1. **Problem statement:** Calculate the average number of plates, the standard deviation of the average number of plates, and the average plate height for a 40 cm packed column using the given gas-liquid chromatography data. 2. **Given data:** - Column length, $L = 40$ cm - Compounds with retention times $t_R$ and peak widths $W$: - Methylcyclohexane: $t_R = 10.0$ min, $W = 0.76$ min - Methylcyclohexene: $t_R = 10.9$ min, $W = 0.82$ min - Toluene: $t_R = 13.4$ min, $W = 1.06$ min 3. **Formula for number of theoretical plates $N$:** $$N = 16 \left(\frac{t_R}{W}\right)^2$$ This formula relates the retention time and peak width to the efficiency of the column. 4. **Calculate $N$ for each compound:** - For Methylcyclohexane: $$N_1 = 16 \left(\frac{10.0}{0.76}\right)^2 = 16 \times \left(13.1579\right)^2 = 16 \times 173.11 = 2770$$ - For Methylcyclohexene: $$N_2 = 16 \left(\frac{10.9}{0.82}\right)^2 = 16 \times \left(13.29\right)^2 = 16 \times 176.68 = 2827$$ - For Toluene: $$N_3 = 16 \left(\frac{13.4}{1.06}\right)^2 = 16 \times \left(12.64\right)^2 = 16 \times 159.7 = 2555$$ 5. **Calculate average number of plates $\bar{N}$:** $$\bar{N} = \frac{N_1 + N_2 + N_3}{3} = \frac{2770 + 2827 + 2555}{3} = \frac{8152}{3} = 2717.33$$ 6. **Calculate standard deviation $s$ of $N$ values:** $$s = \sqrt{\frac{(N_1 - \bar{N})^2 + (N_2 - \bar{N})^2 + (N_3 - \bar{N})^2}{3 - 1}}$$ Calculate each squared difference: $$(2770 - 2717.33)^2 = 2760.44$$ $$(2827 - 2717.33)^2 = 11971.11$$ $$(2555 - 2717.33)^2 = 26244.44$$ Sum: $$2760.44 + 11971.11 + 26244.44 = 40976$$ Divide by 2: $$\frac{40976}{2} = 20488$$ Take square root: $$s = \sqrt{20488} = 143.2$$ 7. **Calculate average plate height $H$:** $$H = \frac{L}{\bar{N}} = \frac{40}{2717.33} = 0.0147 \text{ cm}$$ **Final answers:** - Average number of plates: $2717$ - Standard deviation: $143$ - Average plate height: $0.0147$ cm