Problem 81 - Chemistry and Chemical Reactivity

Question

You are asked to calibrate a spectrophotometer in the laboratory and collect the following data. Plot the data with concentration on the \(x\) -axis and absorbance on the \(y\) -axis. Draw the best straight line using the points on the graph (or do a least-squares or linear regression analysis using a computer program) and then write the equation for the resulting straight line. What is the slope of the line? What is the concentration when the absorbance is \(0.635 ?\) $$\begin{array}{ll} \hline \text { Concentration }(\mathrm{M}) & \text { Absorbance } \\ \hline 0.00 & 0.00 \\ 1.029 \times 10^{-3} & 0.257 \\ 2.058 \times 10^{-3} & 0.518 \\ 3.087 \times 10^{-3} & 0.771 \\ 4.116 \times 10^{-3} & 1.021 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Slope is 0.248. Concentration for absorbance 0.635 is approximately 2.56 × 10^{-3} M.

Step 1: Understanding the Task

We need to plot concentration vs. absorbance data, find the line of best fit, and determine the slope and intercept of the line. Then, use the equation to find the concentration for a given absorbance.

Step 2: Plotting the Data

Plot concentration ( M) on the x-axis and absorbance on the y-axis. The data points are (0, 0), (1.029×10^{-3}, 0.257), (2.058×10^{-3}, 0.518), (3.087×10^{-3}, 0.771), (4.116×10^{-3}, 1.021).

Step 3: Drawing the Best Fit Line

Use a linear regression tool to draw the best fit line through the data points. You may also use software like Excel for precise fitting.

Step 4: Finding the Line Equation

The regression analysis will provide a line equation in the form \( y = mx + c \). Typically, software will output \( m \) (slope) and \( c \) (y-intercept).

Step 5: Calculating the Slope and Intercept

From linear regression analysis, the slope \( m \) is found to be approximately 0.248, and the intercept \( c \) is approximately 0 (close to zero).

Step 6: Using the Line Equation

The line equation becomes \( y = 0.248x \). Given absorbance \( y = 0.635 \), we solve for \( x \):

Step 7: Solving for Concentration

Set 0.635 = 0.248x. Solve for \( x \) by dividing both sides by 0.248: \( x = \frac{0.635}{0.248} \approx 2.56 \times 10^{-3}\ M \).

Key Concepts

Linear Regression Analysis

Linear regression analysis is a powerful statistical tool used to find the relationship between two quantitative variables. In the context of spectrophotometer calibration, it helps determine the best fit line that shows how absorbance changes with concentration.

To perform a linear regression, one needs to plot the given data points onto a graph and calculate the line that minimizes the distance from all data points to the line. This line is commonly called the line of best fit.

Using software, like Excel, or statistical calculators, you can carry out linear regression analysis efficiently. The tool calculates the slope (\(m\)) and the intercept (\(c\)) of the line, yielding an equation in the form \(y = mx + c\). This equation helps predict the absorbance for unknown concentrations, providing a straightforward method to interpret spectrophotometer data.

Absorbance Measurement

Absorbance measurement is essential in determining how much light a solution absorbs at a specific wavelength. In the spectrophotometer calibration, you measure absorbance for solutions of known concentrations to establish a correlation.

Each data point on our graph of concentration versus absorbance represents a solution's visually measured absorbance. By plotting these points, the aim is to see how absorbance increases as concentration increases. Generally, this relationship is linear, meaning the plot should closely form a straight line.

Since the initial concentration is zero, the absorbance should naturally start at zero. As concentration rises, absorbance data points will climb, showing the trend that will aid in forming the best fit line through our key data points.

Concentration Determination

Determining the concentration of an unknown sample using a spectrophotometer involves using the equation derived from linear regression. With the established line equation, concentration determination becomes a calculation.

For instance, if we know an absorbance value and wish to find the corresponding concentration, we rearrange the line equation \(y = 0.248x\) to solve for \(x\) (concentration). Using \(0.635\) as the absorbance, you replace \(y\) and solve: \(x = \frac{0.635}{0.248}\).

This gives you the concentration \(x\), which is approximately\(2.56 \times 10^{-3}M\). Utilizing this method ensures that even if the samples vary in absorbance, their concentration can be accurately calculated.