Question

The distribution coefficient for extraction of a metal complex from aqueous to organic solvents is $\mathbf{D}\mathbf{=}{\left[\mathrm{total}\mathrm{metal}\right]}_{\mathbf{oeg}\mathbf{}}\mathbf{f}{\left[\mathrm{total}\mathrm{metal}\right]}_{\mathbf{aq}}$localid="1654846486193" $\mathbf{D}\mathbf{=}\mathbf{}{\mathbf{[}\mathbf{}\mathbf{total}\mathbf{}\mathbf{metal}\mathbf{}\mathbf{]}}_{\mathbf{oeg}}\mathbf{}\mathbf{f}{\mathbf{[}\mathbf{}\mathbf{total}\mathbf{}\mathbf{metal}\mathbf{}\mathbf{]}}_{\mathbf{aq}}$Give physical reasons why localid="1654846489683" $\mathbf{\beta }$ and localid="1654846493841" ${\mathbf{K}}_{\mathbf{\alpha }}$appear in the numerator of Equation 23-13,but localid="1654846498058" ${\mathbf{K}}_{\mathbf{L}}$and localid="1654846502645" ${\mathbf{\left[}{\mathbf{H}}^{\mathbf{+}}\mathbf{\right]}}_{\mathbf{aq}}$appear in the denominator.

Short answer

For $\mathbf{\beta }$and ${\mathbf{K}}_{\mathrm{a}}$, ${\mathrm{ML}}_{\mathrm{n}}$formation is favoured whereas in ${\mathrm{K}}_{\mathrm{L}}$and $\left[{\mathrm{H}}^{+}\right]$, the formation of ${\mathrm{ML}}_{\mathrm{n}}$is not favoured.

Step-by-step solution

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• The distribution coefficient for extraction of a metal complex from aqueous to organic solvents is given as $\mathrm{D}={[\mathrm{total}\mathrm{metal}]}_{\mathrm{org}}{[\mathrm{total}\mathrm{metal}]}_{\mathrm{aq}}$.
• Here, let us derive physical reasons of why $\mathbf{\beta }$and ${\mathbf{K}}_{\mathbf{\alpha }}$would appear in the numerator of equation 23-13, where ${\mathrm{K}}_{\mathrm{L}}$and $\left[{\mathrm{H}}^{+}\right]$appear in the denominator.

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• The equation for distribution of metal-chelate complex between phases is:

$\mathrm{D}=\frac{{\mathrm{K}}_{\mathrm{M}}{\mathrm{\beta K}}_{\mathrm{a}}^{\mathrm{n}}}{{\mathrm{K}}_{\mathrm{a}}^{\mathrm{n}}}\frac{{\left[\mathrm{HL}\right]}_{\mathrm{org}}^{\mathrm{n}}}{{\left[{\mathrm{H}}^{+}\right]}_{\mathrm{aq}}^{\mathrm{n}}}$

• ${\mathrm{ML}}_{\mathrm{n}}$is the form extracted into organic solvent.

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• For $\mathbf{\beta }$and ${\mathbf{K}}_{\mathrm{a}}$, formation is favoured by the increase of these (an increase in ${\mathrm{K}}_{\mathrm{a}}$ would cause an increase in ${\mathrm{L}}^{-}$formation which necessary for complex formation).
• For ${\mathrm{K}}_{\mathrm{L}}$and $\left[{\mathrm{H}}^{+}\right]$, the formation of ${\mathrm{ML}}_{\mathrm{n}}$is not favoured by the increase of these (as they increase while the formation of ${\mathrm{ML}}_{\mathrm{n}}$decreases).
• Thus, an increase in the concentration of $\left[{\mathrm{H}}^{+}\right]$would cause a decrease in the concentration of ${\mathrm{L}}^{-}$from weak acid in the form HL (if there is less ${\mathrm{L}}^{-}$it becomes harder for complex to form).

RESULT

${\mathrm{L}}^{-}$formation is needed for complex formation of $\mathbf{\beta }$and ${\mathbf{K}}_{\mathbf{a}}$, whereas ${\mathrm{ML}}_{\mathrm{n}}$is not favoured by the increase of these in ${\mathrm{K}}_{\mathrm{L}}$and $\left[{\mathrm{H}}^{+}\right]$as they increase while the formation of ${\mathrm{ML}}_{\mathrm{n}}$decreases.