Problem 92 - Chemistry: The Central Science

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 \(0.100 ?\)

Step-by-Step Solution

Answer

(a) The initial concentration of the colored reactant is \(c = \frac{0.605}{(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)}\). (b) The rate constant is \(k = \frac{\ln(\frac{0.605}{(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)})}{(30.0 \times 60)}\). (c) The half-life of the reaction is \(t_{1/2} = \frac{\ln(2)}{k}\). (d) The time it takes for the absorbance to fall to \(0.100\) is \(t = \frac{\ln(\frac{c_{initial}}{c_{target}})}{k}\).

Step 1: Determine the initial concentration

Using the Beer-Lambert Law, we can find the initial concentration of the colored reactant by: \(Absorbance = \epsilon \times b \times c\) where \(\epsilon\) = molar absorptivity constant (\(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1}\)) \(b\) = path length (\(1.00 \thinspace cm\)) \(c\) = initial concentration Given that the initial absorbance is \(0.605\), we can solve for the initial concentration: \(0.605 = (5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)c\) Solving for \(c\), we get: \(c = \frac{0.605}{(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)}\)

Step 2: Calculate the rate constant

We are given that the absorbance falls to \(0.250\) within \(30.0\) minutes. In order to determine the rate constant, we first need to find the concentrations corresponding to these absorbances. As described in Step 1: \(c_{initial} = \frac{0.605}{(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)}\) \(c_{final} = \frac{0.250}{(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)}\) Now, we can plug those values into the formula for a first-order reaction: \(\ln(\frac{c_{initial}}{c_{final}}) = k \times t\) where \(k\) = rate constant \(t\) = time in seconds (\(30.0 \times 60\)) Solving for \(k\), we get: \(k = \frac{\ln(\frac{0.605}{(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)})}{(30.0 \times 60)}\)

Step 3: Calculate the half-life of the reaction

With the rate constant from Step 2, we can find the half-life of the reaction using the first-order half-life formula: \(t_{1/2} = \frac{\ln(2)}{k}\)

Step 4: Calculate the time for the absorbance to fall to a specific value

Now, we will determine the time it takes for the absorbance to fall to \(0.100\). First, we will calculate the concentration corresponding to this absorbance: \(c_{target} = \frac{0.100}{(5.60 \times 10^3 \mathrm{cm}^{-1}\mathrm{M}^{-1})(1.00 \thinspace cm)}\) Now, we can use the first-order reaction formula to find the time: \(\ln(\frac{c_{initial}}{c_{target}}) = k \times t\) Solving for \(t\), we get: \(t = \frac{\ln(\frac{c_{initial}}{c_{target}})}{k}\)

Key Concepts

Beer-Lambert Law

The Beer-Lambert Law, a fundamental principle in spectroscopy, connects the absorption of light to the properties of the material it passes through. This law is crucial for determining concentrations of solutions in chemistry. It states that absorbance (A) is directly proportional to the concentration (c) of the absorbing species in the sample and the path length (b) of the light through the sample: \[A = \epsilon \times b \times c\]where \(\epsilon\) is the molar absorptivity or extinction coefficient, which signifies how strongly a substance absorbs light at a given wavelength. The initial exercise utilized this law to determine the concentration of a colored reactant, allowing students to calculate effectively using given absorbance and path length values.

Molar absorptivity

Molar absorptivity, represented by \(\epsilon\), is a measure of how well a chemical species absorbs light at a particular wavelength. It plays a crucial role in the Beer-Lambert Law for calculating concentrations. In the context of the exercise, with a molar absorptivity of \(5.60 \times 10^3\, \text{cm}^{-1}\text{M}^{-1}\), it helped determine the initial concentration of the colored reactant from the absorbance. Here are a few important points about molar absorptivity:

• Units are \(\text{cm}^{-1}\text{M}^{-1}\), ensuring consistency across measurements.
• Higher \(\epsilon\) values indicate stronger absorption, useful in identifying and quantifying substances.

Conveying the substance's efficiency in absorbing light, this concept is pivotal for quantitative spectroscopic analysis.

Half-life of reaction

The half-life of a reaction is the time required for half of the reactant to be consumed in a chemical reaction. For first-order reactions, the half-life is independent of the initial concentration and can be calculated using the rate constant \(k\). The formula for the half-life of a first-order reaction is:\[t_{1/2} = \frac{\ln(2)}{k}\]In the exercise, once the rate constant was found, this formula was used to ascertain the time it takes for the absorbance to fall to half its initial value. This concept helps chemists understand reaction dynamics and predict how a reaction proceeds over time, making it especially useful when observing concentration changes.

Rate constant

The rate constant \(k\) is an essential factor in the rate laws of chemical reactions, defining the speed of a reaction. For first-order reactions, the rate constant can be determined by tracking how the concentration of a reactive substance decreases over time. In the exercise, the relation:\[\ln\left(\frac{c_{\text{initial}}}{c_{\text{final}}}\right) = k \times t\]was used to calculate \(k\) given the decrease in absorbance over a specific period. Understanding this constant is crucial in predicting reaction rates, which are fundamental for processes in industry and research.

• Units for first-order reactions are \(\text{s}^{-1}\), ensuring time is appropriately considered.
• Its value gives insight into how quickly a reaction approaches completion.

Spectroscopy in chemistry

Spectroscopy is a powerful analytical tool in chemistry, utilized for observing and quantifying substances. It involves measuring how a sample interacts with electromagnetic radiation, often leading to insights into a molecule’s structure or concentration. In the original problem, spectroscopy followed the absorption of a colored reactant to study a first-order reaction. Spectroscopy enhanced understanding by:

• Providing information on concentration changes over time.
• Allowing calculation of essential kinetic parameters like the rate constant and half-life.

Due to its versatility and non-destructive nature, spectroscopy is widely used in both qualitative and quantitative analysis, significantly progressing research and application in various chemical fields.