Problem 101 - Chemistry The Central Science In Si Units

Question

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorbance of a colored reactant at \(520 \mathrm{nm}\). The reaction occurs in a \(1.00-\mathrm{cm}\) sample cell, and the only colored species in the reaction has an extinction coefficient of \(5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{~cm}^{-1}\) at \(520 \mathrm{nm}\). (a) Calculate the initial concentration of the colored reactant if the absorbance is 0.605 at the beginning of the reaction. (b) The absorbance falls to 0.250 at \(30.0 \mathrm{~min}\). Calculate the rate constant in units of \(\mathrm{s}^{-1}\), (c) Calculate the half-life of the reaction. (d) How long does it take for the absorbance to fall to \(0.100 ?\)

Step-by-Step Solution

Answer

(a) Initial concentration: \(1.08 \times 10^{-4} \mathrm{M}\). (b) Rate constant: \(6.42 \times 10^{-4} \mathrm{s}^{-1}\). (c) Half-life: \(18.0 \) minutes. (d) Time to absorbance 0.100: \(48.0 \) minutes.

Step 1: Calculate Initial Concentration using Beer's Law

Beer's Law states \( A = \varepsilon cl \), where \( A \) is absorbance, \( \varepsilon \) is the extinction coefficient, \( c \) is the concentration, and \( l \) is the path length. Given \( A = 0.605 \), \( \varepsilon = 5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1} \), and \( l = 1.00 \mathrm{cm} \), we solve for \( c \): \[ c = \frac{A}{\varepsilon l} = \frac{0.605}{5.60 \times 10^{3} \times 1.00} \] \( c = 1.08 \times 10^{-4} \mathrm{M} \).

Step 2: Calculate Rate Constant Using First-Order Kinetics

First-order reactions follow the equation \( \ln \left( \frac{[A]_{t}}{[A]_{0}} \right) = -kt \), where \([A]_t\) is the concentration at time \( t \), and \([A]_0\) is the initial concentration. Using concentrations calculated from absorbances (same calculation as Step 1), and knowing absorptions at \( t = 30 \text{ min} = 1800 \text{ s} \), we find:\[ [A]_0 = 1.08 \times 10^{-4} \mathrm{M} \, , \, A = 0.250 \, \rightarrow \, [A]_t = \frac{0.250}{5.60 \times 10^{3} \times 1.00} = 4.46 \times 10^{-5} \mathrm{M} \]\[ \ln \left( \frac{4.46 \times 10^{-5}}{1.08 \times 10^{-4}} \right) = -k \times 1800 \] Solving, \( k = 6.42 \times 10^{-4} \mathrm{s}^{-1} \).

Step 3: Calculate Half-Life of the Reaction

For a first-order reaction, the half-life \( t_{1/2} \) is given by \( t_{1/2} = \frac{\ln(2)}{k} \). Using the rate constant \( k = 6.42 \times 10^{-4} \mathrm{s}^{-1} \):\[ t_{1/2} = \frac{\ln(2)}{6.42 \times 10^{-4}} \]\( t_{1/2} \approx 1080 \text{ seconds} \) or \( 18.0 \text{ minutes} \).

Step 4: Determine Time for Absorbance to Fall to 0.100

Using the same first-order equation, \( \ln \left( \frac{[A]_{t}}{[A]_{0}} \right) = -kt \), and given \( A = 0.100 \), solve for \( t \): \[ [A]_t = \frac{0.100}{5.60 \times 10^{3} \times 1.00} = 1.79 \times 10^{-5} \mathrm{M} \] \[ \ln \left( \frac{1.79 \times 10^{-5}}{1.08 \times 10^{-4}} \right) = -k \times t \] Solving, \( t \approx 2880 \text{ seconds} \) or \( 48.0 \text{ minutes} \).

Key Concepts

Beer's Law

Beer's Law, also known as the Beer-Lambert Law, describes the relationship between absorbance and concentration in a sample. The law is represented by the equation: \[ A = \varepsilon c l \]Here, \( A \) stands for absorbance, \( \varepsilon \) is the molar extinction coefficient, \( c \) represents the concentration, and \( l \) is the path length through which the light travels in the sample.In the context of the problem, absorbance is observed through spectroscopy of a colored reactant. By rearranging Beer's Law, we solve for the concentration:\[ c = \frac{A}{\varepsilon l} \]This means you can determine how much of a substance is present just by measuring how much light it absorbs, provided you know the extinction coefficient and path length. It's like reading the story of a compound's presence and quantity through its interactions with light.

Absorbance spectroscopy

Absorbance spectroscopy is a highly useful technique in chemistry for analyzing the composition and concentration of substances. When a substance absorbs light, it occurs at specific wavelengths, and this forms the basis of absorbance measurements. In the example given, the reactant absorbs light at 520 nm.Through this technique, absorbance (\( A \)) is measured. The absorbance value changes as the concentration of the colored reactant changes during a chemical reaction. The spectrophotometer's job is to determine how much light is absorbed by a sample at 520 nm. Key aspects to note:

• Wavelength: Different substances absorb light at different wavelengths because of their unique structures. In our case, it's 520 nm.
• Comparison: Monitoring how the absorbance value changes over time lets us track the reaction's progress.

Absorbance spectroscopy thus serves as an essential tool in chemical kinetics, allowing scientists to study the dynamics of reactions efficiently.

Rate constant calculation

For first-order reactions, the rate constant \( k \) is a fundamental parameter that provides insight into how quickly a reaction occurs. This can be determined using the first-order reaction equation:\[ \ln \left( \frac{[A]_t}{[A]_0} \right) = -kt \]where \([A]_t\) is the concentration at a later time, and \([A]_0\) is the initial concentration.When conducting experiments, like in our particular exercise, we measure absorbance to infer changes in concentration. By substituting absorbance values into Beer's Law, we can compute these concentrations. Once you have the concentrations, you can plot \( \ln([A]_t/[A]_0) \) against time, or directly calculate \( k \) as shown:\[ k = \frac{-\ln \left( \frac{[A]_t}{[A]_0} \right)}{t} \]Here, \( t \) is the time interval between measurements.

• The calculated rate constant (from the original exercise) was \( 6.42 \times 10^{-4} \mathrm{s}^{-1} \), indicating the speed at which the reaction progresses, crucial for understanding reaction dynamics.
• Knowing \( k \) also aids in determining the reaction's half-life, which provides information on how long it takes for a reactant concentration to reduce by half.

Hence, calculating the rate constant helps in forecasting the reaction's future behavior and verifying its first-order kinetic nature.