Problem 11 - Analytical Chemistry

Question

Prepare a spreadsheet for a van Deemter plot for the following hypothetical, \(A\), \(B,\) and \(C\) terms: \(A=0.5 \mathrm{~mm}, B=30 \mathrm{~mm} \cdot \mathrm{mL} / \mathrm{min},\) and \(C=0.05 \mathrm{~mm} \cdot \mathrm{min} / \mathrm{mL}\) Plot \(H\) vs. \(\bar{u}\) at linear velocities of 4,8,12,20,28,40,80 and \(120 \mathrm{~mL} / \mathrm{min}\). Also, on the same chart, plot \(A\) vs. \(\bar{u}, B / \bar{u}\) vs. \(\bar{u},\) and \(C \bar{u}\) vs. \(\bar{u},\) and note how they change with the linear velocity, that is, how their contributions to \(H\) change. Calculate the hypothetical \(H_{\min }\) and \(\bar{u}_{\text {opt }}\) and compare with the \(H_{\min }\) on the chart. Also calculate \(B / \bar{u}_{\text {of }}\) and \(C \bar{u}_{\text {ont }}\) Look on the chart and see where the \(B / \bar{u}\) and \(C \bar{u}\) lines cross. Check your results with those in your CD, Chapter \(19 .\)

Step-by-Step Solution

Answer

Use the van Deemter equation to calculate and plot H vs. velocity, identifying H_min and u_opt, and note crossover points of B/u and Cu.

Step 1: Understanding the van Deemter Equation

The van Deemter equation is expressed as: \( H = A + \frac{B}{\bar{u}} + C \cdot \bar{u} \) where \( H \) is the height equivalent to a theoretical plate, \( A \), \( B \), and \( C \) are constants related to different dispersion processes, and \( \bar{u} \) is the linear velocity.

Step 2: Set Up Spreadsheet

Open a spreadsheet application like Excel or Google Sheets. Label your columns for calculations: \( \bar{u} \), \( H \), \( A \), \( \frac{B}{\bar{u}} \), and \( C \cdot \bar{u} \). Input the given linear velocities: 4, 8, 12, 20, 28, 40, 80, and 120 mL/min in the first column.

Step 3: Calculate H Values

For each linear velocity, calculate \( H \) using the van Deemter equation. For example, at \( \bar{u} = 4 \), \( H = 0.5 + \frac{30}{4} + 0.05 \times 4 \). Do this for each velocity value and fill in the \( H \) column.

Step 4: Calculate A, B/u, and C*u Components

In the spreadsheet, repeat calculations for the \( A \) term (which remains constant at 0.5 mm), the \( \frac{B}{\bar{u}} \) term by dividing \( B = 30 \) by each \( \bar{u} \), and \( C \cdot \bar{u} \) by multiplying \( C = 0.05 \) with each \( \bar{u} \). Record these results in their respective columns.

Step 5: Plot the Graphs

Use the spreadsheet's graphing tools to create a plot. Plot \( H \), \( A \), \( \frac{B}{\bar{u}} \), and \( C \cdot \bar{u} \) against the linear velocities \( \bar{u} \) on the same chart. Label the axes appropriately and ensure each line is distinct.

Step 6: Calculate Hmin and u_opt

Calculate the hypothetical minimum \( H \) (\( H_{min} \)) and its corresponding optimal velocity \( \bar{u}_{opt} \) using the expressions: \( H_{min} = A + 2\sqrt{BC} \) and \( \bar{u}_{opt} = \sqrt{\frac{B}{C}} \). Use these formulas to determine actual points and compare.

Step 7: Determine Crossover Points

In the graph, observe the points where \( \frac{B}{\bar{u}} \) and \( C \cdot \bar{u} \) intersect. Verify the crossover points and compare them to \( H_{min} \) observed on the chart.

Key Concepts

Chromatography

Chromatography is a crucial analytical technique used to separate components in mixtures. It works by distributing these components between two phases: a stationary phase and a mobile phase. The stationary phase remains fixed inside a column, while the mobile phase moves through the column. As the sample components travel with the mobile phase at different rates, they separate based on their interactions with the stationary phase and their different affinities for it. Chromatography is widely used in laboratories for analysis and purification purposes. It can handle complex mixtures and is crucial in applications such as pharmaceuticals, environmental testing, and food safety.

Plate Height

The concept of plate height, denoted by \(H\), is a parameter in chromatography that measures the efficiency of a column. It is often described as the height equivalent to a theoretical plate, where one plate represents one complete separation of analyte molecules. The smaller the plate height, the more efficient the column is, resulting in better resolution between separated compounds.

• A small \(H\) value indicates better separation efficiency.
• It is calculated using the van Deemter equation: \(H = A + \frac{B}{\bar{u}} + C \cdot \bar{u}\).
• Plate height is affected by dispersion processes represented by constants \(A\), \(B\), and \(C\), and the linear velocity \(\bar{u}\) of the mobile phase.

Linear Velocity

Linear velocity (\(\bar{u}\)) is a crucial factor in chromatography, representing the speed at which the mobile phase travels through the column. It impacts how quickly sample components pass through and separate.

• It is measured in units such as \(\text{mL/min}\).
• Affects the van Deemter equation, influencing the efficiency of separation and plate height.
• Changing linear velocity can alter the balance of the \(A\), \(B\), and \(C\) terms, thereby affecting the overall separation efficiency.
• Finding the optimal \(\bar{u}\) is critical to achieving the best separation of components.

Dispersion Processes

Dispersion processes are factors contributing to band broadening in chromatographic column analysis. The van Deemter equation notably categorizes these into three main processes defined by constants $A$, $B$, and $C$.

• Eddy Diffusion ($A$ term): This occurs due to the multiple pathways a molecule can take through the packed column, causing it to spread over time.
• Longitudinal Diffusion ($B$ term): This process involves the spread of solute molecules along the column axis from areas of higher to lower concentration.
• Mass Transfer ($C$ term): This describes the time taken for a solute to equilibrate between the mobile and stationary phases, with diffusion within the stationary phase being limiting.

Overall, by understanding dispersion processes, chromatographers can optimize column performance by minimizing band broadening to achieve precise separations.