{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 
{CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 
256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 105 "this is id
entical to the A-n-V minimization script, except that here the propert
ies are normalized by n+V" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Compute the L-V cu
rve for water as predicted by the Van der Waals EoS" }}{PARA 0 "" 0 "
" {TEXT -1 53 "by minimization of the Helmoltz Free Energy A(V,T,n):" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Van der
 Waals EoS is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "pvdw := nRT/(V-b) - a/V^2;" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We are concerned with isotherma
l minimization and our EoS predicts the properties of " }}{PARA 0 "" 
0 "" {TEXT -1 35 "both phase, therefore we can write " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "A = A0 - int(p,V)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "where A0 \+
is an arbitrary constant that we can set to 0" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := -int(pvdw,V);
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Next \+
we need the Van der Waals constants, we can evaluate these by making" 
}}{PARA 0 "" 0 "" {TEXT -1 85 "use of the fact that at the critical po
int the first and second derivatives of p with" }}{PARA 0 "" 0 "" 
{TEXT -1 82 "respect to V must vanish. And noting that the critical po
int is at 374 C, 212 bar," }}{PARA 0 "" 0 "" {TEXT -1 36 "with R = 8.3
14 J/mol, and Vcrit ~ 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 30 "avdw := solve(diff(pvdw,V),a):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 52 "bvdwn := solve(subs (a=avdw, V=3,diff(pvdw,V,V
)),b);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "avdwn := subs(V=3,nRT=(37
4+273)*8.314,b=bvdwn,avdw);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 81 "lastly we set up the minimization using pseudocompounds to repr
esent A(V,n), note" }}{PARA 0 "" 0 "" {TEXT -1 40 "we specify the temp
erature the next line" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "nRTK := (324 + 273)*8.314;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "pvdwn := subs(a=avdwn,b=bvdwn,nRT=nRTK, pvdw);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Anum := subs(a=avdwn,b=bvdwn,nRT=nR
TK, A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(pvdwn,V=1.1..16);" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(Anum,V=1.1...20);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "dVol := 00.2: pi := 0: Vol := 1.1:j := 'j
':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "while Vol < 20 do:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "   pi := pi + 1:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "   nh2o[pi] := 1/(1 + Vol);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "   vh2o[pi] := 1-nh2o[pi];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "   ah2o[pi] := subs(V=Vol,Anum)/(1 + Vol);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "   Vol := Vol + dVol;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "end do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Atot :
= sum(ah2o[j]*alpha[j],j=1..pi):" }{MPLTEXT 0 21 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "ntot := sum(nh2o[j]*alpha[j],j=1..pi):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "vtot := sum(vh2o[j]*alpha[j],j=1..pi):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "constraints := \{vtot = 3/4, ntot \+
= 1/4\}:" }{MPLTEXT 0 21 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "wit
h(simplex):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 20 "do the minimization:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 50 "ans := minimize( Atot, constraints, NONNE
GATIVE ):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
85 "filter through the results to get the non-zero alphas, the index p
oints to the volume" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 15 "i := 0: j := 0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "while i < pi do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "   i := i +
 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "   if (rhs(ans[i]) > 0) then
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "      j := j + 1;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "      rans[j] := ans[i];" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "   end if:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end
 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rans[1];rans[2];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "use the volumes identifie
d by the indexes on the alpha's to back calculate the pressure:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "i1 \+
:= 5; i2 := 26;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "subs(V=vh2o[i1]*
(1-vh2o[i1]),pvdwn);subs(V=vh2o[i2]*(1-vh2o[i2]),pvdwn);" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "next back calculat
e by solving A = n*mu - p*V" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "avec := vector(2,[ah2o[i1],ah2o[i2]]):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "mat := matrix(2,2,[-vh2o[i1],nh2o[i1],-vh2o[i2
],nh2o[i2]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "linsolve(mat,avec)
;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "and \+
lastly compare the roots of the vdw at the pressure obtained by minimi
zation to the" }}{PARA 0 "" 0 "" {TEXT -1 38 "volumes of the stable ps
eudocompounds:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "solve(pvdwn-483);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "vh2o[i1]/(1-vh2o[i1]),vh2o[i2]/(1-vh2o[i2]);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 16 "refine estimate:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "vmn1 :=vh2o[i1-1]/(
1-vh2o[i1-1]);vmx1 :=vh2o[i1+1]/(1-vh2o[i1+1]);dv1 := (vmx1-vmn1)/20;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "vmn2 :=vh2o[i2-1]/(1-vh2o[i1-1]
);vmx2 :=vh2o[i2+1]/(1-vh2o[i2+1]);dv2 := (vmx2-vmn2)/20;" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "pi := 0: Vol := vmn1:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "while Vol < vmx1 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "   pi := pi + 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "   nh2o[p
i] := 1/(1 + Vol);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "   vh2o[pi] :
= 1-nh2o[pi];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "   ah2o[pi] := sub
s(V=Vol,Anum)/(1 + Vol);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   Vol \+
:= Vol + dv1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "Vol := vmn2:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "while Vol < vmx2 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "   \+
pi := pi + 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "   nh2o[pi] := 1/(
1 + Vol);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "   vh2o[pi] := 1-nh2o[
pi];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "   ah2o[pi] := subs(V=Vol,A
num)/(1 + Vol);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   Vol := Vol + \+
dv2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "j := 'j';" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Atot
 := sum(ah2o[j]*alpha[j],j=1..pi):" }{MPLTEXT 0 21 0 "" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "ntot := sum(nh2o[j]*alpha[j],j=1..pi):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "vtot := sum(vh2o[j]*alpha[j],j=1..p
i):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "constraints := \{vtot = 3/4,
 ntot = 1/4\}:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 20 "do the minimization:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ans := minimize( At
ot, constraints, NONNEGATIVE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "
i := 0: j := 0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "while i < pi do:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "   i := i + 1:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "   if (rhs(ans[i]) > 0) then" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "      j := j + 1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "      rans[j] := ans[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " \+
  end if:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "rans[1];rans[2];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "i1 := 4
0; i2 := 10;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "subs(V=vh2o[i1]/(1-
vh2o[i1]),pvdwn);subs(V=vh2o[i2]/(1-vh2o[i2]),pvdwn);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 2 0" 73 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }