\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)

Titration Curve:(M(n+) with EDTA ⇒Y(4-) )
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M(n+)+Y(4-) ↔ MY(n-4)
Ca(2+) → n=2 → Ca(2+)+Y(4-) = CaY(2-4)=CaY(2-)
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[email protected], 05/10/2016
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Definition :  me=[M(n+)] = (Fraction remaining * initial concentration *dilutionfactor)=f1*f2*f3
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                                      volume , concentration
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             give : M(n+) → [M(n+)] , yy ml , U Molair conc (erlenmeyer flask)
                    added
                    EDTA  → [y(4-]  , xx ml , W Molair conc (buret)
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             kf  = formation constant = [MY(n-4)]/([M(n+)]*[y(4-)])
                   edta = polyprotic acid (= tetraprotic weak acid)
                   [Y(4-)] = alpha(4) * [EDTA]   ,alpha(4) = f(h,ka1,ka2,ka3,ka4)=g(pH,ka1,ka2,ka3,ka4)
             rem : kfc ,conditional formation for complexes.
                   kfc = kf * alpha(4) = [MY(n-4)]/([M(n+)]*[EDTA])=(my)/(me*edta) = kf *alpha4
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Before EP:
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xx=0 ml,w no EDTA ,yy ml concentration U

find general equation ,yy,u,w ( rule : v*t = v'*t', v' = (v*t)/t',from analytical chemistry, v'=z )

(%i1) kill(all)$
(%i1) p1 : z = (yy*u)/w;
\[\tag{p1}\label{p1}z=\frac{u\,\mathit{yy}}{w}\]
(%i5) p2:f1=(z-xx)/z;
p3:f2= u;
p4:f3= yy/(yy+xx);
p5:f = rhs(p2)*rhs(p3)*rhs(p4);
\[\tag{p2}\label{p2}\mathit{f1}=\frac{z-\mathit{xx}}{z}\] \[\tag{p3}\label{p3}\mathit{f2}=u\] \[\tag{p4}\label{p4}\mathit{f3}=\frac{\mathit{yy}}{\mathit{yy}+\mathit{xx}}\] \[\tag{p5}\label{p5}f=\frac{u\,\mathit{yy}\,\left( z-\mathit{xx}\right) }{\left( \mathit{yy}+\mathit{xx}\right) z}\]
(%i6) p6:me=ev(rhs(p5),p1);
\[\tag{p6}\label{p6}\mathit{me}=\frac{w\,\left( \frac{u\,\mathit{yy}}{w}-\mathit{xx}\right) }{\mathit{yy}+\mathit{xx}}\]
(%i7) p7:ev(rhs(p6),xx=0);
\[\tag{p7}\label{p7}u\]

xx < > 0,ml,w EDTA,yy concentration  U, ( use: z-xx >0),FreeM_before = me

(%i8) p8:p6;
\[\tag{p8}\label{p8}\mathit{me}=\frac{w\,\left( \frac{u\,\mathit{yy}}{w}-\mathit{xx}\right) }{\mathit{yy}+\mathit{xx}}\]

By  EP:
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xx=z ml,w EDTA,yy ml concentration U

(%i9) p9:me=ev(rhs(p5),xx=z);
\[\tag{p9}\label{p9}\mathit{me}=0\]

me = 0  ? all MY_VAL
    M(n+)+EDTA ↔ MY
      0    0       my_val ?

(%i11) p10:my_val=u*yy/(yy+z);
p11: kfc=kf*alpha4;
\[\tag{p10}\label{p10}\mathit{my\_ val}=\frac{u\,\mathit{yy}}{z+\mathit{yy}}\] \[\tag{p11}\label{p11}\mathit{kfc}=\mathit{alpha4}\,\mathit{kf}\]

find free [M(n+)] :freeM =[M(n+)]
                M(n+) + EDTA ↔ MY
conc initial      0       0      my_val
conc formation    freeM  freeM   my_val - freeM

(%i15) p12: my_ep = my_val - freeM;
p13: me_ep = freeM;
p14: edta_ep = freeM;
p15: kfc = my_ep/(me_ep*edta_ep);
\[\tag{p12}\label{p12}\mathit{my\_ ep}=\mathit{my\_ val}-\mathit{freeM}\] \[\tag{p13}\label{p13}\mathit{me\_ ep}=\mathit{freeM}\] \[\tag{p14}\label{p14}\mathit{edta\_ ep}=\mathit{freeM}\] \[\tag{p15}\label{p15}\mathit{kfc}=\frac{\mathit{my\_ ep}}{\mathit{edta\_ ep}\mathit{me\_ ep}}\]
(%i16) p16:my_ep=ev(rhs(p12),p10);
\[\tag{p16}\label{p16}\mathit{my\_ ep}=\frac{u\,\mathit{yy}}{z+\mathit{yy}}-\mathit{freeM}\]
(%i17) p17:kfc=ev(rhs(p15),p16);
\[\tag{p17}\label{p17}\mathit{kfc}=\frac{\frac{u\,\mathit{yy}}{z+\mathit{yy}}-\mathit{freeM}}{\mathit{edta\_ ep}\mathit{me\_ ep}}\]
(%i18) p18:kfc=ev(rhs(p17),p13,p14);
\[\tag{p18}\label{p18}\mathit{kfc}=\frac{\frac{u\,\mathit{yy}}{z+\mathit{yy}}-\mathit{freeM}}{{{\mathit{freeM}}^{2}}}\]
(%i19) p19:solve(p18,freeM);
\[\tag{p19}\label{p19}[\mathit{freeM}=-\frac{\sqrt{{{z}^{2}}+\left( 4\mathit{kfc}u+2\right) \,\mathit{yy}z+\left( 4\mathit{kfc}u+1\right) \,{{\mathit{yy}}^{2}}}+z+\mathit{yy}}{2\mathit{kfc}z+2\mathit{kfc}\,\mathit{yy}},\mathit{freeM}=\frac{\sqrt{{{z}^{2}}+\left( 4\mathit{kfc}u+2\right) \,\mathit{yy}z+\left( 4\mathit{kfc}u+1\right) \,{{\mathit{yy}}^{2}}}-z-\mathit{yy}}{2\mathit{kfc}z+2\mathit{kfc}\,\mathit{yy}}]\]

only positive solution ?

(%i21) p20:part(p19,2);
\[\tag{p20}\label{p20}\mathit{freeM}=\frac{\sqrt{{{z}^{2}}+\left( 4\mathit{kfc}u+2\right) \,\mathit{yy}z+\left( 4\mathit{kfc}u+1\right) \,{{\mathit{yy}}^{2}}}-z-\mathit{yy}}{2\mathit{kfc}z+2\mathit{kfc}\,\mathit{yy}}\]

after EP:
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xx > Z ml:w EDTA ⇒ z < x_excess=xxx,      freeM_after = [M(n+)]

(%i26) p21:edta_after=w*(xxx/(yy+z+xxx));
p22:my_after  =u*(yy/(yy+z+xxx));
p23:kfc = my_after/(freeM_after*edta_after);
\[\tag{p21}\label{p21}\mathit{edta\_ after}=\frac{w\,\mathit{xxx}}{z+\mathit{yy}+\mathit{xxx}}\] \[\tag{p22}\label{p22}\mathit{my\_ after}=\frac{u\,\mathit{yy}}{z+\mathit{yy}+\mathit{xxx}}\] \[\tag{p23}\label{p23}\mathit{kfc}=\frac{\mathit{my\_ after}}{\mathit{edta\_ after}\mathit{freeM\_ after}}\]
(%i27) p24:kfc=ev(rhs(p23),p21,p22);
\[\tag{p24}\label{p24}\mathit{kfc}=\frac{u\,\mathit{yy}}{\mathit{freeM\_ after}w\,\mathit{xxx}}\]
(%i29) p25:part(solve(p24,freeM_after),1);
\[\tag{p25}\label{p25}\mathit{freeM\_ after}=\frac{u\,\mathit{yy}}{\mathit{kfc}w\,\mathit{xxx}}\]
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