1. **State the problem:** Calculate the average number of plates, the standard deviation of the number of plates, and the average plate height for a 40 cm packed column using the given gas-liquid chromatography data. 2. **Formulas and important rules:** - Number of plates $N$ is calculated by the formula: $$N = 16 \left(\frac{t_R}{W}\right)^2$$ where $t_R$ is the retention time and $W$ is the peak width. - Standard deviation $\sigma$ of $N$ is calculated using the sample standard deviation formula: $$\sigma = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (N_i - \bar{N})^2}$$ - Average plate height $H$ is: $$H = \frac{L}{\bar{N}}$$ where $L$ is the column length (40 cm). 3. **Calculate number of plates $N$ for each compound with $W$ given:** - Methylcyclohexane: $$N = 16 \left(\frac{10.0}{0.76}\right)^2 = 16 \times (13.1579)^2 = 16 \times 173.11 = 2770$$ - Methylcyclohexene: $$N = 16 \left(\frac{10.9}{0.82}\right)^2 = 16 \times (13.29)^2 = 16 \times 176.68 = 2827$$ - Toluene: $$N = 16 \left(\frac{13.4}{1.06}\right)^2 = 16 \times (12.64)^2 = 16 \times 159.7 = 2555$$ 4. **Calculate average number of plates $\bar{N}$:** $$\bar{N} = \frac{2770 + 2827 + 2555}{3} = \frac{8152}{3} = 2717$$ 5. **Calculate standard deviation $\sigma$ of $N$:** $$\sigma = \sqrt{\frac{(2770 - 2717)^2 + (2827 - 2717)^2 + (2555 - 2717)^2}{3 - 1}}$$ $$= \sqrt{\frac{53^2 + 110^2 + (-162)^2}{2}} = \sqrt{\frac{2809 + 12100 + 26244}{2}} = \sqrt{20576.5} = 143.5$$ 6. **Calculate average plate height $H$:** $$H = \frac{L}{\bar{N}} = \frac{40}{2717} = 0.0147\ \text{cm} = 0.147\ \text{mm}$$ **Final answers:** - Average number of plates: $2717$ - Standard deviation: $143.5$ - Average plate height: $0.0147$ cm (or $0.147$ mm)