Question

From the data in Problem $27-22$, calculate

($a$) the retention factor $k$for each component.

($b$) selectivity factor $\alpha$ for each adjacent pair of compounds.

($c$) the average number of theoretical plates and plate height for the column.

($d$) the resolution for each adjacent pair of compounds.

Short answer

$a$) methyl acetate $(5.6)$methyl propionate $(12.86);$ methyl butyrate $(25.43)$

$b$) propionate-acetate $(2.3);$ n-butyrate-propionate$\left(2\right)$

$c$) avg. number of plates$\left(1723\right);$ average plate height $(0.06\mathrm{cm})$

$d$) propionate-acetate $(7.51);$ butyrate-propionate$(6.38)$

Step-by-step solution

Retention factor

a) Retention factor $\left(\mathrm{k}\right)$can be calculated via relation $\left(1\right):$

$k=\frac{t_R-t_M}{t_M}$

which where ${\mathrm{t}}_R$is retention time, ${\mathrm{t}}_M$is time of the air from the data and is $0.3$min. The retention factor is calculated in the table for all esters.

Table $1$: Measurement result of retention factor $\mathrm{k}$

Adjacent compound.

$b$) Selectivity factor can be calculated between two adjacent compounds by the formula :

${\alpha }_{n,N}=\frac{k_n}{k_N}$

which uses the retention factor of two compounds. $\mathrm{n}$is for a compound that comes out after $\mathrm{N}$, and we can calculate selectivity factors:

${\alpha }_{2,1}=\frac{k_2}{k_1}=\frac{12.86}{5.6}=2.3$

${\alpha }_{3,2}=\frac{k_3}{k_2}=\frac{25.43}{12.86}=2$

Tables of compounds.

c) Peak widths are stated in the problem and shown in the table for compounds. A number of theoretical plates is calculated by the formula :

$N=16{\left(t_R/W\right)}^2$

where $\mathrm{W}$is peak width and${\mathrm{t}}_R$is given in the table at the beginning of the solution.

Table $1$: Measurement result of plate number

Average number of plates that's calculated is :

$N_{avg}=\frac{1738+1820+1612}{3}=1723$

Average height of plate.

and with column length of $110\mathrm{cm}$we can calculate the average height of the plate :

$H_{avg}=\frac{L}{N}=\frac{110\mathrm{cm}}{1723}=0.0638\mathrm{cm}$

$d$) Resolution is calculated by the formula :

$R_{2,1}:\frac{2\left[{\left(t_R\right)}_2-{\left(t_R\right)}_1\right]}{W_2+W_1}$ ($2$)

which gives the following for methyl acetate $1$and methyl propionate $2$:

$R_{2,1}:\frac{2[4.16-1.98]}{0.19+0.39}=7.51$

and following result for methyl n-butyrate (localid="1650891542448" $3$) and methyl propionate (localid="1650891550664" $2$):

$R_{3,2}:\frac{2[7.93-4.16]}{0.79+0.39}=6.38$