Question

Solvent occupies15% of the volume of a chromatography column whose inner diameter is 3mm. If the volume flow rate is 0.2mL / min, find the linear velocity.

Short answer

Linear velocity is${\mathrm{u}}_{\mathrm{x}}=18.87\mathrm{cm}/\min$

Step-by-step solution

Given Information:

${\mathrm{V}}_{\mathrm{solvent}}=0.15{\mathrm{V}}_{\mathrm{column}},\mathrm{d}=3\mathrm{mm}=0.3\mathrm{cm},\mathrm{r}=\mathrm{d}/2=0.15\mathrm{cm},{\mathrm{u}}_{\mathrm{V}}=0.2\mathrm{mL}/\min$

To find the volume of solvent:Volume of solvent can be expressed as per length of a column as

$$\begin{gathered} {\mathrm{V}}_{\mathrm{solvent}/\mathrm{cm}}=0.15{\mathrm{V}}_{\mathrm{column}} \\ {\mathrm{V}}_{\mathrm{solvent}/\mathrm{cm}}=0.15\times \left({\mathrm{\pi r}}^2\mathrm{I}\right) \\ \mathrm{As}\mathrm{length}\mathrm{I}=1\mathrm{cm} \\ {\mathrm{V}}_{\mathrm{solvent}/\mathrm{cm}}=0.15\times (\mathrm{\pi }\times {\left(0.15\mathrm{cm}\right)}^2\times 1\mathrm{cm} \\ {\mathrm{V}}_{\mathrm{solvent}/\mathrm{cm}}=0.0106{\mathrm{cm}}^3 \\ {\mathrm{V}}_{\mathrm{solvent}/\mathrm{cm}}=0.0106\mathrm{mL} \\ {\mathrm{V}}_{\mathrm{solvent}/\mathrm{cm}}=0.0106\mathrm{mL}/1\mathrm{cm} \end{gathered}$$

To find the linear velocity:

$$\begin{gathered} {\mathrm{u}}_{\mathrm{x}}=\frac{{\mathrm{u}}_{\mathrm{V}}}{{\mathrm{V}}_{\mathrm{solvent}/\mathrm{cm}}} \\ {\mathrm{u}}_{\mathrm{x}}=\frac{0.2\mathrm{mL}/\min }{0.0106\mathrm{mL}/\mathrm{cm}} \\ {\mathrm{u}}_{\mathrm{x}}=18.87\mathrm{cm}/\min \end{gathered}$$