Chapter 23, Problem 23.32P

(a)

Interpretation Introduction

Interpretation:

The column with more number of theoretical plates has to be identified.

Concept introduction:

Plates of column:

The relationship between $\text{N}\text{and}{\text{w}}_{\text{1/2}}$

$\begin{array}{l}\text{N}\text{=}\frac{\text{5}\text{.55}{\text{t}}_{\text{r}}{}^{\text{2}}}{{\text{w}}_{\text{1/2}}{}^{\text{2}}}\\ \text{Where,}\\ \text{N}\text{-}\text{number}\text{of}\text{plates}\text{on}\text{column}\\ {\text{t}}_{\text{r}}\text{-}\text{retention}\text{time}\\ {\text{w}}_{\text{1/2}}\text{-}\text{half-width}\end{array}$

Answer to Problem 23.32P

• Column 1 has more number of theoretical plates.

Explanation of Solution

Given information,

Figure 1

The relationship between $\text{N}\text{and}{\text{w}}_{\text{1/2}}$

$\begin{array}{l}\text{N}\text{=}\frac{\text{5}\text{.55}{\text{t}}_{\text{r}}{}^{\text{2}}}{{\text{w}}_{\text{1/2}}{}^{\text{2}}}\\ \text{Where,}\\ \text{N}\text{-}\text{number}\text{of}\text{plates}\text{on}\text{column}\\ {\text{t}}_{\text{r}}\text{-}\text{retention}\text{time}\\ {\text{w}}_{\text{1/2}}\text{-}\text{half-width}\end{array}$

The above relationship shows that column 1 has more number of theoretical plates.

(b)

Interpretation Introduction

Interpretation:

The column with larger plate height has to be identified.

Concept introduction:

Plate height:

The relationship between $\text{H}\text{and}\text{L}$ is given by

$\text{H}\text{=}\frac{\text{Length}\text{of}\text{the}\text{column}}{\text{Number}\text{of}\text{plates}\text{on}\text{column}}\text{=}\frac{\text{L}}{\text{N}}$

Answer to Problem 23.32P

• Column 2 has larger plate height.

Explanation of Solution

Given information,

Figure 1

The relationship between $\text{H}\text{and}\text{L}$ is given by

$\text{H}\text{=}\frac{\text{L}}{\text{N}}\text{=}\frac{\text{Length}\text{of}\text{the}\text{column}}{\text{Number}\text{of}\text{plates}\text{on}\text{column}}$

The relationship between $\text{N}\text{and}{\text{w}}_{\text{1/2}}$

$\begin{array}{l}\text{N}\text{=}\frac{\text{5}\text{.55}{\text{t}}_{\text{r}}{}^{\text{2}}}{{\text{w}}_{\text{1/2}}{}^{\text{2}}}\\ \text{Where,}\\ \text{N}\text{-}\text{number}\text{of}\text{plates}\text{on}\text{column}\\ {\text{t}}_{\text{r}}\text{-}\text{retention}\text{time}\\ {\text{w}}_{\text{1/2}}\text{-}\text{half-width}\end{array}$

The above relationship shows that column 1 has more number of theoretical plates.

Column 2 has larger plate height.  Since column 1 has higher value of $\text{N}$.

(c)

Interpretation Introduction

Interpretation:

The column which will give higher resolution has to be identified.

Concept introduction:

Purnell equation for resolution:

The relationship between $\text{H}\text{and}\text{α}$ is given by Purnell equation,

$\begin{array}{l}\text{Resolution}\text{=}\frac{\sqrt{\text{N}}}{\text{4}}\frac{\left(\text{α}\text{-}\text{1}\right)}{\text{α}}\left(\frac{{\text{k}}_{\text{2}}}{\text{1}\text{+}\text{}{\text{k}}_{\text{2}}}\right)\\ \text{Where,}\\ \text{N}\text{-}\text{number}\text{of}\text{theoretical}\text{plates}\\ \text{α}\text{-}\text{relative}\text{retention}\text{of}\text{the}\text{two}\text{peaks}\\ {\text{k}}_{\text{2}}\text{-}\text{retention}\text{factor}\text{of}\text{the}\text{more}\text{retained}\text{component}\end{array}$

Answer to Problem 23.32P

• Column 1 will give a greater resolution.

Explanation of Solution

Given information,

Figure 1

The relationship between $\text{H}\text{and}\text{α}$ is given by Purnell equation,

$\begin{array}{l}\text{Resolution}\text{=}\frac{\sqrt{\text{N}}}{\text{4}}\frac{\left(\text{α}\text{-}\text{1}\right)}{\text{α}}\left(\frac{{\text{k}}_{\text{2}}}{\text{1}\text{+}\text{}{\text{k}}_{\text{2}}}\right)\\ \text{Where,}\\ \text{N}\text{-}\text{number}\text{of}\text{theoretical}\text{plates}\\ \text{α}\text{-}\text{relative}\text{retention}\text{of}\text{the}\text{two}\text{peaks}\\ {\text{k}}_{\text{2}}\text{-}\text{retention}\text{factor}\text{of}\text{the}\text{more}\text{retained}\text{component}\end{array}$

The above equation shows that the value of resolution is proportional to $\sqrt{\text{N}\text{.}}$

The above relationship and figure 1 shows that column 1 will gives a greater resolution.

(d)

Interpretation Introduction

Interpretation:

The column which will give a greater relative retention has to be identified.

Concept introduction:

Relative retention:

The ratio of the adjusted retention time of two components is given by relative retention $\text{α}$

$\begin{array}{l}\text{α}\text{=}\frac{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}}{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}}\\ \text{Where,}\\ \text{α}\text{-}\text{relative}\text{retention}\left(\text{also}\text{called}\text{separation}\text{factor}\right)\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{2}\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{1}\\ \text{The}\text{separation}\text{between}\text{the}\text{two}\text{component}\text{will}\text{be}\text{higher}\text{when}{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}\text{>}{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}\text{,}\text{so}\text{α}\text{>}\text{1}\end{array}$

Answer to Problem 23.32P

• Neither column 1 nor column 2 will give a greater relative retention.

Explanation of Solution

Given information,

Figure 1

The ratio of the adjusted retention time of two components is given by relative retention $\text{α}$

$\begin{array}{l}\text{α}\text{=}\frac{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}}{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}}\\ \text{Where,}\\ \text{α}\text{-}\text{relative}\text{retention}\left(\text{also}\text{called}\text{separation}\text{factor}\right)\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{2}\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{1}\end{array}$

Calculate the value of relative retention

$\begin{array}{l}\text{α}\text{=}\frac{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}}{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}}\text{=}\frac{{\text{t}}_{\text{r2}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{r1}}\text{-}{\text{t}}_{\text{m}}}\\ \\ \text{=}\frac{\text{10}\text{-}\text{1}\text{.3}}{\text{8}\text{-}\text{1}\text{.3}}\text{=}\frac{\text{8}\text{.7}}{\text{6}\text{.7}}\text{=}\text{1}\text{.2985}\text{=}\text{1}\text{.3}\end{array}$

The numerical value of the relative retention is $\text{1}\text{.3}$.

The above relationship shows that neither column 1 nor column 2 will gives higher relative retention.

(e)

Interpretation Introduction

Interpretation:

The compound with higher retention factor has to be identified.

Concept introduction:

Retention factor:

The retention factor $\left(\text{k}\right)$ is given by

$\begin{array}{l}\text{k}\text{=}\frac{{\text{t}}_{\text{r}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{m}}}\\ \text{Where,}\\ \text{k}\text{-}\text{retention}\text{factor}\\ {\text{t}}_{\text{r}}\text{-}\text{time}\text{required}\text{to}\text{elute}\text{the}\text{peak}\\ {\text{t}}_{\text{m}}\text{-}\text{time}\text{required}\text{for}\text{mobile}\text{phase}\text{to}\text{pass}\text{through}\text{the}\text{column}\text{.}\end{array}$

Answer to Problem 23.32P

• Compound B has higher retention factor.

Explanation of Solution

Given information,

Figure 1

The retention factor $\left(\text{k}\right)$ is given by

$\begin{array}{l}\text{k}\text{=}\frac{{\text{t}}_{\text{r}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{m}}}\\ \text{Where,}\\ \text{k}\text{-}\text{retention}\text{factor}\\ {\text{t}}_{\text{r}}\text{-}\text{time}\text{required}\text{to}\text{elute}\text{the}\text{peak}\\ {\text{t}}_{\text{m}}\text{-}\text{time}\text{required}\text{for}\text{mobile}\text{phase}\text{to}\text{pass}\text{through}\text{the}\text{column}\text{.}\end{array}$

The above relationship shows that compound B has higher retention factor.

(f)

Interpretation Introduction

Interpretation:

The compound with greater partition coefficient has to be identified.

Concept introduction:

Partition coefficient:

Partition of a solute is depends on “like dissolves like” concept.  This means the solubility of solute is more in a solvent whose polarity is similar to that of the solute.

The partition coefficient of solute $\left(\text{S}\right)$ is given by

$\text{Solute}\left(\text{in}\text{phase}\text{1}\right)\stackrel{}{\rightleftharpoons }\text{Solute}\left(\text{in}\text{phase}\text{2}\right)$

$\begin{array}{l}\text{K=}\frac{{\text{A}}_{{\text{S}}_{\text{2}}}}{{\text{A}}_{{\text{S}}_{\text{1}}}}\approx \frac{{\left[\text{S}\right]}_{\text{2}}}{{\left[\text{S}\right]}_{\text{1}}}\\ \text{where,}\\ \text{K}\text{-}\text{partition}\text{coefficient}\\ {\text{A}}_{{\text{S}}_{\text{1}}}\text{-}\text{Activity}\text{of}\text{solute}\text{in}\text{phase}\text{1}\\ {\text{A}}_{{\text{S}}_{\text{2}}}\text{-}\text{Activity}\text{of}\text{solute}\text{in}\text{phase}\text{2}\\ {\left[\text{S}\right]}_{\text{1}}\text{-}\text{Concentration}\text{of}\text{solute}\text{in}\text{phase}\text{1}\\ {\left[\text{S}\right]}_{\text{2}}\text{-}\text{Concentration}\text{of}\text{solute}\text{in}\text{phase}\text{2}\end{array}$

Answer to Problem 23.32P

• Compound B has higher partition coefficient.

Explanation of Solution

Given information,

Figure 1

Partition of a solute is depends on “like dissolves like” concept.  This means the solubility of solute is more in a solvent whose polarity is similar to that of the solute.

The partition coefficient $\left(\text{K}\right)$ of solute $\left(\text{S}\right)$ is given by

$\begin{array}{l}\text{K=}\frac{{\text{A}}_{{\text{S}}_{\text{2}}}}{{\text{A}}_{{\text{S}}_{\text{1}}}}\approx \frac{{\left[\text{S}\right]}_{\text{2}}}{{\left[\text{S}\right]}_{\text{1}}}\\ \text{where,}\\ \text{K}\text{-}\text{partition}\text{coefficient}\\ {\text{A}}_{{\text{S}}_{\text{1}}}\text{-}\text{Activity}\text{of}\text{solute}\text{in}\text{phase}\text{1}\\ {\text{A}}_{{\text{S}}_{\text{2}}}\text{-}\text{Activity}\text{of}\text{solute}\text{in}\text{phase}\text{2}\\ {\left[\text{S}\right]}_{\text{1}}\text{-}\text{Concentration}\text{of}\text{solute}\text{in}\text{phase}\text{1}\\ {\left[\text{S}\right]}_{\text{2}}\text{-}\text{Concentration}\text{of}\text{solute}\text{in}\text{phase}\text{2}\end{array}$

The above relationship shows that Compound B has higher partition coefficient.

(g)

Interpretation Introduction

Interpretation:

The numerical value of the retention factor has to be calculated for peak A.

Concept introduction:

Retention factor:

The retention factor $\left(\text{k}\right)$ is given by

$\begin{array}{l}\text{k}\text{=}\frac{{\text{t}}_{\text{r}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{m}}}\\ \text{Where,}\\ \text{k}\text{-}\text{retention}\text{factor}\\ {\text{t}}_{\text{r}}\text{-}\text{time}\text{required}\text{to}\text{elute}\text{the}\text{peak}\\ {\text{t}}_{\text{m}}\text{-}\text{time}\text{required}\text{for}\text{mobile}\text{phase}\text{to}\text{pass}\text{through}\text{the}\text{column}\text{.}\end{array}$

Answer to Problem 23.32P

• The numerical value of the retention factor for peak A is $\text{5}\text{.2}$.

Explanation of Solution

Given information,

Figure 1

The retention factor $\left(\text{k}\right)$ is given by

$\begin{array}{l}\text{k}\text{=}\frac{{\text{t}}_{\text{r}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{m}}}\\ \\ \text{=}\frac{\text{8}\text{-}\text{1}\text{.3}}{\text{1}\text{.3}}\text{=}\text{5}\text{.1538}\text{=}\text{5}\text{.2}\text{.}\end{array}$

The numerical value of the retention factor for peak A is $\text{5}\text{.2}$.

(h)

Interpretation Introduction

Interpretation:

The numerical value of the retention factor has to be calculated for peak B.

Concept introduction:

Retention factor:

The retention factor $\left(\text{k}\right)$ is given by

$\begin{array}{l}\text{k}\text{=}\frac{{\text{t}}_{\text{r}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{m}}}\\ \text{Where,}\\ \text{k}\text{-}\text{retention}\text{factor}\\ {\text{t}}_{\text{r}}\text{-}\text{time}\text{required}\text{to}\text{elute}\text{the}\text{peak}\\ {\text{t}}_{\text{m}}\text{-}\text{time}\text{required}\text{for}\text{mobile}\text{phase}\text{to}\text{pass}\text{through}\text{the}\text{column}\text{.}\end{array}$

Answer to Problem 23.32P

• The numerical value of the retention factor for peak B is $\text{6}\text{.7}$.

Explanation of Solution

Given information,

Figure 1

The retention factor $\left(\text{k}\right)$ is given by

$\begin{array}{l}\text{k}\text{=}\frac{{\text{t}}_{\text{r}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{m}}}\\ \\ \text{=}\frac{\text{10}\text{-}\text{1}\text{.3}}{\text{1}\text{.3}}\text{=}\text{6}\text{.6923}\text{=}\text{6}\text{.7}\text{.}\end{array}$

The numerical value of the retention factor for peak B is $\text{6}\text{.7}$.

(i)

Interpretation Introduction

Interpretation:

The numerical value of the relative retention has to be identified.

Concept introduction:

Relative retention:

The ratio of the adjusted retention time of two components is given by relative retention $\text{α}$

$\begin{array}{l}\text{α}\text{=}\frac{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}}{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}}\\ \text{Where,}\\ \text{α}\text{-}\text{relative}\text{retention}\left(\text{also}\text{called}\text{separation}\text{factor}\right)\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{2}\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{1}\\ \text{The}\text{separation}\text{between}\text{the}\text{two}\text{component}\text{will}\text{be}\text{higher}\text{when}{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}\text{>}{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}\text{,}\text{so}\text{α}\text{>}\text{1}\end{array}$

Answer to Problem 23.32P

• The numerical value of the relative retention is $\text{1}\text{.3}$.

Explanation of Solution

Given information,

Figure 1

The ratio of the adjusted retention time of two components is given by relative retention $\text{α}$

$\begin{array}{l}\text{α}\text{=}\frac{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}}{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}}\\ \text{Where,}\\ \text{α}\text{-}\text{relative}\text{retention}\left(\text{also}\text{called}\text{separation}\text{factor}\right)\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{2}\\ {\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}\text{-}\text{adjusted}\text{retention}\text{time}\text{of}\text{component}\text{1}\end{array}$

Calculate the value of relative retention

$\begin{array}{l}\text{α}\text{=}\frac{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{2}}}{{\text{t}}_{\text{r}}^{\text{'}}{}_{\text{1}}}\text{=}\frac{{\text{t}}_{\text{r2}}\text{-}{\text{t}}_{\text{m}}}{{\text{t}}_{\text{r1}}\text{-}{\text{t}}_{\text{m}}}\\ \\ \text{=}\frac{\text{10}\text{-}\text{1}\text{.3}}{\text{8}\text{-}\text{1}\text{.3}}\text{=}\frac{\text{8}\text{.7}}{\text{6}\text{.7}}\text{=}\text{1}\text{.2985}\text{=}\text{1}\text{.3}\end{array}$

The numerical value of the relative retention is $\text{1}\text{.3}$.