Question

Absorbance values for five standard solutions of a colored solute were determined at \(410 \mathrm{nm}\) with a \(1.00-\mathrm{cm}\) path length, giving the following table of data: \begin{tabular}{cc} Solute concentration \((M)\) & \multicolumn{1}{c} { A } \\ \hline 0.250 & 0.165 \\ 0.500 & 0.317 \\ 0.750 & 0.510 \\ 1.000 & 0.650 \\\ 1.250 & 0.837 \end{tabular} The absorbance of a solution of unknown concentration containing the same solute was 0.400 . (a) What is the concentration of the unknown solution? (b) Determine the absorbance values you would expect for solutions with the following concentrations: \(0.4 M, 0.6 M, 0.8 M\), 1\. \(1 M\). (c) Calculate the average molar absorptivity of the compound and determine the units of molar absorptivity.

Short answer

(a) Calculate concentration from line equation. (b) Use equation for absorbance values. (c) Molar absorptivity is the slope.

Step-by-step solution

Plot the Calibration Curve

We will plot the calibration curve using the given solute concentration and absorbance data. This forms a linear relationship where absorbance is on the y-axis and concentration is on the x-axis. The given data points are: (0.250 M, 0.165), (0.500 M, 0.317), (0.750 M, 0.510), (1.000 M, 0.650), and (1.250 M, 0.837).

Determine the Line Equation

Using the plotted points, we find the equation of the line that best fits the data using linear regression. The relationship between absorbance (A) and concentration (C) is often linear: \( A = \varepsilon C \), where \( \varepsilon \) is the molar absorptivity, and the line equation is often found to be \( A = mC + b \). From calculations, the line equation can be determined.

Calculate the Unknown Concentration

With the line equation from Step 2, plug in the absorbance (0.400) of the unknown solution to solve for its concentration (C). This requires substituting A = 0.400 into the equation and solving for C.

Predict Absorbance for Given Concentrations

Using the line equation from Step 2, substitute the concentrations 0.4 M, 0.6 M, 0.8 M, and 1.0 M into the equation to predict the absorbance values for these concentrations.

Calculate Molar Absorptivity

The molar absorptivity \( \varepsilon \) is calculated from the slope (m) of the line equation \( A = \varepsilon C \). The units of molar absorptivity are determined based on the relationship \( \varepsilon = \frac{A}{C} \), which leads to the units being \( \text{M}^{-1}\text{cm}^{-1} \).

Key concepts

Molar Absorptivity

Molar absorptivity, also known as the molar extinction coefficient, is a crucial concept in understanding how substances interact with light. This parameter is part of Beer's Law, which helps us understand how much light a chemical solution can absorb. In formula terms, it's represented by \( \varepsilon \). This term is defined as the amount of light absorbed per unit concentration and path length. It is usually expressed in units of \( \text{M}^{-1}\text{cm}^{-1} \), which means it accounts for the solute concentration (\( \text{M} \)) and the path length (\( \text{cm} \)) the light travels through. To find molar absorptivity:

• Plot the calibration curve of absorbance vs. concentration.
• Determine the slope of this linear plot.
• This slope represents the molar absorptivity.

The larger the \( \varepsilon \) value, the greater the light absorption, indicating a stronger interaction with light.

Calibration Curve

A calibration curve is a graphical representation that helps you determine the concentration of an unknown sample based on a set of standard solutions. In this context, it's used to show how absorbance changes with known concentrations of a solution. To create a calibration curve:

• Measure the absorbance of various concentrations of the same solute.
• Plot these absorbance values on the y-axis against their concentrations on the x-axis.
• The plot should form a straight line if Beer's Law is obeyed.

The calibration curve is crucial because it forms the basis for determining unknown concentrations—it’s essentially a roadmap showing the relationship between concentration and absorbance.

Linear Regression

Linear regression is a statistical method used to find the best-fitting straight line through a set of data points. When we apply this to absorbance and concentration data, we're looking for a line that best summarizes this relationship.The equation of the line is generally expressed as \( A = mC + b \), where:

• \( A \) is the absorbance.
• \( C \) is the concentration of the solute.
• \( m \) is the slope, equivalent to molar absorptivity \( \varepsilon \).
• \( b \) is the y-intercept.

Linear regression allows us to mathematically describe the relationship and use this to predict unknown values like concentration or absorbance, enhancing our ability to work quickly and accurately with chemical solutions.

Absorbance

Absorbance, in the context of Beer's Law, is a measure of how much light is absorbed by a solution. It is dimensionless but crucial for calculating concentrations using a calibration curve.Absorbance is calculated using the path length and molar absorptivity:

• Given by the formula \( A = \varepsilon C l \), where \( A \) is absorbance, \( \varepsilon \) is molar absorptivity, \( C \) is the concentration, and \( l \) is the path length.

Absorbance is measured using a spectrophotometer, and its main advantage is that it provides a simple way to quantify how a chemical solution interacts with light. Lower absorbance means less light absorbed; higher absorbance indicates more light absorbed. Using this, we can infer the concentration of unknown solutions just by measuring their absorbance against known standards.