Two solutes, A and B, with distribution ratios of 9 and 4, respectively, are to be

Question

Two solutes, A and B, with distribution ratios of 9 and 4, respectively, are to be separated by a countercurrent extraction in which the volumes of the upper and lower phases are equal. After 100 steps, determine the [latex]99 \%[/latex] confidence interval for the location of each solute.

Accepted Answer

The fraction, [latex]q[/latex], of each solute remaining in the lower phase is calculated using equation A6.1.

[latex]\left(q_{\mathrm{aq}} ight)_1=\frac{( ext { moles aq })_1}{( ext { moles aq })_0}=\frac{V_{\mathrm{aq}}}{D V_{\mathrm{org}}+V_{\mathrm{aq}}}[/latex] (A6.1)

Since the volumes of the lower and upper phases are equal, we get

[latex] \begin{aligned} & q_{\mathrm{A}}=\frac{1}{D_{\mathrm{A}}+1}=\frac{1}{9+1}=0.10 \ & q_{\mathrm{B}}=\frac{1}{D_{\mathrm{B}}+1}=\frac{1}{4+1}=0.20 \end{aligned} [/latex]

and, consequently, [latex]p_{\mathrm{A}}[/latex] is 0.90 and [latex]p_{\mathrm{B}}[/latex] is 0.80 . The mean and standard deviation for the distribution of solutes A and B after 100 steps of the countercurrent extraction are

[latex] \begin{array}{ll} \mu_{\mathrm{A}}=n p_{\mathrm{A}}=(100)(0.90)=90 & \sigma_{\mathrm{A}}=\sqrt{n p_{\mathrm{A}} q_{\mathrm{A}}}=\sqrt{(100)(0.90)(0.10)}=3 \ \mu_{\mathrm{B}}=n p_{\mathrm{B}}=(100)(0.80)=80 & \sigma_{\mathrm{B}}=\sqrt{n p_{\mathrm{B}} q_{\mathrm{B}}}=\sqrt{(100)(0.80)(0.20)}=4 \end{array} [/latex]

The confidence interval for a normally distributed solute is

[latex]r=\mu \pm z \sigma[/latex]

where [latex]r[/latex] is the number of the tube, and the value of [latex]z[/latex] is determined by the chosen significance level. For a [latex]99 \%[/latex] confidence interval, the value of [latex]z[/latex] is 2.58 (Appendix 1B); thus,

[latex] \begin{aligned} & r_{\mathrm{A}}=90 \pm(2.58)(3)=90 \pm 8 \ & r_{\mathrm{B}}=80 \pm(2.58)(4)=80 \pm 10 \end{aligned} [/latex]

Since the two confidence intervals overlap, a complete separation of the two solutes cannot be achieved in a 100-step countercurrent extraction. The complete distribution of the solutes is shown in Figure A6.4.

Question A6.2: Two solutes, A and B, with distribution ratios of 9 and 4, r...

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