+PPH*rho^3*c^2-189*c^2*rho^2-54*c^3*rho^2+27*rho^3*c^2-216*rho^2*c+54*rho^3*c-36*PPH*c^2-54*PPH*c-8*PPH*c^3-27*PPH),rho);factor(-54*PPH*c-27*PPH-36*PPH*c^2-8*PPH*c^3);PPress:=subs(c=c,Press/(2*c+3)^3);r3:=collect(factor(coeff(PPress,rho^3)),c);r2:=collect(factor(coeff(PPress,rho^2)),c);r1:=collect(factor(coeff(PPress,rho^1)),c);CC:=solve(r3=1,c);factor(subs(c=CC[3],r3));PPress:=subs(c=c,Press/(2*c+3)^3);r3:=collect(factor(coeff(PPress,rho^3)),c);r2:=collect(factor(coeff(PPress,rho^2)),c);r1:=collect(factor(coeff(PPress,rho^1)),c);CC:=solve(r3=1,c);factor(subs(c=CC[3],r3));factor(subs(c=CC[3],r2));factor(subs(c=CC[3],r1));r0:=factor(PPress-r3*rho^3-r2*rho^2-r1*rho);pppp:=subs(c=c,(r3*rho^3+r2*rho^2+r1*rho+r0));VW:=subs(c=0,pppp);subs(c=1/2,pppp);subs(c=1,pppp);subs(c=3/2,pppp);subs(c=-1,pppp);subs(TTH=1,PPH=1,pppp);factor(8*c^3+36*c^2+54*c+27);factor(-12*c^2-36*c-27);solve(VW,PPH);factor(diff(pppp,PPH));factor(diff(pppp,TTH));Q:=factor(collect(factor(subs(c=-1,rho=y,pppp)),y));subs(c=1/2,pppp);subs(c=4,RATIO);plot(1/RATIO,c= 0..10,v=0.25..0.4,color=black,thickness =2,labels=[C,`1/Omega`]);PP:=rho^3-3*rho^2+(P+8*T)*rho/3-P=0;PPP:=solve(PP,P);`CT1.2`:=subs(T=1.2,PPP);`CT1.15`:=subs(T=1.15,PPP);`CT1.1`:=subs(T=1.1,PPP);`CT1.05`:=subs(T=1.05,PPP);`CT1`:=subs(T=1,PPP);`CT.95`:=subs(T=.95,PPP);`CT.9`:=subs(T=0.9,PPP);`CT.85`:=subs(T=0.85,PPP);`CT.8`:=subs(T=0.8,PPP);`CT.75`:=subs(T=0.75,PPP);`CT.7`:=subs(T=0.7,PPP);`CT.6`:=subs(T=0.6,PPP);`CT.4`:=subs(T=0.4,PPP);`CT.2`:=subs(T=0.2,PPP);`CT.0`:=subs(T=0,PPP);plot([`CT1`,`CT.85`,`CT.4`,`CT.2`,`CT.0`],rho=-20..20,view=-300..100,thickness =1,numpoints=1000);plot([`CT1`,`CT.85`,`CT.75`,`CT.6`,`CT.0`],rho=-0..4,view=-0..5,thickness =1,numpoints=1000);plot([`CT1`,`CT.85`,`CT.75`,`CT.6`,`CT.0`],rho=-0..4,view=-5..5,thickness =1,numpoints=1000);plot([`CT1`,`CT.85`,`CT.75`,`CT.6`,`CT.0`],rho=-3..4,view=-5..5,thickness =1,numpoints=1000);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