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

find : alpha4;see before (verion 1.0)
[email protected],09/10/2016

definition : log to base 10

(%i1) log10(x):= log(x)/log(10.0);
\[\tag{\%{}o1}\label{o1} \operatorname{log10}(x):=\frac{\log{(x)}}{\log{\left( 10.0\right) }}\]
(%i2) log10(100.0);
\[\tag{\%{}o2}\label{o2} 2.0\]

EDTA = H4(Y) , m=4
alpha4 = (ka(1)*ka(2)*ka(3)*ka(4))/
        (h^4+ka(1)*h^3+ka(1)*ka(2)*h^2+ka(1)*ka(2)*ka(3)*h+..
         +ka(1)*ka(2)*ka(3)*ka(4))
ka(1) = ka_edta(1)=1.02e-2
ka(2) = ka_edta(2)=2.14e-3
ka(3) = ka_edta(3)=6.92e-7
ka(4) = ka_edta(4)=5.50e-11

p1:1*2*3*4*5 = 120
p2:1+2+3+4+5 = 15

(%i4) p1:'product(k, k, 1, 5)=product(k, k, 1, 5);
p2:'sum(k,k,1,5)=sum(k,k,1,5);
\[\tag{p1}\label{p1}\prod_{k=1}^{5}{\left. k\right.}=120\] \[\tag{p2}\label{p2}\sum_{k=1}^{5}{\left. k\right.}=15\]
(%i5) alpha(h,i,n):='product(ka_edta[j],j,1,i)/
(h^n+sum(h^(n-k)*product(ka_edta[j],j,1,k),k,1,n));
\[\tag{\%{}o5}\label{o5} \alpha\left( h,i,n\right) :=\frac{\prod_{j=1}^{i}{\left. {{\mathit{ka\_ edta}}_{j}}\right.}}{{{h}^{n}}+\sum_{k=1}^{n}{\left. {{h}^{n-k}}\,\prod_{j=1}^{k}{\left. {{\mathit{ka\_ edta}}_{j}}\right.}\right.}}\]
(%i6) alpha4:alpha(h,4,4);
\[\tag{alpha4}\label{alpha4}\frac{\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.}}{{{h}^{4}}+{{\mathit{ka\_ edta}}_{1}}{{h}^{3}}+{{\mathit{ka\_ edta}}_{1}}{{\mathit{ka\_ edta}}_{2}}{{h}^{2}}+{{\mathit{ka\_ edta}}_{1}}{{\mathit{ka\_ edta}}_{2}}{{\mathit{ka\_ edta}}_{3}}h+{{\mathit{ka\_ edta}}_{1}}{{\mathit{ka\_ edta}}_{2}}{{\mathit{ka\_ edta}}_{3}}{{\mathit{ka\_ edta}}_{4}}}\]
(%i10) ka_edta[1]:1.02e-2$
ka_edta[2]:2.14e-3$
ka_edta[3]:6.92e-7$
ka_edta[4]:5.50e-11$
(%i11) alpha4:(alpha(h,4,4));
\[\tag{alpha4}\label{alpha4}\frac{\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.}}{{{h}^{4}}+0.0102{{h}^{3}}+2.1828{{10}^{-5}}\,{{h}^{2}}+1.5104976{{10}^{-11}}h+8.307736800000001{{10}^{-22}}}\]
(%i12) alpha4:alpha(h,4,4);
\[\tag{alpha4}\label{alpha4}\frac{\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.}}{{{h}^{4}}+0.0102{{h}^{3}}+2.1828{{10}^{-5}}\,{{h}^{2}}+1.5104976{{10}^{-11}}h+8.307736800000001{{10}^{-22}}}\]

pH =10,Y(m-) ,m=4,see :truebasic graph

(%i13) alpha(10^(-10),4,4);
\[\tag{\%{}o13}\label{o13} 4.270785644654524{{10}^{20}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.}\]
(%i14) float(4.270785644654524*10^20*product(ka_edta[j],j,1,4)), numer;
\[\tag{\%{}o14}\label{o14} 0.3548056306500811\]

for n:1 thru 5 do print(n);

(%i15) for n:1 thru 2
do print(n);
\[1 2 \] \[\tag{\%{}o15}\label{o15} \mathit{done}\]
(%i16) conch(pH):=float(10^(-pH));
\[\tag{\%{}o16}\label{o16} \operatorname{conch}\left( \mathit{pH}\right) :=\operatorname{float}\left( {{10}^{-\mathit{pH}}}\right) \]
(%i19) pH:10;
h:conch(pH);
float(alpha(h,4,4));
\[\tag{pH}\label{pH}10\] \[\tag{h}\label{h}1.0{{10}^{-10}}\] \[\tag{\%{}o19}\label{o19} 4.270785644654524{{10}^{20}}\,\prod_{j=1.0}^{4.0}{\left. {{\mathit{ka\_ edta}}_{j}}\right.}\]
(%i20) ratsimp(4.270785644654524*10^20*product(ka_edta[j],j,1.0,4.0))$
\[\mbox{}\\\mbox{rat: replaced 4.0 by 4/1 = 4.0}\mbox{}\\\mbox{rat: replaced 3.0 by 3/1 = 3.0}\mbox{}\\\mbox{rat: replaced 2.0 by 2/1 = 2.0}\mbox{}\\\mbox{rat: replaced 1.0 by 1/1 = 1.0}\mbox{}\\\mbox{rat: replaced 4.270785644654524E+20 by 427078564465452384256/1 = 4.270785644654524E+20}\]
(%i21) float(427078564465452384256*ka_edta[1]*ka_edta[2]*ka_edta[3]*ka_edta[4]), numer;
\[\tag{\%{}o21}\label{o21} 0.3548056306500811\]
(%i22) for pH:1 thru 14 do
(hh:conch(pH),alpha(hh,4,4),
vv:alpha(hh,4,4),   
print(pH,hh,vv));
\[1 0.1 9056.471322685169\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 2 0.01 4.467686132078125{{10}^{7}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 3 0.001 3.026349977404308{{10}^{10}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 4 1.0{{10}^{-4}} 4.346116014437968{{10}^{12}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 5 1.0{{10}^{-5}} 4.266101328336141{{10}^{14}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 6 1.0{{10}^{-6}} 2.706798910735476{{10}^{16}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 7 1.0{{10}^{-7}} 5.781621538756646{{10}^{17}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 8 1.0{{10}^{-8}} 6.490836657283224{{10}^{18}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 9 1.0{{10}^{-9}} 6.266615268394575{{10}^{19}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 10 1.0{{10}^{-10}} 4.270785644654524{{10}^{20}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 11 1.0{{10}^{-11}} 1.018510799451495{{10}^{21}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 12 1.0{{10}^{-12}} 1.182202632850073{{10}^{21}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 13 1.0{{10}^{-13}} 1.201512688396125{{10}^{21}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} 14 1.0{{10}^{-14}} 1.20347844297065{{10}^{21}}\,\prod_{j=1}^{4}{\left. {{\mathit{ka\_ edta}}_{j}}\right.} \] \[\tag{\%{}o22}\label{o22} \mathit{done}\]

pH =14,Y(m-) ,m=4,see :truebasic graph

(%i23) float(1203478442970649985024*product(ka_edta[j],j,1,4)), numer;
\[\tag{\%{}o23}\label{o23} 0.9998182148673971\]

pH =10,Y(m-) ,m=4,see :truebasic graph

(%i24) float(427078564465452384256*product(ka_edta[j],j,1,4)), numer;
\[\tag{\%{}o24}\label{o24} 0.3548056306500811\]
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