Question

Two components in an HPLC separation have retention times that differ by $15\mathrm{s}$. The first peak elutes in$9.0\min$and the peak widths are approximately equal. The dead time$f_{\mathrm{M}}$was $65\mathrm{s}$. Use a spreadsheet to find the minimal number of theoretical plates needed to achieve the following resolution $R_{\mathrm{n}}$values: $0.50,0.75,0.90,1.0,1.10,1.25,1.50,1.75,2.0,2.5$. How would the results change if peak $2$were twice as broad as peak $1$?

Short answer

When the peak is twice as broad as the peak, R and N grow smaller. As a result, the thin peak is not visible.

Step-by-step solution

Given information

HPLC, or high-pressure liquid chromatography, is an analytical chemistry technique used to separate, identify, and quantify each component in a mixture.

Given:

Time is taken for the first peak $t_R=9.0\text{min}$

Deadtime $t_M=65\text{sec}$

The number of theoretical plates from the given resolution,

localid="1646195084905" $$\begin{gathered} R_s=\frac{1}{4}\sqrt{N}\left(\frac{a-1}{a}\right)\left(\frac{k}{k+1}\right)\dots \dots \left(1\right) \\ \text{where,a=Separationfactor} \\ \text{k=Retentionfactor} \end{gathered}$$

Explanation

Rewrite equation (1), and taking square on both sides,

$N=16R_S{\left(\frac{a}{a-1}\right)}^2{\left(\frac{1+k}{k}\right)}^2$

Then, a and k is calculated by the separating formula, they are given below,

$$\begin{gathered} a=\frac{t_{R2}-t_M}{t_{R1}-t_M} \\ k=\frac{t_{R2}-t_M}{t_M} \end{gathered}$$

Calculate the difference in the retention time,

$$\begin{gathered} t_{R1}-t_{R2}=15\text{sec} \\ {t}_{R1}-t_{R2}=\frac{15}{60}\text{min} \\ =0.25\text{min} \end{gathered}$$

$$\begin{gathered} t_{R2}=9.0+0.25 \\ =9.25min \end{gathered}$$

Explanation

Calculate all the values by the use of excel,

tR1	9
tR2	9.25
tM	1.0833
K	7.53846
a	1.03158
Rs	N
0.25	5476
0.75	12321
1.25	34222.78993
1.5	148986.2218
2	87610.34221
2.5	136891.1597
3	197123.27
3.5	144.0052205
4	350441.3688
4.5	443527.3574

Spreadsheet Documentation

The above table shows the number of plates at various goals.

R_S and N become more modest when peak $2$is twice as wide as pinnacle $1$. Thus, there is no perception of the thin peak.