Problem 77 - Chemistry and Chemical Reactivity

Question

A solution of a dye was analyzed by spectrophotometry, and the following calibration data were collected. $$\begin{array}{|c|c|}\hline \text { Dye Concentration } & \begin{array}{c}\text { Absorbance }(A) \\\\\text { at } 475\mathrm{nm} \end{array} \\\\\hline 0.50 \times 10^{-6} \mathrm{M} & 0.24 \\\1.5 \times 10^{-6} \mathrm{M} & 0.36 \\\2.5 \times 10^{-6} \mathrm{M} & 0.44 \\\\\hline 3.5 \times 10^{-6} \mathrm{M} & 0.59 \\\4.5 \times 10^{-6} \mathrm{M} & 0.70\end{array}$$ (a) Construct a calibration plot, and determine the slope and intercept. (b) What is the dye concentration in a solution with \(A=0.52 ?\)

Step-by-Step Solution

Answer

The dye concentration is approximately \(3.00 \times 10^{-6} \mathrm{M}\).

Step 1: Understand the Calibration Plot

A calibration plot is a graph of absorbance versus concentration. We plot the given concentrations on the x-axis and the corresponding absorbance values on the y-axis to create a linear plot. If a straight line can fit the data, the relationship between absorbance and concentration can be described using Beer's Law: \( A = ext{slope} \times C + ext{intercept} \).

Step 2: Plot the Calibration Data

Plot the data points \((0.50 \times 10^{-6}, 0.24), (1.5 \times 10^{-6}, 0.36), (2.5 \times 10^{-6}, 0.44), (3.5 \times 10^{-6}, 0.59), (4.5 \times 10^{-6}, 0.70)\) on a graph. The concentration units on the x-axis should be consistent (\(\times 10^{-6} \, \text{M}\)), and the absorbance on the y-axis is unitless.

Step 3: Determine the Equation of the Line

Using a method such as least squares regression, determine the best-fit line for the plotted points. The line should be in the form \( A = mC + b \), where \(m\) is the slope and \(b\) is the y-intercept. In this case, the slope (\(m\)) is approximately 0.114 and the intercept (\(b\)) is approximately 0.178.

Step 4: Calculate the Unknown Concentration

With the determined values for the slope \(m\) and y-intercept \(b\), and the relationship \( A = mC + b \), substitute \(A = 0.52\) to solve for the dye concentration \(C\): \(0.52 = 0.114C + 0.178\). Rearrange to find \(C\): \(C = \frac{0.52 - 0.178}{0.114}\). Upon calculation, \(C \approx 3.00 \times 10^{-6} \mathrm{M}\).

Key Concepts

Calibration Plot

A calibration plot is a fundamental tool in spectrophotometry, helping us understand how a substance's concentration relates to its absorbance. By plotting a graph of absorbance versus concentration, we can visually interpret this relationship. In our example, concentration values are placed on the x-axis, while absorbance values are on the y-axis. This forms the basis for creating a linear graph.

Once plotted, we look for a straight line that best fits through the data points. If such a line exists, it indicates a linear relationship, which is crucial for using Beer's Law. The equation of this line takes the form of \(A = mC + b\), where \(A\) is absorbance, \(C\) is concentration, \(m\) is the slope of the line, and \(b\) is the intercept on the y-axis. These parameters allow us to predict unknown concentrations by simply knowing their absorbance and vice versa.

Beer's Law

Beer's Law, also known as Beer-Lambert's Law, defines the linear relationship between absorbance and concentration of a solution. Mathematically, it is described as \(A = \epsilon \times l \times C\). Here, \(A\) represents absorbance, \(\epsilon\) is the molar absorptivity (a constant that indicates how strongly the substance absorbs light), \(l\) is the path length of light through the solution (typically measured in centimeters), and \(C\) is the concentration of the solution.

This law is incredibly useful for determining an unknown concentration when the slope (\(m\)) and intercept (\(b\)) from the calibration plot are known. The simple direct proportionality makes it easy to interpolate values within the range of measured data, ensuring more precise analytical results.

• For our exercise, with the slope \(m\) calculated as 0.114 and the intercept \(b\) as 0.178, Beer's Law simplifies to the linear form \(A = 0.114C + 0.178\). Using this, an absorbance value helps us quickly determine the concentration of the dye in solution.

Absorbance

Absorbance is a measure of the quantity of light absorbed by a solution. It is unitless and provides a way to quantify how much light at a particular wavelength is absorbed by the substance in question. In spectrophotometry, absorbance measurements are crucial, as they are directly related to the concentration of a solution through Beer's Law.

In our specific exercise, we used absorbance at 475 nm to track how the dye solution responds to light. As the concentration of the dye increases, more light is absorbed, leading to higher absorbance values. This relationship is not only straightforward but intuitive, meaning that by observing changes in absorbance, we can infer changes in concentration. These absorbance readings allowed us to plot them against concentration to help create the calibration plot, which is central to our calculations.