Question A6.2: Two solutes, A and B, with distribution ratios of 9 and 4, r...
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 99 \% confidence interval for the location of each solute.
The fraction, q, of each solute remaining in the lower phase is calculated using equation A6.1.
\left(q_{\mathrm{aq}}\right)_1=\frac{(\text { moles aq })_1}{(\text { moles aq })_0}=\frac{V_{\mathrm{aq}}}{D V_{\mathrm{org}}+V_{\mathrm{aq}}} (A6.1)
Since the volumes of the lower and upper phases are equal, we get
\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}
and, consequently, p_{\mathrm{A}} is 0.90 and p_{\mathrm{B}} is 0.80 . The mean and standard deviation for the distribution of solutes A and B after 100 steps of the countercurrent extraction are
\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}
The confidence interval for a normally distributed solute is
r=\mu \pm z \sigma
where r is the number of the tube, and the value of z is determined by the chosen significance level. For a 99 \% confidence interval, the value of z is 2.58 (Appendix 1B); thus,
\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}
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.
| Appendix 1B t-Table^a |
||||
| Value of t for confidence interval of : Critical value of |t| for α values of : Degrees of Freedom |
90% 0.10 |
95 % 0.05 |
98 % 0.02 |
99 % 0.01 |
| 1 | 6.31 | 12.71 | 31.82 | 63.66 |
| 2 | 2.92 | 4.30 | 6.96 | 9.92 |
| 3 | 2.35 | 3.18 | 4.54 | 5.84 |
| 4 | 2.13 | 2.78 | 3.75 | 4.60 |
| 5 | 2.02 | 2.57 | 3.36 | 4.03 |
| 6 | 1.94 | 2.45 | 3.14 | 3.71 |
| 7 | 1.89 | 2.36 | 3.00 | 3.50 |
| 8 | 1.86 | 2.31 | 2.90 | 3.36 |
| 9 | 1.83 | 2.26 | 2.82 | 3.25 |
| 10 | 1.81 | 2.23 | 2.76 | 3.17 |
| 12 | 1.78 | 2.18 | 2.68 | 3.05 |
| 14 | 1.76 | 2.14 | 2.62 | 2.98 |
| 16 | 1.75 | 2.12 | 2.58 | 2.92 |
| 18 | 1.73 | 2.10 | 2.55 | 2.88 |
| 20 | 1.72 | 2.09 | 2.53 | 2.85 |
| 30 | 1.70 | 2.04 | 2.46 | 2.75 |
| 50 | 1.68 | 2.01 | 2.40 | 2.68 |
| \infty | 1.64 | 1.96 | 2.33 | 2.58 |
| ^aThe t-values in this table are for a two-tailed test. For a one-tailed test, the α values for each column are half of the stated value. For example, the first column for a one-tailed test is for the 95% confidence level, α = 0.05. |
||||
