Question

(a) Write the meaning of the retention factor, , in terms of time spent by solute in each phase. (b) Write an expression in terms of for the fraction of time spent by a solute molecule in the mobile phase. (c) The retention ratio in chromatography is defined as

$\mathbf{R}\mathbf{=}\frac{\mathbf{time}\mathbf{}\mathbf{for}\mathbf{}\mathbf{solvent}}{\mathbf{time}\mathbf{}\mathbf{for}\mathbf{}\mathbf{solute}\mathbf{}\mathbf{to}\mathbf{}\mathbf{pass}\mathbf{}\mathbf{through}\mathbf{}\mathbf{column}}\mathbf{=}\frac{{\mathbf{t}}_{\mathbf{m}}}{{\mathbf{t}}_{\mathbf{r}}}$Show that is related to the retention factor by the equation$\mathbf{R}\mathbf{=}\mathbf{1}\mathbf{/}\mathbf{k}\mathbf{+}\mathbf{1}$

Short answer

$$\begin{gathered} \mathrm{a})\mathrm{k}=\frac{{\mathrm{t}}_{\mathrm{s}}}{{\mathrm{t}}_{\mathrm{m}}} \\ \mathrm{b})\Rightarrow \frac{1}{1+\mathrm{k}} \\ \mathrm{c})\mathrm{R}=\frac{1}{1+\mathrm{k}} \end{gathered}$$

Step-by-step solution

Define Retention Factor:

The retention factor, also known as the capacity factor (k), in column

chromatography is defined as the ratio of time an analyte is retained in the

stationary phase to time

To find the retention Factor :

a) To calculate the retention factor as

$$\begin{gathered} \mathrm{k}=\frac{\mathrm{t}\left(\mathrm{solute}\mathrm{in}\mathrm{the}\mathrm{stationary}\mathrm{phase}\right)}{\mathrm{t}(\mathrm{solute}\mathrm{in}\mathrm{the}\mathrm{mobile}\mathrm{phase})} \\ \mathrm{k}=\frac{{\mathrm{t}}_{\mathrm{r}}-{\mathrm{t}}_{\mathrm{m}}}{{\mathrm{t}}_{\mathrm{m}}} \\ \mathrm{k}=\frac{{\mathrm{t}}_{\mathrm{s}}}{{\mathrm{t}}_{\mathrm{m}}} \end{gathered}$$

${\mathrm{t}}_{\mathrm{r}}$is actually the retention time of solute of interest

To find the fraction of time spent in the mobile phase:

b) Fraction of time spent in the mobile phase ${\mathrm{t}}_{\mathrm{m}}$would be

$$\begin{gathered} \frac{{\mathrm{t}}_{\mathrm{m}}}{{\mathrm{t}}_{\mathrm{m}}+{\mathrm{t}}_{\mathrm{s}}} \\ \mathrm{k}=\frac{{\mathrm{t}}_{\mathrm{s}}}{{\mathrm{t}}_{\mathrm{m}}} \\ {\mathrm{t}}_{\mathrm{s}}=\mathrm{k}\times {\mathrm{t}}_{\mathrm{m}} \end{gathered}$$

Substitute the expression

$$\begin{gathered} \Rightarrow \frac{{\mathrm{t}}_{\mathrm{m}}}{{\mathrm{t}}_{\mathrm{m}}+{\mathrm{t}}_{\mathrm{s}}} \\ \Rightarrow \frac{{\mathrm{t}}_{\mathrm{m}}}{{\mathrm{t}}_{\mathrm{m}}+\mathrm{k}\times {\mathrm{t}}_{\mathrm{m}}} \\ \Rightarrow \frac{{\mathrm{t}}_{\mathrm{m}}}{{\mathrm{t}}_{\mathrm{m}}+\mathrm{k}\times {\mathrm{t}}_{\mathrm{m}}} \\ \frac{1}{1+\mathrm{k}\times 1} \\ \frac{1}{1+\mathrm{k}} \end{gathered}$$

To find the retention Factor in chromatography:

c) The retention ration in chromatography is defined as

$$\begin{gathered} \mathrm{R}=\frac{\mathrm{time}\mathrm{for}\mathrm{solvent}}{\mathrm{time}\mathrm{for}\mathrm{solute}\mathrm{to}\mathrm{pass}\mathrm{through}} \\ \mathrm{R}=\frac{{\mathrm{t}}_{\mathrm{m}}}{{\mathrm{t}}_{\mathrm{r}}} \\ \mathrm{R}=\frac{{\mathrm{t}}_{\mathrm{m}}}{{\mathrm{t}}_{\mathrm{m}}+{\mathrm{t}}_{\mathrm{s}}} \\ \mathrm{R}=\frac{1}{1+\mathrm{k}} \end{gathered}$$