Chapter 25, Problem 25.13P

(a)

Interpretation Introduction

Interpretation:

The plate number has to be calculated for the two peaks given in the chromatogram.

Explanation of Solution

For L-enantiomer:

Retention time ( $t_r$) for L-enantiomer is given as $4.70\min$.

Half-width is given as $0.28\min$.

The relation between plate number, retention time and half-width is given as,

$N=5.54{\left(\frac{t_r}{w_{1/2}}\right)}^2$

Plate number can be calculated as shown below,

$\begin{array}{l}N=5.54\times \frac{t_r{}^2}{w_{1/2}{}^2}\\ =5.54\times \frac{{\left(4.70\right)}^2}{{\left(0.28\right)}^2}\\ =5.54\times \frac{22.09}{0.0784}\\ =5.54\times 281.76\\ =1560.95\\ =1561\end{array}$

The plate number is calculated as 1561 for L-enantiomer.

For D-enantiomer:

Retention time ( $t_r$) for L-enantiomer is given as $5.37\min$.

Half-width is given as $0.35\min$.

The relation between plate number, retention time and half-width is given as,

$N=5.54{\left(\frac{t_r}{w_{1/2}}\right)}^2$

Plate number can be calculated as shown below,

$\begin{array}{l}N=5.54\times \frac{t_r{}^2}{w_{1/2}{}^2}\\ =5.54\times \frac{{\left(5.37\right)}^2}{{\left(0.35\right)}^2}\\ =5.54\times \frac{28.8369}{0.1225}\\ =5.54\times 235.40\\ =1304.11\\ =1304\end{array}$

The plate number is calculated as 1304 for D-enantiomer.

(b)

Interpretation Introduction

Interpretation:

The resolution has to be calculated for the two peaks given in the chromatogram.

Explanation of Solution

Retention time ( $t_r$) for L-enantiomer is given as $4.70\min$.

Retention time ( $t_r$) for L-enantiomer is given as $5.37\min$.

Half-width for L-enantiomer is given as $0.28\min$ and therefore width is equal to $0.56\min$.

Half-width for D-enantiomer is given as $0.35\min$ and therefore width is equal to $0.70\min$.

The resolution can be calculated using the formula as given below,

$\begin{array}{l}\text{Resolution}\text{=}\frac{{\left(t_r\right)}_{\text{D}}-{\left(t_r\right)}_{\text{L}}}{\left(W_D+W_L\right)/2}\\ =\frac{5.37-4.70}{\left(0.56+0.70\right)/2}\\ =\frac{0.67}{\left(1.26\right)/2}\\ =\frac{0.67}{0.63}\\ =1.06\end{array}$

(c)

Interpretation Introduction

Interpretation:

The resolution has to be predicted for the average plate number in the chromatogram.

Explanation of Solution

Retention time ( $t_r$) for L-enantiomer is given as $4.70\min$.

Retention time ( $t_r$) for L-enantiomer is given as $5.37\min$.

$t_m$ is given as $1.62\min$.

The average plate number for the D-enantiomer and L-enantiomer can be calculated as shown below,

$\begin{array}{l}\text{Average plate number}\text{=}\frac{1561+1304}{2}\\ =\frac{2865}{2}\\ =1432\end{array}$

Retention factor can be calculated as shown below,

$\begin{array}{l}k=\frac{t_r-t_m}{t_m}\\ =\frac{5.37-1.62}{1.62}\\ =\frac{3.75}{1.62}\\ =2.31\end{array}$

Relative retention is calculated as shown below,

$\begin{array}{l}\alpha =\frac{5.37\min }{4.70\min }\\ =1.142\end{array}$

The resolution can be calculated using the formula as given below,

$\text{Resolution}\text{=}\frac{\sqrt{N}}{4}\times \frac{\alpha -1}{\alpha }\times \left(\frac{k}{1+k}\right)$

Where,

$N$ is the plate number

$\alpha$ is the relative retention

$k$ is the retention factor

$\begin{array}{l}\text{Resolution}\text{=}\frac{\sqrt{1432}}{4}\times \frac{1.142-1}{1.142}\times \left(\frac{2.31}{1+2.31}\right)\\ =\frac{37.84}{4}\times \frac{0.142}{1.142}\times \left(\frac{2.31}{3.31}\right)\\ =9.46\times 0.124\times 0.697\\ =0.817\end{array}$