Problem 102 - Chemistry The Central Science In Si Units

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 \(1-\mathrm{cm}\) cuvette in a spectrometer and obtain the following data: \begin{tabular}{cc} \hline Time (min) & Absorbance at \(608 \mathrm{nm}\) \\ \hline 0 & 1.254 \\ 30 & 0.941 \\ 60 & 0.752 \\ 90 & 0.672 \\ 120 & 0.545 \\ \hline \end{tabular} From these data, determine the rate law for the reaction "dye \(\longrightarrow\) product" and determine the rate constant.

Step-by-Step Solution

Answer

Determine reaction order by plotting concentration, ln(concentration), and 1/concentration over time. Calculate rate constant using the order-dependent plot's slope.

Step 1: Understanding Beer-Lambert Law

The Beer-Lambert Law relates absorbance to concentration: \(A = \epsilon c l\), where \(A\) is the absorbance, \(\epsilon\) is the molar extinction coefficient, \(c\) is the concentration of the solution, and \(l\) is the path length of the cuvette. Given the absorption readings at different times, we can calculate the concentration at each time point by rearranging the formula to \(c = \frac{A}{\epsilon l}\).

Step 2: Initial Concentration Calculation

The initial absorbance at \(t=0\) is 1.254. Using \(\epsilon = 4.7 \times 10^4 \ \mathrm{M}^{-1}\mathrm{cm}^{-1}\) and \(l = 1\ \mathrm{cm}\), the initial concentration \(c_0\) is calculated as follows:\[c_0 = \frac{A_0}{\epsilon l} = \frac{1.254}{4.7 \times 10^4 \times 1} = 2.67 \times 10^{-5} \ \mathrm{M}\].

Step 3: Calculation of Concentrations at Different Times

Calculate the concentrations for each recorded absorbance value: - At 30 min: \(c_{30} = \frac{0.941}{4.7 \times 10^4} = 2.00 \times 10^{-5} \ \mathrm{M}\)- At 60 min: \(c_{60} = \frac{0.752}{4.7 \times 10^4} = 1.60 \times 10^{-5} \ \mathrm{M}\)- At 90 min: \(c_{90} = \frac{0.672}{4.7 \times 10^4} = 1.43 \times 10^{-5} \ \mathrm{M}\)- At 120 min: \(c_{120} = \frac{0.545}{4.7 \times 10^4} = 1.16 \times 10^{-5} \ \mathrm{M}\)"."

Step 4: Determining Reaction Order from Plot

To determine the reaction order, plot the concentration (\([ ext{Dye}]\)), ln(\([ ext{Dye}]\)), and \(1/[ ext{Dye}]\) over time. - Zero-order: \([ ext{Dye}] = -kt + [ ext{Dye}]_0\), graph should be linear.- First-order: \( ext{ln}([ ext{Dye}]) = -kt + ext{ln}([ ext{Dye}]_0)\), graph should be linear.- Second-order: \(1/[ ext{Dye}] = kt + 1/[ ext{Dye}]_0\), graph should be linear. The correct graph should be identified to determine the order.

Step 5: Calculate the Rate Constant (k)

Based on the reaction order identified, use the corresponding linear plot equation to solve for the rate constant \(k\). For instance, if the reaction is found to be first-order, \(k\) can be obtained from the slope of the ln(\([ ext{Dye}]\)) vs. time plot. The unit of \(k\) will depend on the reaction order: - Zero order: \(\text{M/min}\) - First order: \(\text{min}^{-1}\) - Second order: \(\text{M}^{-1}\text{min}^{-1}\)".

Key Concepts

Beer-Lambert Law

The Beer-Lambert Law is a fundamental principle in chemistry that relates the absorbance of light to the properties of the material through which the light is traveling. The formula is given by:

• \(A = \epsilon c l\)

Here:- \(A\) represents the absorbance, a measure of the amount of light absorbed by the solution.- \(\epsilon\) is the molar extinction coefficient, a constant indicating how strongly a substance absorbs light at a given wavelength.- \(c\) is the concentration of the absorbing species.- \(l\) is the length of the path that the light travels through.
This law is crucial in spectroscopy as it allows us to calculate the concentration of a substance in solution from the absorbance. By rearranging the formula to \(c = \frac{A}{\epsilon l}\), we can determine how much of the substance is present when given the absorbance, molar extinction coefficient, and path length. This makes the Beer-Lambert Law highly beneficial for tracking the progress of reactions that involve absorbing species.

Reaction Order

The reaction order is an important concept in chemical kinetics. It describes how the rate of a chemical reaction depends on the concentration of the reactants. To determine the reaction order, we observe how the reaction progresses over time and analyze plots of concentration data.
The reaction order can be determined using the following characteristics:

• Zero-Order: The rate is constant and does not depend on the concentration of the reactant. A plot of concentration vs. time is linear.
• First-Order: The rate is directly proportional to the concentration of one reactant. A plot of the natural log of concentration vs. time is linear.
• Second-Order: The rate is proportional to the square of the concentration of one reactant or to the product of two reactant concentrations. A plot of the inverse of concentration vs. time is linear.

Understanding the reaction order is critical to finding the rate law and predicting how changes in concentration can affect the rate of the reaction.

Rate Constant

In kinetics, the rate constant \(k\) is a crucial parameter of the rate law. It quantifies the speed of a chemical reaction and varies depending on the reaction order.

• For a first-order reaction, the rate constant \(k\) has the units of \(\text{min}^{-1}\).
• For zero-order reactions, \(k\) is expressed in \(\text{M/min}\).
• For second-order reactions, \(k\) is in \(\text{M}^{-1}\text{min}^{-1}\).

To determine \(k\), one must first identify the order of the reaction by analyzing different concentration plots. Once the correct plot type is determined, the slope of this plot gives the value of \(k\). The units of \(k\) will vary based on the reaction order, signifying how changes in reactant concentration affect the rate. Knowing the rate constant is essential for predictive modelling of chemical reactions over time.

Spectroscopy

Spectroscopy refers to the study of the interaction between matter and electromagnetic radiation. It is fundamentally used to analyze the structure and concentration of substances. In the context of this exercise, spectroscopy is employed to understand how light interacts with the dye compound as it decomposes.
The spectrometer measures the absorbance of light at a specific wavelength, which, for the dye in question, is 608 nm. By tracking changes in absorbance over time, one can infer changes in concentration during chemical reactions. This technique is invaluable as it provides a non-intrusive method to monitor reactions accurately. For students, mastering the use of spectroscopy not only aids in lab work but also deepens comprehension of how molecules react and change.

Absorbance

Absorbance is a key concept in understanding how spectroscopy is used to track chemical reactions. It indicates the amount of light absorbed by a sample solution and is directly related to concentration.
High absorbance means more light is absorbed and suggests a higher concentration of the absorbing species, while low absorbance indicates less absorption and therefore, a lower concentration.

• Absorbance is given by the formula: \(A = \epsilon c l\).
• Changes in absorbance over time can reveal how a reaction proceeds.

For recording changes accurately, spectrometry is used to measure absorbance at specific wavelengths, illustrating how much of the light is absorbed by dyes or other absorbing compounds. Understanding absorbance is essential for determining reaction kinetics and for applications such as quantifying the progress of a reaction or concentration changes.