Problem 101 - Chemistry The Central Science

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

The short answers are: a) The initial concentration of the colored reactant is \(1.08 \times 10^{-4} \mathrm{M}\). b) The rate constant k is \(4.89 \times 10^{-4} \mathrm{s}^{-1}\). c) The half-life of the reaction is approximately \(1416.61 \mathrm{s}\). d) It takes approximately \(4890.38 \mathrm{s}\) for the absorbance to fall to 0.100.

Step 1: Use Beer-Lambert's law

Beer-Lambert's law relates the absorbance (A) to the concentration (c) of the colored reactant, the path length (l), and the extinction coefficient (ε). The formula is given by: \[A = ε × c × l\] In this case, \(A = 0.605\), \(ε = 5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1}\), and \(l = 1.00 \mathrm{cm}\). We will now solve for the concentration (c).

Step 2: Calculate the concentration (c)

Using the given values, we can solve for c: \[\begin{aligned} 0.605 &= (5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1}) \times c \times (1.00 \mathrm{cm}) \\ c &= \frac{0.605}{5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1} \times 1.00 \mathrm{cm}} \\ c &= 1.08 \times 10^{-4} \mathrm{M} \end{aligned}\] So, the initial concentration of the colored reactant is \(1.08 \times 10^{-4} \mathrm{M}\). #b) Calculating the rate constant#

Step 3: Use the first-order reaction rate expression

For a first-order reaction, the rate expression is given by: \[ln\frac{c_{0}}{c_{t}} = kt\] Where \(c_0\) is the initial concentration, \(c_t\) is the concentration after time t, k is the rate constant, and t is the time. Since we are given the absorbances at the start and after 30 minutes, we can determine the concentrations and solve for the rate constant (k).

Step 4: Determine the concentration after 30 minutes

Using Beer-Lambert's law, we can determine the concentration (\(c_{30}\)) after 30 minutes when the absorbance is 0.250: \[\begin{aligned} 0.250 &= (5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1}) \times c_{30} \times (1.00 \mathrm{cm}) \\ c_{30} &= \frac{0.250}{5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1} \times 1.00 \mathrm{cm}} \\ c_{30} &= 4.46 \times 10^{-5} \mathrm{M} \end{aligned}\]

Step 5: Calculate the rate constant (k)

Now, we can use the concentrations and given time (t) of 30 minutes (1800 seconds) to solve for the rate constant (k) in the expression \(ln\frac{c_{0}}{c_{t}} = kt\): \[\begin{aligned} ln\frac{1.08 \times 10^{-4}}{4.46 \times 10^{-5}} &= k \times 1800 \\ k &= \frac{ln(1.08 \times 10^{-4}) - ln(4.46 \times 10^{-5})}{1800} \\ k &= 4.89 \times 10^{-4} \mathrm{s}^{-1} \end{aligned}\] So, the rate constant k is \(4.89 \times 10^{-4} \mathrm{s}^{-1}\). #c) Calculating the half-life#

Step 6: Use the first-order half-life expression

For a first-order reaction, the half-life (t₁/₂) is given by the expression: \[t_{1/2} = \frac{ln(2)}{k}\] We will use the calculated rate constant (k) to find the half-life.

Step 7: Calculate the half-life (t₁/₂)

Plugging the rate constant (k) in the half-life expression: \[\begin{aligned} t_{1/2} &= \frac{ln(2)}{4.89 \times 10^{-4} \mathrm{s}^{-1}} \\ t_{1/2} &= 1416.61 \mathrm{s} \end{aligned}\] So, the half-life of the reaction is approximately \(1416.61 \mathrm{s}\). #d) Calculating the time for the absorbance to fall to 0.100#

Step 8: Determine the concentration corresponding to the given absorbance

Using Beer-Lambert's law, we can determine the concentration (\(c_{new}\)) for the given absorbance (0.100): \[\begin{aligned} 0.100 &= (5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1}) \times c_{new} \times (1.00 \mathrm{cm}) \\ c_{new} &= \frac{0.100}{5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{cm}^{-1} \times 1.00 \mathrm{cm}} \\ c_{new} &= 1.79 \times 10^{-5} \mathrm{M} \end{aligned}\]

Step 9: Calculate the time required for the absorbance to fall to 0.100

We can use the first-order reaction rate expression \(ln\frac{c_{0}}{c_{new}} = kt\), and solve for the time (t): \[\begin{aligned} ln\frac{1.08 \times 10^{-4}}{1.79 \times 10^{-5}} &= (4.89 \times 10^{-4} \mathrm{s}^{-1}) \times t \\ t &= \frac{ln(1.08 \times 10^{-4}) - ln(1.79 \times 10^{-5})}{4.89 \times 10^{-4} \mathrm{s}^{-1}} \\ t &= 4890.38 \mathrm{s} \end{aligned}\] So, it takes approximately \(4890.38 \mathrm{s}\) for the absorbance to fall to 0.100.

Key Concepts

Understanding Beer-Lambert's Law

In chemistry, Beer-Lambert's Law is a fundamental principle that explains the relationship between the absorbance of light by a substance and the concentration of that substance. This law is particularly useful in spectroscopy, a technique often employed to monitor the progress of chemical reactions, as seen in the given problem where it was used to follow a first-order reaction.

Beer-Lambert's Law is mathematically expressed as: \[\begin{equation} A = \text{ε} \times c \times l,\text{\}\text{\}\text{\}\end{equation}\] where

• (\( A \)) is the absorbance of the substance, which is a measure of how much light is absorbed as it passes through the sample,
• (\( \text{ε} \)) (epsilon) is the molar extinction coefficient or absorptivity that represents how strongly the substance absorbs light at a particular wavelength,
• (\( c \)) is the concentration of the substance, and
• (\( l \)) is the path length, the distance the light travels through the sample.

The absorbance is directly proportional to the concentration and the path length, meaning that as the concentration of the substance increases, or as the light travels a longer path through the substance, the absorbance will increase proportionally.

The Role of Absorbance in Monitoring Reactions

Moving on to absorbance, it is critical to understand this concept when observing chemical reactions via spectroscopy. Absorbance (\( A \)) is a dimensionless unit that measures the amount of light absorbed by a sample. In the context of the exercise, absorbance was measured to follow the concentration of a colored reactant over time. As the reaction progresses, the concentration of the reactant changes, causing a corresponding change in absorbance.

The calculation of the initial concentration of the reactant is based on the measured initial absorbance; a lower absorbance later in the reaction indicates that the colored reactant is being consumed. By regularly measuring absorbance at a specific wavelength, chemists can obtain a detailed picture of the reaction kinetics. It was observed that the absorbance fell from 0.605 to 0.250, which played a key role in calculating the rate constant and half-life of the reaction.

Determining the Rate Constant in Reaction Kinetics

The rate constant, denoted by (\( k \)), is a quantifier of the rate at which a chemical reaction proceeds. For first-order reactions, the rate constant reflects the proportionality between the reaction rate and the reactant concentration.

The exercise demonstrates the use of absorbance measurements to calculate the rate constant. After establishing the relationship between absorbance and concentration via Beer-Lambert's Law, the rate constant can be derived using the first-order rate expression:\[\begin{equation} ln\frac{c_{0}}{c_{t}} = kt,\text{\}\text{\}\text{\}\text{\}\text{\}\end{equation}\] where

• (\( c_{0} \)) is the initial concentration,
• (\( c_{t} \)) is the concentration at time \( t \),
• and \( t \) is the time that has passed.

By substituting the known concentrations and elapsed time into this equation, we can solve for (\( k \)), thus providing insight into the speed of the reaction. The calculated rate constant helps in predicting reaction behavior and is essential for comparison with theoretical values or other experimental results.

Calculating Reaction Half-Life

When discussing kinetic studies, the term reaction half-life (\( t_{1/2} \)) is frequently used. It is defined as the time required for the concentration of a reactant to decrease to half its initial value. For first-order reactions, the half-life is a constant, meaning it is independent of the initial concentration.

The formula for calculating the half-life of a first-order reaction is given by:\[\begin{equation} t_{1/2} = \frac{ln(2)}{k},\text{\}\text{\}\text{\}\end{equation}\] where (\( k \)) is the rate constant, determined earlier in our problem. The natural logarithm of 2 arises from the fact that we are specifically dealing with a half-reduction in concentration. Using the calculated rate constant, the half-life can be computed, giving us an important parameter that characterizes the timescale of the reaction. In our exercise, this value aided in determining how quickly the reaction progresses, further enhancing our understanding of the reaction dynamics.