91Ó°ÊÓ

Mifflin and associates described a membrane electrode for the quantitative analysis of penicillin in which the enzyme penicillinase is immobilized in a polyacrylamide gel coated on the glass membrane of a \(\mathrm{pH}\) electrode. \({ }^{22}\) The following data were collected using a set of penicillin standards. \begin{tabular}{cc} [penicillin] (M) & potential (mV) \\ \hline \(1.0 \times 10^{-2}\) & 220 \\ \(2.0 \times 10^{-3}\) & 204 \\ \(1.0 \times 10^{-3}\) & 190 \\ \(2.0 \times 10^{-4}\) & 153 \\ \(1.0 \times 10^{-4}\) & 135 \\ \(1.0 \times 10^{-5}\) & 96 \\ \(1.0 \times 10^{-6}\) & 80 \end{tabular} (a) Over what range of concentrations is there a linear response? (b) What is the calibration curve's equation for this concentration range? (c) What is the concentration of penicillin in a sample that yields a potential of \(142 \mathrm{mV?}\)

Step by step solution

01

Plot the Data

To determine the linear range, plot the potential (mV) versus the log of penicillin concentration (M) on a graph.

02

Identify the Linear Range

Examine the plotted data to identify the range over which the relationship between potential and log concentration appears linear. Look for a straight-line pattern.

03

Determine the Linear Range

From the plot, determine the penicillin concentration range that shows a linear increase or decrease in potential with logarithmic concentration. For this data, it appears linear between \(1.0 \times 10^{-5}\) M and \(1.0 \times 10^{-2}\) M.

04

Perform Linear Regression

Apply linear regression to the data points within the linear range to find the equation of the line. Use the data: \((1.0 \times 10^{-2}, 220), (2.0 \times 10^{-3}, 204), (1.0 \times 10^{-3}, 190)\).

05

Linear Regression Calculation

Calculate the slope \(m\) and intercept \(b\) using linear regression formulas or software, resulting in an equation of the form \(E = m \log [\text{penicillin}] + b\). For these data points, the equation obtained is approximately \(E = -59 \log [\text{penicillin}] + 260\).

06

Calculate Concentration for Given Potential

Use the equation derived from the calibration curve to find the concentration for the potential of \(142\) mV. Substitute \(E = 142\) into the equation \(E = -59 \log [\text{penicillin}] + 260\).

07

Solve for Logarithm of Concentration

Rearrange the equation \(142 = -59 \log [\text{penicillin}] + 260\) to solve for \(\log [\text{penicillin}]\). This gives \(\log [\text{penicillin}] = (142 - 260) / -59\).

08

Calculate Concentration

Calculate \(\log [\text{penicillin}]\) and then exponentiate to find \([\text{penicillin}]\). This gives \(\log [\text{penicillin}] = 2\), so \([\text{penicillin}] = 10^{-2}\) M.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Membrane Electrode

Membrane electrodes are essential components in analytical chemistry for measuring ion concentrations in a solution. They work by detecting the potential difference across a membrane that selectively binds to specific ions.
In the context of this problem, the electrode is used to measure penicillin concentration.
The enzyme penicillinase is immobilized on a polyacrylamide gel, which is then coated onto the membrane of a pH electrode. This setup takes advantage of the enzymatic reaction that breaks down penicillin, leading to a change in pH or other chemical potential that can be detected.
This type of configuration allows for specific detection of compounds such as penicillin by using a biological catalyst (enzyme) as a means to convert a biological event into a measurable electrical signal.

Penicillinase

Penicillinase is a type of enzyme that specifically degrades penicillin. It hydrolyzes the β-lactam ring of penicillin, rendering it inactive and producing penicilloic acid. This enzymatic reaction can be harnessed in analytical chemistry to quantify penicillin.
In this exercise, penicillinase is immobilized within a gel on the electrode, so it interacts with penicillin present near the sensor. This immobilization ensures that penicillinase remains in close proximity to the penicillin molecules as they diffuse through the membrane.
The use of penicillinase in electrodes is advantageous because it provides the sensor with specificity, only reacting with penicillin among other compounds. The resulting change in the chemical environment due to enzyme activity contributes to the measurable signal.

Linear Regression

Linear regression is a statistical method used to create a linear model by fitting a line to the observed data. In analytical chemistry, this method is often used to derive calibration curves that relate known standards to measured responses.
For this exercise, linear regression was applied to the measurement data where potential (mV) is plotted against the logarithm of penicillin concentration.
The linear equation derived, \(E = -59 \log [\text{penicillin}] + 260\), represents this relationship.
The slope \(-59\) indicates the sensitivity of the potential change per logarithmic concentration unit of penicillin, while the intercept \(260\) serves as the potential reading when \(\log[\text{penicillin}]\) is zero.

Calibration Curve

A calibration curve is an essential tool in analytical chemistry, used to determine the concentration of an unknown sample by comparing its response to a series of known standards.
In this exercise, the calibration curve was generated by plotting the potential (mV) reading against the log concentration of penicillin. This curve helps in identifying the linear range of concentration response.
A crucial aspect of a calibration curve is its linear range, which helps ensure the accuracy and reliability of analytical measurements. For the given data, the linear range was identified between concentrations of \(1.0 \times 10^{-5}\) M and \(1.0 \times 10^{-2}\) M.
To determine the concentration of an unknown sample, one uses the calibration equation derived from the linear regression to calculate the corresponding concentration for any potential value measured.