Problem 102 - Chemistry: The Central Science

Question

A colored dye compound decomposes to give a colorless product. The original dye absorbs at \(608 \mathrm{~nm}\) and has an extinction coefficient of \(4.7 \times 10^{4} \mathrm{M}^{-1} \mathrm{~cm}^{-1}\) at that wavelength. You perform the decomposition reaction in a 1 -cm cuvette in a spectrometer and obtain the following data: From these data, determine the rate law for the reaction "dye \(\longrightarrow\) product" and determine the rate constant.

Step-by-Step Solution

Answer

#ShortAnswer# The rate law for the decomposition reaction of the colored dye compound is given by: \[\frac{d[\text{dye}]}{dt} = k[\text{dye}]\] Using the provided data and Beer-Lambert Law, the rate constant \(k\) can be calculated as: \[k = \frac{\frac{d[\text{dye}]_{30}}{d[\text{dye}]_{0}}}{[\text{dye}]_{30}}\] Plug in the values and solve to find the rate constant \(k\).

Step 1: Understand the Beer-Lambert Law

The Beer-Lambert Law describes the relationship between the absorbance (A) of light by a solution and the properties of the solution, given as: \[A = \epsilon \times c \times l \] where: - \(A\) is the absorbance at a specific wavelength, - \(\epsilon\) is the molar extinction coefficient with units \(\mathrm{M}^{-1} \mathrm{cm}^{-1}\), - \(c\) is the concentration of the solution in molarity (\(\mathrm{M}\)), and - \(l\) is the path length of the cuvette in centimeters (\(\mathrm{cm}\)). Given the extinction coefficient (\(\epsilon\)) and path length (\(l\)), we can calculate the concentration (\(c\)) of the dye at different times using the provided absorbance values.

Step 2: Calculate the concentration at each time point

Using the Beer-Lambert Law, we can find the concentration (\(c\)) at each time point from the absorbance values. \[c = \frac{A}{\epsilon \times l} \] For the given data, Time (min) | Absorbance | Concentration (\(\mathrm{M}\)) --- | --- | --- 0 | 1.254 | \(c_0 = \frac{1.254}{4.7 \times 10^{4} \times 1} \) 30 | 0.941 | \(c_{30} = \frac{0.941}{4.7 \times 10^{4} \times 1} \) 60 | 0.752 | \(c_{60} = \frac{0.752}{4.7 \times 10^{4} \times 1} \) 90 | 0.672 | \(c_{90} = \frac{0.672}{4.7 \times 10^{4} \times 1} \) 120 | 0.545 | \(c_{120} = \frac{0.545}{4.7 \times 10^{4} \times 1} \)

Step 3: Determine the rate law using the method of initial rates

To determine the rate law, let's assume the general form of a first-order reaction: \[\frac{d[\text{dye}]}{dt} = k[\text{dye}]^n\] Using the method of initial rates, we'll compare the ratio of initial rates and initial concentrations between two different time points (0 and 30 minutes). \[\frac{\frac{d[\text{dye}]_{30}}{dt}}{\frac{d[\text{dye}]_{0}}{dt}} = \frac{k[\text{dye}]_{30}^n}{k[\text{dye}]_{0}^n}\] Since the reaction is first-order, \(n = 1\), and we can simplify the expression as: \[\frac{d[\text{dye}]_{30}}{d[\text{dye}]_{0}} = \frac{[\text{dye}]_{30}}{[\text{dye}]_{0}}\]

Step 4: Calculate the rate constant k using the initial rates

Now that we have the rate law, we can find the rate constant \(k\), as it's the only unknown. We can find the rate of change in dye concentration from the provided data: \[\frac{d[\text{dye}]_{30}}{d[\text{dye}]_{0}} = \frac{[\text{dye}]_{30} - [\text{dye}]_{0}}{30}\] And use the simplified expression to determine \(k\): \[k = \frac{\frac{d[\text{dye}]_{30}}{d[\text{dye}]_{0}}}{[\text{dye}]_{30}}\] Now, we can plug in the values to calculate the rate constant \(k\).

Key Concepts

Reaction Rate Law

The reaction rate law is a fundamental concept in chemical kinetics that expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. In the context of the exercise, we are dealing with the decomposition of a dye. The rate law for this reaction provides insight into how quickly the dye breaks down into the colorless product over time.

To determine the rate law, we typically start with a general expression. For our dye decomposition, we assume the form:

• \( \frac{d[\text{dye}]}{dt} = k[\text{dye}]^n \)

where \( \frac{d[\text{dye}]}{dt} \) represents the rate of change of the dye concentration, \( k \) is the rate constant, and \( n \) is the order of the reaction, a value we need to determine.
In many cases, the order \( n \) can be found experimentally using the method of initial rates, by observing how the rate of reaction changes as the concentration of a reactant is varied. For a first-order reaction, \( n = 1 \), indicating that the rate is directly proportional to the concentration of the dye. This conclusion can be supported by comparing concentration ratios at successive time intervals and noting that the concentration halves over equal time periods, consistent with first-order kinetics.

Absorbance and Concentration Relationship

The relationship between absorbance and concentration is described by the Beer-Lambert Law, which connects how much light a solution absorbs to the amount of the absorbing species present in that solution. This is particularly useful in monitoring reactions that involve colored compounds, such as the dye decomposition in the exercise.

The Beer-Lambert Law is expressed as:

• \( A = \epsilon \times c \times l \)

where \( A \) is the absorbance, \( \epsilon \) is the molar extinction coefficient, \( c \) is the concentration, and \( l \) is the path length.
In this reaction, knowing the extinction coefficient \( \epsilon \), the path length \( l \) (usually the cuvette's width), and measuring absorbance \( A \), we can compute the concentration \( c \) at any given time. This is crucial for monitoring the rate at which the dye decomposes. Essentially, the higher the absorbance, the higher the concentration of the dye.
Determining the concentration at various times allows us to plot concentration versus time and analyze the change over time to further understand the reaction kinetics.

First-Order Reaction Kinetics

First-order reaction kinetics describes how the concentration of reactants decreases over time at a rate proportional to their concentration. For the dye decomposition to a colorless product, as observed, this typically implies:

• The reaction rate is directly proportional to the concentration of the dye.
• The logarithm of the concentration decreases linearly over time.

A first-order rate law is given by:

• \( \frac{d[\text{dye}]}{dt} = k[\text{dye}] \)

It's useful to note that in a first-order reaction, the half-life (time for half of the substance to react) is constant and independent of its initial concentration, making it simpler to predict how concentration drops over time.

Using absorbance data and the Beer-Lambert Law, as seen in the exercise, allows you to transform absorbance values to concentration values. Then, by plotting these concentrations logarithmically against time, you’ll observe a straight line if the reaction is first-order. The slope of this line gives the negative rate constant \( k \), thus fully describing the kinetics for practical applications like predicting reaction times and concentrations.