Problem 76 - Chemistry and Chemical Reactivity

Question

The nitrite ion is involved in the biochemical nitrogen cycle. You can determine the nitrite ion content of a sample using spectrophotometry by first using several organic compounds to form a colored compound from the ion. The following data were collected. $$\begin{array}{cc} \begin{array}{c} \mathrm{NO}_{2} \text { - Ion } \\ \text { Concentration } \end{array} & \begin{array}{c} \text { Absorbance of Solution } \\ \text { at } 550 \mathrm{nm} \end{array} \\ \hline 2.00 \times 10^{-6} \mathrm{M} & 0.065 \\ 6.00 \times 10^{-6} \mathrm{M} & 0.205 \\ 10.00 \times 10^{-6} \mathrm{M} & 0.338 \\ 14.00 \times 10^{-6} \mathrm{M} & 0.474 \\ 18.00 \times 10^{-6} \mathrm{M} & 0.598 \\ \text { Unknown solution } & 0.402 \end{array}$$ (a) Construct a calibration plot, and determine the slope and intercept. (b) What is the nitrite ion concentration in the unknown solution?

Step-by-Step Solution

Answer

The slope is derived from the calibration plot and used to find the unknown concentration.

Step 1: Organize the Data

Record the given data for nitrite ion concentration and absorbance in a table format. You will use these values to create a calibration plot. Make it clear which data belong to the known samples and which belongs to the unknown sample.

Step 2: Create the Calibration Plot

Plot the known concentrations on the x-axis and the corresponding absorbance values on the y-axis. This should form a straight line since Beer's Law implies a linear relationship between concentration and absorbance for dilute solutions.

Step 3: Determine the Equation of the Line

Use linear regression or a graphing tool to find the best fit line through the data points. The equation of the line will be in the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. Calculate these values.

Step 4: Slope and Intercept Calculation

Based on the linear regression results, determine the slope (\(m\)) and the y-intercept (\(b\)) of the line you have plotted. For example, if you find the line equation as \(y = 0.033x + 0.027\), then the slope is 0.033 and the y-intercept is 0.027.

Step 5: Calculate the Unknown Concentration

Use the calibration equation \(y = mx + b\) to solve for \(x\) (concentration) when \(y\) (absorbance) of the unknown solution is 0.402. Rearrange the equation: \(x = \frac{y - b}{m}\), and substitute the values to find \(x\).

Step 6: Solve for X

Substitute \(y = 0.402\), \(m = 0.033\), and \(b = 0.027\) into the equation \(x = \frac{y - b}{m}\). Calculate \(x\) to find the nitrite ion concentration in the unknown solution.

Key Concepts

Beer's Law

Beer's Law, also known as the Beer-Lambert Law, is an important principle in spectrophotometry that describes how the absorbance of a solution is linearly related to the concentration of a solute. Simply put, this law allows us to deduce the concentration of an unknown sample by measuring its absorbance. The relationship can be expressed with the formula: \[ A = \varepsilon \, b \, c \]where:

• \(A\) is the absorbance.
• \(\varepsilon\) represents the molar absorptivity or extinction coefficient.
• \(b\) is the path length of the sample cell in centimeters.
• \(c\) is the concentration of the compound in solution.

According to Beer's Law, if the concentration of a solute in a solution increases, its absorbance also increases, provided that the path length and molar absorptivity remain constant. This linear relationship, however, holds true mainly for dilute solutions, where interactions between molecules do not affect measurement significantly. Understanding Beer's Law is critical when analyzing data graphically through calibration plots and determining unknown concentrations.

Nitrite Ion Concentration

Determining the nitrite ion concentration of an unknown sample involves forming a calibration curve using known concentrations. Spectrophotometry is used to assess the absorbance of solutions at a specific wavelength, which, for this experiment, is 550 nm. Absorbance values from known nitrite ion concentrations allow the creation of a graph.
To discover the concentration of an unknown nitrite sample, its absorbance is compared to the calibration curve by using the plotted line's equation. Through the equation of the best fit line, \[ y = mx + b \]

• \(y\) represents the absorbance of the solution.
• \(m\) is the slope of the line, indicating how absorbance changes with concentration.
• \(x\) is the concentration of the nitrite ion.
• \(b\) is the y-intercept, which accounts for any absorbance not due to the nitrite ion concentration.

For an unknown sample with a known absorbance value (0.402 in this case), the concentration \(x\) can be determined by rearranging the line equation to \(x = \frac{y - b}{m}\). This allows us to accurately calculate the unknown concentration based on its absorbance measurement.

Calibration Plot

A calibration plot is a graphical representation of Beer's Law, showing the correlation between the concentration of a substance and its absorbance. This graph is essential for interpreting spectrophotometric data and determining unknown concentrations. When constructing a calibration plot, concentrations of known standards are placed on the x-axis, while their corresponding absorbance values are on the y-axis.
This usually results in a straight line due to the linear relationship Beer's Law suggests. The slope of this line, calculated via linear regression, indicates how sensitive the absorbance is to changes in concentration. The y-intercept shows the baseline absorbance level that isn't directly linked to the solute concentration.
To effectively use the calibration plot:

• Ensure all data points are accurately plotted.
• Use the graphing tool to draw the best fit line through the points.
• Extract the slope (\(m\)) and y-intercept (\(b\)) to form the line equation \(y = mx + b\).
• Apply this equation to find concentrations for unknown samples by solving for \(x\) using the known \(y\) (absorbance).

Calibration plots are a powerful tool in chemical analysis, providing reliable insight into the composition of unknown samples. Accuracy in plotting and calculation is key to obtaining meaningful results.