Problem 75 - Chemistry and Chemical Reactivity

Question

Spectrophotometry A solution of a dye was analyzed by spectrophotometry, and the following calibration data were collected. $$\begin{array}{cc} \text { Dye Concentration } & \text { Absorbance }(A) \text { at } 475 \mathrm{nm} \\ \hline 0.50 \times 10^{-6} \mathrm{M} & 0.24 \\ 1.5 \times 10^{-6} \mathrm{M} & 0.36 \\ 2.5 \times 10^{-6} \mathrm{M} & 0.44 \\ 3.5 \times 10^{-6} \mathrm{M} & 0.59 \\ 4.5 \times 10^{-6} \mathrm{M} & 0.70 \\ \hline \end{array}$$ (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 2.44 x 10⁻⁶ M.

Step 1: Plot the Calibration Graph

Plot the __Dye Concentration__ on the x-axis and __Absorbance (A)__ on the y-axis. Each pair data point (Dye Concentration, Absorbance) will be plotted as follows: - (0.50 x 10⁻⁶ M, 0.24) - (1.5 x 10⁻⁶ M, 0.36) - (2.5 x 10⁻⁶ M, 0.44) - (3.5 x 10⁻⁶ M, 0.59) - (4.5 x 10⁻⁶ M, 0.70) Draw a line of best fit through these data points on the graph.

Step 2: Calculate the Slope and Intercept

Using the line of best fit, apply the equation of a straight line, \[ A = m imes ext{Concentration} + b \]To find the slope (m), select two data points on the line, such as (0.50, 0.24) and (4.5, 0.70). Calculate:\[ m = \frac{(0.70 - 0.24)}{(4.5 - 0.50) \times 10^{-6}} = \frac{0.46}{4 \times 10^{-6}} = 0.115 \]For the intercept (b), use one point and the slope:\[ 0.24 = 0.115 \times 0.50 \times 10^{-6} + b \]\[ b = 0.24 - 0.0575 \times 10^{-6} \approx 0.240 \]

Step 3: Use the Calibration Equation to Find Unknown Concentration

With the known value of absorbance \(A = 0.52\), plug this into the linear equation to solve for the unknown concentration.\[ 0.52 = 0.115 \times ext{Concentration} + 0.24 \]Rearrange to find:\[ ext{Concentration} = \frac{0.52 - 0.24}{0.115} = \frac{0.28}{0.115} = 2.435 \times 10^{-6} \text{ M} \]

Key Concepts

Calibration Plot

Creating a calibration plot is an essential part of spectrophotometric analysis. It helps in determining the relationship between absorbance and concentration of a solution. To create such a plot:

• Plot the dye concentration on the x-axis.
• Plot the corresponding absorbance values on the y-axis.

Connect these data points to form a line, known as the line of best fit. This line should ideally be straight, indicating a linear relationship between absorbance and concentration. The slope and intercept of this line are crucial, as they represent how much absorbance changes with concentration. By using these, you can predict unknown concentrations from known absorbances.

Beer-Lambert Law

The Beer-Lambert Law is fundamental in spectrophotometry. It explains how the concentration of a substance is directly proportional to its absorbance. Mathematically, it is expressed as:\[A = \varepsilon \cdot l \cdot c\]Where:

• \(A\) is absorbance
• \( \varepsilon \) is molar absorptivity (a constant for each substance)
• \(l\) is the path length of the cuvette (usually in cm)
• \(c\) is the concentration of the substance

This law shows that absorbance increases with higher concentration and longer path length. Molar absorptivity is unique for each substance at a particular wavelength, being a key factor in determining concentration through absorbance.

Absorbance Measurements

Absorbance measurements are the cornerstone of spectrophotometric analysis. This method measures the amount of light absorbed by a solution at a specific wavelength. The absorbance reading reflects how much light does not pass through the solution. Using the Beer-Lambert Law, one can take the measured absorbance to determine the concentration of a solute in a solution. Always ensure your spectrophotometer is calibrated before taking measurements to enhance accuracy. Each measurement provides a data point which will be crucial in constructing a calibration curve.

Concentration Determination

Determining concentration using a calibration plot involves plugging the absorbance value of an unknown sample into the linear equation derived from the line of best fit:\[A = m \cdot \text{Concentration} + b\]Here, \(m\) is the slope, \(b\) is the intercept, and \(A\) is the known absorbance of the unknown sample.Rearrange the equation to solve for the unknown concentration.This calculated concentration allows us to identify how much of a particular substance is present, making it crucial in applications ranging from chemistry labs to environmental testing.Remember, precise calibration and careful measurements deliver the most accurate concentration values.