Question

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorption of a colored reactant. The reaction occurs in a \(1.00-\mathrm{cm}\) sample cell, and the only colored species in the reaction has a molar absorptivity constant of \(5.60 \times 10^{3} \mathrm{~cm}^{-1} \mathrm{M}^{-1}\). (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 \(0.250\) within \(30.0\) 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 \(0.100 ?\)

Short answer

The initial concentration of the colored reactant is \(1.08 \times 10^{-4} \mathrm{M}\). The rate constant is \(1.32 \times 10^{-4} \mathrm{s^{-1}}\). The half-life of the reaction is 5240 seconds. Finally, it takes 14700 seconds for the absorbance to fall to 0.100.

Step-by-step solution

(a) Beer-Lambert Law and Initial Concentration Calculation

To find the initial concentration of the colored reactant, we'll need to use the Beer-Lambert law: \(A = \epsilon c l\) Where \(A\) is the absorbance, \(\epsilon\) is the molar absorptivity constant, \(c\) is the concentration of the colored reactant, and \(l\) is the path length. We are given the absorbance (\(A = 0.605\)), molar absorptivity constant (\(\epsilon = 5.60 \times 10^{3} \mathrm{cm^{-1}M^{-1}}\)), and path length (\(l = 1.00 \mathrm{cm}\)) and asked to find the concentration \(c\). Rearrange the equation to solve for \(c\): \(c = \frac{A}{\epsilon l}\) Now, substitute the given values into the equation: \(c = \frac{0.605}{(5.60 \times 10^{3} \mathrm{cm^{-1}M^{-1}})(1.00 \mathrm{cm})} = 1.08 \times 10^{-4} \mathrm{M}\) So, the initial concentration of the colored reactant is \(1.08 \times 10^{-4} \mathrm{M}\).

(b) First-Order Rate Law Integration and Rate Constant Calculation

To find the rate constant, we need to use the integrated first-order rate law: \(A_{t} = A_{0}e^{-kt}\) Where \(A_{t}\) is the absorbance at time \(t\), \(A_{0}\) is the initial absorbance, \(k\) is the rate constant, and \(t\) is the reaction time. We are given the initial absorbance (\(A_{0} = 0.605\)), the absorbance at 30.0 minutes (\(A_{t} = 0.250\)), and the time (\(t = 30.0 \mathrm{~min} = 1800 \mathrm{~s}\)), and are asked to find the rate constant (\(k\)). First, rearrange the equation to solve for \(k\): \(k = \frac{-\ln (A_{t}/A_{0})}{t}\) Next, substitute the given values into the equation: \(k = \frac{-\ln(0.250/0.605)}{1800 \mathrm{s}}= 1.32 \times 10^{-4} \mathrm{s^{-1}}\) So, the rate constant is \(1.32 \times 10^{-4} \mathrm{s^{-1}}\).

(c) Half-Life Calculation

To calculate the half-life for a first-order reaction, we can use the formula: \(t_{1/2} = \frac{0.693}{k}\) Since we have calculated the rate constant in the previous step (\(k = 1.32 \times 10^{-4} \mathrm{s^{-1}}\)), we only need to substitute it into the equation: \(t_{1/2} = \frac{0.693}{1.32 \times 10^{-4} \mathrm{s^{-1}}}= 5240 \mathrm{~s}\) So, the half-life of the reaction is 5240 seconds.

(d) Time Calculation for Absorbance Reduction

To find the time it takes for the absorbance to fall to 0.100, we can use the integrated first-order rate law again: \(A_{t} = A_{0}e^{-kt}\) We are given the initial absorbance (\(A_{0} = 0.605\)), the desired absorbance (\(A_{t} = 0.100\)), and the rate constant (\(k = 1.32 \times 10^{-4} \mathrm{s^{-1}}\)), and are asked to find the time (\(t\)). First, rearrange the equation to solve for \(t\): \(t = -\frac{\ln(A_{t}/A_{0})}{k}\) Next, substitute the given values into the equation: \(t = -\frac{\ln(0.100/0.605)}{1.32 \times 10^{-4} \mathrm{s^{-1}}}= 14700 \mathrm{~s}\) So, it takes 14700 seconds for the absorbance to fall to 0.100.

Key concepts

Beer-Lambert Law

The Beer-Lambert Law is a fundamental concept in spectroscopy and is crucial for understanding how we can relate the absorbance of a substance to its concentration. The law is expressed by the equation \( A = \epsilon c l \), where:

• \( A \) is the absorbance of the substance
• \( \epsilon \) is the molar absorptivity constant, which indicates how well the substance absorbs light at a particular wavelength
• \( c \) is the concentration of the substance
• \( l \) is the path length, which is the distance the light travels through the sample

This law helps chemists determine the concentration of a chemical in a solution by measuring how much light is absorbed by the sample. In practice, this involves using a spectrophotometer to measure the absorbance at a known path length. By knowing the molar absorptivity constant, one can easily find the concentration of the substance. If you ever need to calculate initial concentrations in chemistry, this law is a powerful starting point. Always remember that higher absorbance indicates greater concentrations.

Rate Constant Calculation

Calculating the rate constant \( (k) \) for a reaction, especially one that follows first-order kinetics, involves the integrated rate law equation. The equation for a reaction showing first-order kinetics is \( A_t = A_0 e^{-kt} \).

• \( A_t \) is the absorbance at time \( t \)
• \( A_0 \) is the initial absorbance, measured at the start of the reaction
• \( k \) is the rate constant that we want to find
• \( t \) is the time that has passed during the reaction

To solve for \( k \), you can rearrange this formula: \[k = \frac{-\ln (A_t/A_0)}{t}\]After determining all these values, you simply plug them into this formula to find the rate constant. This value tells us how quickly the reaction proceeds and is fundamental in determining other reaction variables, such as half-life and reactant concentration over time.

Half-Life Determination

The half-life \( (t_{1/2}) \) of a reaction is the time it takes for the concentration of a reactant to reduce to half its initial value. For first-order reactions, the half-life is a constant and can be easily calculated using the formula:\[t_{1/2} = \frac{0.693}{k}\]Here, \( k \) is the rate constant of the reaction, determined from the previous calculations. What's interesting about first-order reactions is that their half-lives do not depend on the initial concentration of the reactant. This is particularly useful in many real-world applications where concentrations might change or be unknown. Calculating half-life can help predict how long a reaction requires to reach a particular stage and is essential for planning in both industrial and laboratory settings.

Spectroscopy in Chemistry

Spectroscopy plays a key role in chemistry for analyzing substances. It's a technique that involves measuring how much light a chemical substance absorbs. By using the Beer-Lambert Law, it provides valuable insights into the concentration of analytes in a sample.

• Spectroscopy works by passing light through a sample and recording the absorbance, allowing chemists to identify and quantify molecules.
• It provides a non-destructive way to find the compositions and concentrations of different solutions.
• Advanced spectroscopic techniques can also determine the structure of molecules, types of chemical bonds present, and even the physical properties of substances.

This method is widely used not only in chemical research but also in many industrial processes, quality control, and even in environmental monitoring. With spectroscopy, chemists can achieve precise and reliable results simply and efficiently, making it an indispensable tool in the world of chemistry.