Problem 28 - Analytical Chemistry

Question

Fernández-Abedul and Costa-García developed an FIA method to determine cocaine in samples using an amperometric detector. \(^{27}\) The following signals (arbitrary units) were collected for 12 replicate injections of a \(6.2 \times 10^{-6} \mathrm{M}\) sample of cocaine, \(\mathrm{C}_{17} \mathrm{H}_{21} \mathrm{NO}_{4}\). \(\begin{array}{lll}24.5 & 24.1 & 24.1 \\ 23.8 & 23.9 & 25.1 \\ 23.9 & 24.8 & 23.7 \\ 23.3 & 23.2 & 23.2\end{array}\) (a) What is the relative standard deviation for this sample? (b) The following calibration data are available $$ \begin{array}{cc} \text { [cocaine] }(\mu \mathrm{M}) & \text { signal (arb. units) } \\ \hline 0.18 & 0.8 \\ 0.36 & 2.1 \\ 0.60 & 2.4 \\ 0.81 & 3.2 \\ 1.0 & 4.5 \\ 2.0 & 8.1 \\ 4.0 & 14.4 \\ 6.0 & 21.6 \\ 8.0 & 27.1 \\ 10.0 & 32.9 \end{array} $$ In a typical analysis a 10.0 -mg sample is dissolved in water and diluted to volume in a \(25-\mathrm{mL}\) volumetric flask. A \(125-\mu \mathrm{L}\) aliquot is transferred to a \(25-\mathrm{mL}\) volumetric flask and diluted to volume with a \(\mathrm{pH} 9\) buffer. When injected into the flow injection apparatus a signal of 21.4 (arb. units) is obtained. What is the \(\% \mathrm{w} / \mathrm{w}\) cocaine in the sample?

Step-by-Step Solution

Answer

(a) Relative standard deviation is calculated from mean and standard deviation. (b) Use calibration curve to find concentration, adjust for dilution, and calculate % w/w.

Step 1: Calculate the Mean Signal

First, calculate the mean of the 12 replicate injection signals. The signals are: 24.5, 24.1, 24.1, 23.8, 23.9, 25.1, 23.9, 24.8, 23.7, 23.3, 23.2, 23.2. Sum these values and divide by 12 to get the mean: \[\text{Mean} = \frac{24.5 + 24.1 + 24.1 + 23.8 + 23.9 + 25.1 + 23.9 + 24.8 + 23.7 + 23.3 + 23.2 + 23.2}{12} = 24.02\]

Step 2: Calculate the Standard Deviation

Next, calculate the standard deviation of the signals. Use the formula:\[\sigma = \sqrt{\frac{\sum (x_i - \text{mean})^2}{n-1}}\]where \( x_i \) are the individual signal values, \( \text{mean} = 24.02 \), and \( n = 12 \). Calculate each squared deviation, sum them, and divide by 11 (as \( n - 1 = 11 \)), and take the square root.

Step 3: Determine Relative Standard Deviation (RSD)

Calculate the relative standard deviation (RSD) using the formula:\[\text{RSD} = \left(\frac{\sigma}{\text{mean}}\right) \times 100\%\]

Step 4: Create the Calibration Curve Equation

Use the given calibration data to determine the relationship between concentration and signal. Consider a linear relationship of the form:\[ \text{Signal} = m \cdot [\text{Cocaine}] + b\]where \( m \) is the slope and \( b \) is the y-intercept. Perform a linear regression with the data to find \( m \) and \( b \).

Step 5: Determine Concentration at Given Signal

Use the linear calibration curve equation from Step 4 to solve for the concentration corresponding to the measured signal of 21.4 arbitrary units:\[ [\text{Cocaine}] = \frac{\text{Signal} - b}{m}\]Substitute 21.4 for the signal.

Step 6: Calculate Original Sample Concentration

Calculate the concentration in the original 10.0 mg sample based on the dilution scheme:- The detected concentration from Step 5 corresponds to the diluted sample in the \( 25 \text{ mL} \) flask.- Track the total dilution factor through the steps given in the problem, considering both the first and second dilution.

Step 7: Calculate Percent w/w of Cocaine in Sample

Calculate the weight/weight percentage of cocaine in the original sample:\[\% w/w = \left(\frac{\text{mass of cocaine}}{\text{mass of sample}} ight) \times 100\%\]Use the mass of cocaine determined from the concentration and dilution steps divided by the initial sample mass of 10.0 mg.

Key Concepts

Flow Injection Analysis (FIA)

Flow Injection Analysis (FIA) is an elegant technique in analytical chemistry used to automate analyses and enables rapid determination of substances in samples. The main working principle involves injecting a sample into a continuously flowing carrier stream, which then transports the sample through a flow path where different reactions or detections can take place.

• High throughput: FIA allows multiple samples to be analyzed quickly.
• Reproducibility: Standardizes conditions, which enhances repeatability.
• Low sample and reagent consumption: Ensures efficient usage with minimal waste.

In the context of the exercise, FIA is employed to analyze cocaine concentrations efficiently, with an injection triggering a signal that is measured in arbitrary units.

Amperometric Detection

Amperometric Detection is often coupled with FIA to enhance analytical capabilities. This detection method relies on measuring electric current resulting from oxidation or reduction reactions occurring at the electrode surface. The current is proportional to the concentration of the analyte, making it a sensitive and selective detection technique.
Key advantages include:

• High sensitivity: Capable of detecting very low concentrations.
• Specificity: Tailored to particular analytes depending on the applied potential.

In the exercise, an amperometric detector is used to measure the signal of cocaine as it passes through the system, and this signal is utilized to deduce concentration levels.

Relative Standard Deviation

Relative Standard Deviation (RSD) is a statistical measure commonly used to assess precision in analytical chemistry. It adjusts the standard deviation relative to the mean of the data set, which is expressed as a percentage. The formula for RSD is:
\[ \text{RSD} = \left(\frac{\sigma}{\text{mean}}\right) \times 100\% \] where \( \sigma \) is the standard deviation of the signals and \( \text{mean} \) is their average.
The RSD provides insight into the variability of the measurements, helping to determine the consistency of the results obtained from the analysis. A lower RSD value indicates higher precision, which is essential when evaluating analytical methods like those in the provided exercise.

Calibration Curve

A Calibration Curve is an essential tool in quantitative analysis that relates the instrument response or signal to known concentrations of an analyte. It enables the conversion of arbitrary unit measurements to meaningful concentration values.
To form a calibration curve:

• Prepare solutions with known analyte concentrations.
• Measure their signals using the analytical technique.
• Plot signal versus concentration on a graph.
• Use linear regression to establish an equation.

The calibration curve created in the exercise allows for the calculation of unknown cocaine concentrations from measured signals, enhancing the robustness of the analysis.

Concentration Calculation

Concentration Calculation in the given exercise involves determining the unknown cocaine content in the sample based on the signal detected by the FIA with amperometric detection. This calculation hinges on using the calibration curve to find the analyte's concentration in the original sample.
Steps involve:

• Finding the concentration from the calibration curve using the given signal.
• Compensating for sample dilutions to backtrack to the original sample concentration.
• Calculate the \(\% \text{w/w}\) by comparing the cocaine mass to the total sample mass.

This allows for a complete analysis of the cocaine present in the sample, crucial in fields like forensic and environmental analysis.