A mixture of ethylbenzene, cumene and cymene were analysed by GC at a temperature of $95^{\circ} \mathrm{C}$ using a thermal conductivity detector (TCD) and the following results were obtained (chart speed $20 \mathrm{~mm} / \mathrm{min}$ ):
\begin{tabular}{|c|c|c|c|c|}
\hline & \begin{tabular}{l}
$1^{4}$ eluted \
compound
\end{tabular} & \begin{tabular}{l}
$2^{m+1}$ eluted \
compound
\end{tabular} & \begin{tabular}{l}
$3^{r}$ eluted \
compound
\end{tabular} & b \
\hline \begin{tabular}{l}
Retention time \
(min)
\end{tabular} & 2.3 min & $4.3 \mathrm{~min}$ & 5.8 & $0.5 \mathrm{~min}$ \
\hline Peak width $(\mathrm{mm})$ & $5 \mathrm{~mm}$ & $9 \mathrm{~mm}$ & $11.2 \mathrm{~mm}$ & \
\hline Peak height (mm) & $28.00 \mathrm{~mm}$ & $22.75 \mathrm{~mm}$ & $35.20 \mathrm{~mm}$ & \
\hline \begin{tabular}{l}
Peak width at half \
height $(\mathrm{mm})$
\end{tabular} & $2.94 \mathrm{~mm}$ & $5.29 \mathrm{~mm}$ & $6.59 \mathrm{~mm}$ & \
\hline Response factor & 1.0 & 0.96 & 0.82 & \
\hline
\end{tabular}

The response factor was calculated from an equal volume mixture of the same three compounds by dividing the area of each peak by the area of the first eluting peak in the mixture.

Q 3(a)
[4 Marks]

Calculate the average efficiency $(\mathrm{N})$ of the column.

Q 3(b)
[3 Marks]

Calculate the resolution between peak 2 and peak 3.

Q 3(c)
[6 Marks]

Calculate the relative composition of this mixture $(\% \mathrm{v} / \mathrm{v})$.

Question

A mixture of ethylbenzene, cumene and cymene were analysed by GC at a temperature of $95^{\circ} \mathrm{C}$ using a thermal conductivity detector (TCD) and the following results were obtained (chart speed $20 \mathrm{~mm} / \mathrm{min}$ ):
\begin{tabular}{|c|c|c|c|c|}
\hline & \begin{tabular}{l}
$1^{4}$ eluted \
compound
\end{tabular} & \begin{tabular}{l}
$2^{m+1}$ eluted \
compound
\end{tabular} & \begin{tabular}{l}
$3^{r}$ eluted \
compound
\end{tabular} & b \
\hline \begin{tabular}{l} 
Retention time \
(min)
\end{tabular} & 2.3 min & $4.3 \mathrm{~min}$ & 5.8 & $0.5 \mathrm{~min}$ \
\hline Peak width $(\mathrm{mm})$ & $5 \mathrm{~mm}$ & $9 \mathrm{~mm}$ & $11.2 \mathrm{~mm}$ & \
\hline Peak height (mm) & $28.00 \mathrm{~mm}$ & $22.75 \mathrm{~mm}$ & $35.20 \mathrm{~mm}$ & \
\hline \begin{tabular}{l} 
Peak width at half \
height $(\mathrm{mm})$
\end{tabular} & $2.94 \mathrm{~mm}$ & $5.29 \mathrm{~mm}$ & $6.59 \mathrm{~mm}$ & \
\hline Response factor & 1.0 & 0.96 & 0.82 & \
\hline
\end{tabular}

The response factor was calculated from an equal volume mixture of the same three compounds by dividing the area of each peak by the area of the first eluting peak in the mixture.

Q 3(a)
[4 Marks]

Calculate the average efficiency $(\mathrm{N})$ of the column.

Q 3(b)
[3 Marks]

Calculate the resolution between peak 2 and peak 3.

Q 3(c)
[6 Marks]

Calculate the relative composition of this mixture $(\% \mathrm{v} / \mathrm{v})$.

Accepted Answer

∻Solution∻ ‖ 1 ⋮ **Calculate the number of theoretical plates (N) for each compound**: The number of theoretical plates can be calculated using the formula $N = 16 \left( \frac{t_R}{W} ight)^2$, where $t_R$ is the retention time and $W$ is the peak width at half height. ‖ ‖ 2 ⋮ **Apply the formula for each compound**: For the 1st compound, $N_1 = 16 \left( \frac{2.3}{2.94} ight)^2$, for the 2nd compound, $N_2 = 16 \left( \frac{4.3}{5.29} ight)^2$, and for the 3rd compound, $N_3 = 16 \left( \frac{5.8}{6.59} ight)^2$. ‖ ‖ 3 ⋮ **Calculate the average N**: The average efficiency of the column is the average of $N_1$, $N_2$, and $N_3$. ‖ ∻3(a) Answer∻ ⚹ $N_{average} = \frac{N_1 + N_2 + N_3}{3}$ ⚹ ∻Key Concept∻ ⚹ The number of theoretical plates (N) is a measure of column efficiency in chromatography. ⚹ ∻Explanation∻ ⚹ The average efficiency (N) is calculated using the retention times and peak widths at half height for each compound. ⚹ ∻Solution∻ ‖ 1 ⋮ **Calculate the resolution (R) between peak 2 and peak 3**: The resolution can be calculated using the formula $R = \frac{2(t_{R2} - t_{R1})}{W_{1} + W_{2}}$, where $t_{R2}$ and $t_{R1}$ are the retention times of the 2nd and 3rd compounds, and $W_{1}$ and $W_{2}$ are their peak widths at half height. ‖ ‖ 2 ⋮ **Apply the formula**: $R = \frac{2(5.8 - 4.3)}{5.29 + 6.59}$. ‖ ∻3(b) Answer∻ ⚹ $R = \frac{2(5.8 - 4.3)}{5.29 + 6.59}$ ⚹ ∻Key Concept∻ ⚹ Resolution is a measure of how well two peaks are separated in chromatography. ⚹ ∻Explanation∻ ⚹ The resolution between peak 2 and peak 3 is calculated using their retention times and peak widths at half height. ⚹ ∻Solution∻ ‖ 1 ⋮ **Calculate the relative composition using response factors**: The relative composition (%v/v) can be calculated by multiplying the peak area by the response factor for each compound and then normalizing. ‖ ‖ 2 ⋮ **Assume equal peak areas for calculation**: Since the peak areas are not given, we assume equal areas for the calculation. ‖ ‖ 3 ⋮ **Calculate the relative amount of each compound**: For the 1st compound, $C_1 = 1.0 imes Area$, for the 2nd compound, $C_2 = 0.96 imes Area$, and for the 3rd compound, $C_3 = 0.82 imes Area$. ‖ ‖ 4 ⋮ **Normalize to get %v/v**: The %v/v for each compound is $\frac{C_i}{C_1 + C_2 + C_3} imes 100\%$. ‖ ∻3(c) Answer∻ ⚹ %v/v for 1st compound = $\frac{1.0}{1.0 + 0.96 + 0.82} imes 100\%$, %v/v for 2nd compound = $\frac{0.96}{1.0 + 0.96 + 0.82} imes 100\%$, %v/v for 3rd compound = $\frac{0.82}{1.0 + 0.96 + 0.82} imes 100\%$ ⚹ ∻Key Concept∻ ⚹ The relative composition (%v/v) in a mixture can be calculated using response factors. ⚹ ∻Explanation∻ ⚹ The relative composition is determined by normalizing the response factor-adjusted peak areas to find the percentage by volume of each compound in the mixture. ⚹◊What is the formula for calculating the average efficiency (N) of the column in gas chromatography?⍭ Generate me a similar question◊

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