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: $$ \begin{array}{rl} \hline \text { Time (min) } & \text { Absorbance at } 608 \mathrm{nm} \\ \hline 0 & 1.254 \\ 30 & 0.941 \\ 60 & 0.752 \\ 90 & 0.672 \\ 120 & 0.545 \end{array} $$ From these data, determine the rate law for the reaction "dye \(\longrightarrow\) product" and determine the rate constant.

Short answer

From the given absorbance data and by applying Beer-Lambert Law, we can convert this data into concentration values. Then, by analyzing the relationship between the reaction rate and the concentrations of the dye, we can determine the reaction order. In this case, we can check for 0th, 1st, and 2nd order reactions and calculate reaction half-life for each order to find the best fit. Once we have determined the reaction order, we can use the rate law and the concentration data to determine the rate constant, \(k\), for the decay process.

Step-by-step solution

Convert absorbance data into concentration data

We can use the Beer-Lambert Law to convert the given absorbance data into concentration data. The Beer-Lambert Law is given by: \[A = \epsilon \cdot c \cdot l\] where \(A\) is the absorbance, \(\epsilon\) is the molar absorptivity (extinction coefficient), \(c\) is the concentration, and \(l\) is the path length of the cuvette. We can rearrange the formula to solve for the concentration: \[c = \frac{A}{\epsilon \cdot l}\] For our given exercise, the extinction coefficient \(\epsilon = 4.7 \times 10^4 M^{-1} cm^{-1}\) and the path length \(l = 1 cm\). Using this information, we can convert the absorbance data into concentration data.

Determine the order of the reaction

Next, we have to determine the order of the reaction. To do this, we will have to analyze the relationship between the rate of the reaction and the concentrations of the reactant (dye). We will consider different reaction orders and test them by calculating the reaction half-life. We will examine 0th, 1st, and 2nd order reactions. For each order, we can use the half-life expressions for that order and see which one fits the best with our data.

Calculate the rate constant

Once we have determined the order of the reaction, we can calculate the rate constant using the rate law and the experimental concentration data. The rate law is given by: \[r = k \cdot [Dye]^n\] where \(r\) is the reaction rate, \(k\) is the rate constant, \([Dye]\) is the concentration of the dye, and \(n\) is the order of the reaction. First, find the reaction rate (\(r\)) at each time point using the concentration data. Then, rearrange the rate law to solve for the rate constant \(k\), and finally, calculate the rate constant for each experimental time point.

Key concepts

Beer-Lambert Law

The Beer-Lambert Law is fundamental to understanding how we can use light to determine the concentration of solutions in spectrophotometry. It establishes a direct relationship between the absorbance (A) of light passing through a material and the properties of the material itself—the concentration (c) of the absorbing species and the path length (l) of the sample that light travels through. Specifically, the law is described by the equation: A = ε c l, where ε represents the extinction coefficient, which is an intrinsic property of the substance that quantifies how strongly the substance absorbs light at a particular wavelength.

The importance of this law lies in its ability to enable the conversion of absorbance data to concentration data, which is crucial in many fields, including chemistry and biology. When working with colored dye compounds, as in the exercise described, the Beer-Lambert Law is utilized to monitor the decomposition of the dye to a colorless product by observing changes in absorbance at a specific wavelength.

Reaction Kinetics

Reaction kinetics involves studying the rates of chemical reactions and the factors that affect these rates. It's all about understanding how quickly reactants turn into products and identifying the steps of molecular transformation. When examining reaction kinetics, scientists investigate the relationship between various factors like concentration, temperature, and catalyst presence, and the speed of chemical reactions.

In our textbook exercise, reaction kinetics is applied by observing how the concentration of a colored dye changes over time as it decomposes into a product. By analyzing this data, students can determine the order of the reaction—whether it’s zeroth, first, or second order. Each order of reaction has its own implications on how the concentration of reactants affects the rate of the reaction, and understanding this is key to mastering chemical reactions in a laboratory setting and in industry.

Extinction Coefficient

The extinction coefficient, also known as the molar absorptivity, is a value that indicates how strongly a substance can absorb light at a particular wavelength. It's a constant that reflects the substance's effectiveness in preventing light from passing through a solution at the specified wavelength. The higher the extinction coefficient, the more absorbent the substance is.

In our exercise, the dye’s extinction coefficient is given and is vital for converting absorbance into concentration using the Beer-Lambert Law. With the known extinction coefficient (ε = 4.7 x 10^4 M^{-1} cm^{-1}), we have a key piece of the puzzle allowing us to calculate the concentration of the dye in solution at different times, which subsequently helps in determining the rate law for the dye’s decomposition. The precision of the extinction coefficient is critical as it directly impacts the accuracy of the concentration data obtained from absorbance measurements.

Spectrophotometry

Spectrophotometry is a technique that measures how much a chemical substance absorbs light by measuring the intensity of light as a beam of light passes through sample solution. The basic principle is that each compound absorbs or transmits light over a certain range of wavelengths. This analytical method is commonly used to determine the concentration of substances in solution, especially in reactions like the one in our exercise, where the dye changes color as it reacts.

This technique also allows us to follow the reaction over time by taking periodic measurements of absorbance at specific wavelengths, which can be used to understand the rate at which the dye degrades. By using spectrophotometry, students are able to visually represent and quantify the extent of the reaction, giving them a concrete data set that they can use to extrapolate the kinetics and the mechanics behind the chemical reactions occurring.