Problem 80 - Analytical Chemistry

Question

PROFESSOR'S FAVORITE PROBLEM Contributed by Professor Milton L. Lee, Brigham Young University A student was asked to separate two substances, \(\mathrm{A}\) and \(\mathrm{B},\) on a \(30.0 \mathrm{~cm}\) column. She obtained a chromatogram that gave retention times of 15.80 and 17.23 min for \(A\) and \(B\), respectively, and an elution time of an unretained compound of \(1.60 \mathrm{~min}\). The base peak widths for \(\mathrm{A}\) and \(\mathrm{B}\) were 1.25 and 1.38 min, respectively. Please calculate (a) the average number of theoretical plates for the column, (b) the plate height, (c) the resolution of \(\mathrm{A}\) and \(\mathrm{B},\) (d) the length of column that would be required to achieve a resolution of \(1.5,\) and (e) the time required to elute compounds A and \(B\) on the longer column.

Step-by-Step Solution

Answer

(a) 1580.42 plates, (b) 0.0190 cm/plate, (c) 1.09, (d) 56.5 cm column, (e) 29.77 min for A and 32.49 min for B.

Step 1: Calculate the Number of Theoretical Plates

The number of theoretical plates (N) for each compound can be calculated using the formula: \[ N = 16 \left( \frac{t_R}{W} \right)^2 \]where \( t_R \) is the retention time and \( W \) is the base peak width. Calculate \( N_A \) for compound A:\[ N_A = 16 \left( \frac{15.80}{1.25} \right)^2 = 1608.64 \]Calculate \( N_B \) for compound B:\[ N_B = 16 \left( \frac{17.23}{1.38} \right)^2 = 1552.20 \]Now, find the average number of theoretical plates:\[ N_{avg} = \frac{1608.64 + 1552.20}{2} = 1580.42 \]

Step 2: Calculate the Plate Height

Plate height (H) is given by \( H = \frac{L}{N} \) where \( L \) is the column length in centimeters. Use the average number of theoretical plates from Step 1:\[ H = \frac{30.0 \text{ cm}}{1580.42} = 0.01898 \text{ cm/plate} \approx 0.0190 \text{ cm/plate} \]

Step 3: Calculate the Resolution of A and B

The resolution \( R_s \) between two compounds is expressed as:\[ R_s = \frac{2(t_{R,B} - t_{R,A})}{W_A + W_B} \]where \( t_{R,B} \) and \( t_{R,A} \) are the retention times, and \( W_A \) and \( W_B \) are the base widths. Substituting values:\[ R_s = \frac{2(17.23 - 15.80)}{1.25 + 1.38} = \frac{2(1.43)}{2.63} \approx 1.09 \]

Step 4: Calculate the Required Column Length for Resolution 1.5

To achieve a resolution \( R_s' = 1.5 \), use the formula:\[ L' = L \left( \frac{R_s'}{R_s} \right)^2 \]Substituting the values, we have:\[ L' = 30.0 \left( \frac{1.5}{1.09} \right)^2 \approx 56.47 \text{ cm} \approx 56.5 \text{ cm} \]

Step 5: Calculate the Elution Time on Longer Column

The elution time \( t_R' \) for a column with increased length \( L' \) is proportional to the increase in column length:\[ t_R' = t_R \times \frac{L'}{L} \]For compound A:\[ t_{RA}' = 15.80 \times \frac{56.5}{30.0} = 29.77 \text{ min} \]For compound B:\[ t_{RB}' = 17.23 \times \frac{56.5}{30.0} = 32.49 \text{ min} \]

Key Concepts

Chromatography

Chromatography is a critical analytical method used to separate components in a mixture. In this example, we focus on separating substances A and B using a chromatographic column. The process depends on the differential partitioning between the mobile phase, which travels through the column, and the stationary phase, which is fixed inside the column.
For successful separation, components are carried at different speeds through the column based on their interactions with the stationary phase. By measuring retention times, or the times required for each component to pass through the column, scientists can identify and quantify the substances in a sample.

• Retention time is influenced by both the chemical nature of the compound and the column's conditions.
• Greater differences in retention time indicate better separation between substances.
• The method can be adapted for various kinds of analysis, including liquid and gas chromatography.

Understanding the retention time is vital, as it represents how long a substance is detected in the chromatograph. It provides the basis for calculating other values such as theoretical plates and resolution.

Plate Height

Plate height (H), or Height Equivalent to a Theoretical Plate (HETP), is a measure of column efficiency in chromatography. It provides insight into how well the column separates components over a specific length.
The calculation is straightforward: divide the column's length (in centimeters) by the average number of theoretical plates. Smaller plate heights indicate higher resolution and efficiency, which are desirable for effective separation of analytes.
Mathematically, this is expressed as:

• \[ H = \frac{L}{N} \]
• L = column length,
• N = number of theoretical plates.

A lower plate height optimizes conditions for better separation, with sharper and narrower peaks, which are evident on the chromatogram. As seen in the example, the calculated plate height was approximately 0.019 cm/plate, indicating a fairly effective column for the task at hand.

Resolution in Chromatography

The resolution measuring the ability of a chromatographic system to distinguish between two separated analytes is crucial. In simple terms, it tells how well two substances like A and B are separated in the column.
Resolution (R_s) is generally calculated using:

• \[ R_s = \frac{2(t_{R,B} - t_{R,A})}{W_A + W_B} \]
• t_R,B and t_R,A are retention times for substances B and A,
• W_A and W_B are the base widths of the peaks for substances A and B.

In the practical example, the resolution of substances A and B was calculated as approximately 1.09, which is below the optimal value. Chromatographers often aim for an R_s value of at least 1.5 for a successful separation. Improving resolution can involve altering the column length, mobile phase composition, or temperature settings.

Column Length Calculation

Column length is a key factor impacting both the resolution and efficiency of chromatographic separation. Modifying the column length can help achieve desired separation quality.
To achieve a certain resolution (for example, R_s = 1.5), one might adjust the column length using:

• \[ L' = L \left( \frac{R_s'}{R_s} \right)^2 \]
• L is the initial column length,
• L’ is the new column length needed,
• R_s' is the desired resolution,
• R_s is the observed resolution.

The exercise demonstrated that increasing the column length to approximately 56.5 cm would achieve a resolution of 1.5. Doing so will proportionally extend the retention times for the elution of compounds A and B, but with improved separation quality. Column length adjustments are often required in the practical lab settings to meet specific separation goals.