\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
(%i1) kill(all)$

[email protected],16/06/2017
thermodynamics : 'Equation of State'
1e) virial gas equation state

The virial gas equation of state ( thrid order)
--------------------------------
ideal     gases: P*V=R*T*n ,' n = number mole '
non-ideal gases: Z = (P*V)/(R*T),Z ' = compressibility factor '
                z = Z1 = Z2

(%i1) p1:(a+b+c+d)^3;
\[\tag{p1}\label{p1}{{\left( d+c+b+a\right) }^{3}}\]
(%i2) p2:expand(p1);
\[\tag{p2}\label{p2}{{d}^{3}}+3c\,{{d}^{2}}+3b\,{{d}^{2}}+3a\,{{d}^{2}}+3{{c}^{2}}d+6bcd+6acd+3{{b}^{2}}d+6abd+3{{a}^{2}}d+{{c}^{3}}+3b\,{{c}^{2}}+3a\,{{c}^{2}}+3{{b}^{2}}c+6abc+3{{a}^{2}}c+{{b}^{3}}+3a\,{{b}^{2}}+3{{a}^{2}}b+{{a}^{3}}\]
(%i3) coeff(p2,d);
\[\tag{\%{}o3}\label{o3} 3{{c}^{2}}+6bc+6ac+3{{b}^{2}}+6ab+3{{a}^{2}}\]
(%i4) p11:(P*V)/(R*T);
\[\tag{p11}\label{p11}\frac{PV}{RT}\]

Z(V)=Z1=1+B/V+C/V^2+D/V^3
Z(P)=Z2=1+B1*P+C1*P^2+D1*P^3

(%i6) p3:Z1=1+B/V+C/V^2+D/V^3;
p4:Z2=1+B1*P+C1*P^2+D1*P^3;
\[\tag{p3}\label{p3}\mathit{Z1}=\frac{B}{V}+\frac{C}{{{V}^{2}}}+\frac{D}{{{V}^{3}}}+1\] \[\tag{p4}\label{p4}\mathit{Z2}=\mathit{D1}\,{{P}^{3}}+\mathit{C1}\,{{P}^{2}}+\mathit{B1}P+1\]

? P=P(T,Bi,Ci,Di,B,C,D)  i=1,2,3....

(%i8) p5:ratsubst(p11, Z1, p3);
p6:lhs(p5)*R*T/V=rhs(p5)*R*T/V;
\[\tag{p5}\label{p5}\frac{PV}{RT}=\frac{{{V}^{3}}+B\,{{V}^{2}}+CV+D}{{{V}^{3}}}\] \[\tag{p6}\label{p6}P=\frac{RT\,\left( {{V}^{3}}+B\,{{V}^{2}}+CV+D\right) }{{{V}^{4}}}\]
(%i12) p(i):=coeff(radcan(num(rhs(p6))),V^i)*V^i/denom(rhs(p6));
p00:num(rhs(p6))$
p000:ev(p00,V=0)/denom(rhs(p6))$
pt:p000+p(1)+p(2)+p(3);
\[\tag{\%{}o9}\label{o9} \operatorname{p}(i):=\frac{\operatorname{coeff}\left( \operatorname{radcan}\left( \operatorname{num}\left( \operatorname{rhs}\left( \mathit{p6}\right) \right) \right) ,{{V}^{i}}\right) \,{{V}^{i}}}{\operatorname{denom}\left( \operatorname{rhs}\left( \mathit{p6}\right) \right) }\] \[\tag{pt}\label{pt}\frac{RT}{V}+\frac{BRT}{{{V}^{2}}}+\frac{CRT}{{{V}^{3}}}+\frac{DRT}{{{V}^{4}}}\]

Z2=Z(T,V,Bi,Ci,Di,B,C,D)  i=1,2,3...

(%i13) p7:subst(pt, P, p4);
\[\tag{p7}\label{p7}\mathit{Z2}=\mathit{B1}\,\left( \frac{RT}{V}+\frac{BRT}{{{V}^{2}}}+\frac{CRT}{{{V}^{3}}}+\frac{DRT}{{{V}^{4}}}\right) +\mathit{D1}\,{{\left( \frac{RT}{V}+\frac{BRT}{{{V}^{2}}}+\frac{CRT}{{{V}^{3}}}+\frac{DRT}{{{V}^{4}}}\right) }^{3}}+\mathit{C1}\,{{\left( \frac{RT}{V}+\frac{BRT}{{{V}^{2}}}+\frac{CRT}{{{V}^{3}}}+\frac{DRT}{{{V}^{4}}}\right) }^{2}}+1\]
(%i14) p8:expand(p7);
\[\tag{p8}\label{p8}\mathit{Z2}=\frac{\mathit{B1}RT}{V}+\frac{\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{2}}}+\frac{B\,\mathit{B1}RT}{{{V}^{2}}}+\frac{\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{3}}}+\frac{2B\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{3}}}+\frac{\mathit{B1}CRT}{{{V}^{3}}}+\frac{3B\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{4}}}+\frac{2C\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{4}}}+\frac{{{B}^{2}}\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{4}}}+\frac{\mathit{B1}DRT}{{{V}^{4}}}+\frac{3C\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{5}}}+\frac{3{{B}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{5}}}+\frac{2\mathit{C1}D\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{5}}}+\frac{2BC\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{5}}}+\frac{3D\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{6}}}+\frac{6BC\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{6}}}+\frac{{{B}^{3}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{6}}}+\frac{2B\,\mathit{C1}D\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{6}}}+\frac{{{C}^{2}}\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{6}}}+\frac{6BD\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{7}}}+\frac{3{{C}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{7}}}+\frac{3{{B}^{2}}C\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{7}}}+\frac{2C\,\mathit{C1}D\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{7}}}+\frac{6CD\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{8}}}+\frac{3{{B}^{2}}D\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{8}}}+\frac{3B\,{{C}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{8}}}+\frac{\mathit{C1}\,{{D}^{2}}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{8}}}+\frac{3{{D}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{9}}}+\frac{6BCD\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{9}}}+\frac{{{C}^{3}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{9}}}+\frac{3B\,{{D}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{10}}}+\frac{3{{C}^{2}}D\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{10}}}+\frac{3C\,{{D}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{11}}}+\frac{{{D}^{3}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{12}}}+1\]

new Z2 with old Z1     : coefficient comparing

part 1 : new Z2

(%i16) for i:1 thru 12 do
pp(i):=subst(x^i, 1/V^i, p8)$
ppp(i):=coeff(rhs(pp(i)),x^(i))$
(%i18) pp(2);
ppp(2);
\[\tag{\%{}o17}\label{o17} \mathit{Z2}=\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}\,{{x}^{2}}+B\,\mathit{B1}RT\,{{x}^{2}}+\frac{\mathit{B1}RT}{V}+\frac{\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{3}}}+\frac{2B\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{3}}}+\frac{\mathit{B1}CRT}{{{V}^{3}}}+\frac{3B\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{4}}}+\frac{2C\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{4}}}+\frac{{{B}^{2}}\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{4}}}+\frac{\mathit{B1}DRT}{{{V}^{4}}}+\frac{3C\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{5}}}+\frac{3{{B}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{5}}}+\frac{2\mathit{C1}D\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{5}}}+\frac{2BC\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{5}}}+\frac{3D\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{6}}}+\frac{6BC\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{6}}}+\frac{{{B}^{3}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{6}}}+\frac{2B\,\mathit{C1}D\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{6}}}+\frac{{{C}^{2}}\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{6}}}+\frac{6BD\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{7}}}+\frac{3{{C}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{7}}}+\frac{3{{B}^{2}}C\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{7}}}+\frac{2C\,\mathit{C1}D\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{7}}}+\frac{6CD\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{8}}}+\frac{3{{B}^{2}}D\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{8}}}+\frac{3B\,{{C}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{8}}}+\frac{\mathit{C1}\,{{D}^{2}}\,{{R}^{2}}\,{{T}^{2}}}{{{V}^{8}}}+\frac{3{{D}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{9}}}+\frac{6BCD\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{9}}}+\frac{{{C}^{3}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{9}}}+\frac{3B\,{{D}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{10}}}+\frac{3{{C}^{2}}D\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{10}}}+\frac{3C\,{{D}^{2}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{11}}}+\frac{{{D}^{3}}\,\mathit{D1}\,{{R}^{3}}\,{{T}^{3}}}{{{V}^{12}}}+1\] \[\tag{\%{}o18}\label{o18} \mathit{C1}\,{{R}^{2}}\,{{T}^{2}}+B\,\mathit{B1}RT\]

part 2 : old Z1

(%i20) for i:1 thru 4 do
qq(i):=subst(x^i, 1/V^i, p3)$
qqq(i):=coeff(rhs(qq(i)),x^(i))$
(%i22) qq(2);
qqq(2);
\[\tag{\%{}o21}\label{o21} \mathit{Z1}=C\,{{x}^{2}}+\frac{B}{V}+\frac{D}{{{V}^{3}}}+1\] \[\tag{\%{}o22}\label{o22} C\]

coefficient : old Z2 ( 'p4')

(%i25) q1:part(solve(ppp(1)=qqq(1),B1),1);
q2:part(solve(ppp(2)=qqq(2),C1),1);
q3:part(solve(ppp(3)=qqq(3),D1),1);
\[\tag{q1}\label{q1}\mathit{B1}=\frac{B}{RT}\] \[\tag{q2}\label{q2}\mathit{C1}=-\frac{B\,\mathit{B1}RT-C}{{{R}^{2}}\,{{T}^{2}}}\] \[\tag{q3}\label{q3}\mathit{D1}=-\frac{2B\,\mathit{C1}\,{{R}^{2}}\,{{T}^{2}}+\mathit{B1}CRT-D}{{{R}^{3}}\,{{T}^{3}}}\]
(%i28) qq1:q1;
qq2:ev(q2,q1);
qq3:ev(q3,q1,qq2);
\[\tag{qq1}\label{qq1}\mathit{B1}=\frac{B}{RT}\] \[\tag{qq2}\label{qq2}\mathit{C1}=-\frac{{{B}^{2}}-C}{{{R}^{2}}\,{{T}^{2}}}\] \[\tag{qq3}\label{qq3}\mathit{D1}=-\frac{-D+BC-2B\,\left( {{B}^{2}}-C\right) }{{{R}^{3}}\,{{T}^{3}}}\]

find : Z2=Z(P,T,C,B,D)

(%i29) pq:ev(p4,qq1,qq2,qq3);
\[\tag{pq}\label{pq}\mathit{Z2}=\frac{BP}{RT}-\frac{\left( {{B}^{2}}-C\right) \,{{P}^{2}}}{{{R}^{2}}\,{{T}^{2}}}-\frac{\left( -D+BC-2B\,\left( {{B}^{2}}-C\right) \right) \,{{P}^{3}}}{{{R}^{3}}\,{{T}^{3}}}+1\]

general : v=volume
         p=pressure
         Z=compressibility factor
         n= integer
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Z=1+sum(a(i)*1/v^i,i=1..n)
z=1+sum(a1(i)*p^i ,i=1..n)
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