Problem 38 - Chemistry The Molecular Science

Question

The compound \(\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}\) decomposes in a first-order reaction to elemental Xe with a half-life of \(30 . \mathrm{min}\). If you place \(7.50 \mathrm{mg} \mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}\) in a flask, calculate how long you must wait until only \(0.25 \mathrm{mg} \mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}\) remains.

Step-by-Step Solution

Answer

Wait approximately 147.2 minutes for the compound to decompose to 0.25 mg.

Step 1: Understanding First-Order Reaction Kinetics

In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. The half-life of a first-order reaction is a constant value, given by the formula: \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant of the reaction.

Step 2: Calculate the Rate Constant

Given the half-life (\( t_{1/2} \)) is 30 minutes, we can calculate the rate constant (\( k \)) using the formula: \( k = \frac{0.693}{t_{1/2}} \). Substituting the given half-life, \( k = \frac{0.693}{30} \text{ min}^{-1} \).

Step 3: Apply First-Order Kinetics Equation

The first-order kinetics equation is: \( \ln \left( \frac{[A]_0}{[A]} \right) = kt \). Here, \([A]_0\) is the initial concentration and \([A]\) is the remaining concentration after time \( t \). Convert the masses given into the equation: \( \ln \left( \frac{7.50}{0.25} \right) = kt \).

Step 4: Substitute Values and Solve for Time

Substitute the calculated rate constant \( k = \frac{0.693}{30} \) and solve for time \( t \): \( \ln(30) = \left( \frac{0.693}{30} \right)t \). Calculate \( \ln(30) \) and solve for \( t \): \( t = \frac{\ln(30)}{\frac{0.693}{30}} \).

Step 5: Calculate and Conclude

Calculate \( \ln(30) \approx 3.401 \). Thus, \( t = \frac{3.401}{0.0231} \approx 147.2 \text{ minutes} \). The decomposition will take approximately 147.2 minutes for \(7.50 \text{ mg} \) to decrease to \(0.25 \text{ mg} \).

Key Concepts

Half-Life Calculation

In first-order reaction kinetics, the half-life is the time required for half of the reactant to transform or decompose. The half-life of a first-order reaction remains constant and is independent of the initial concentration of the reactants. This unique feature makes these reactions particularly interesting because it facilitates the prediction of how the concentration of a substance changes over time.
To calculate the half-life, we use the formula: \[ t_{1/2} = \frac{0.693}{k} \]where \( t_{1/2} \) represents the half-life and \( k \) is the rate constant of the reaction. The constant 0.693 is derived from the natural logarithm of 2, as the concept hinges on exponential decay.
Understanding half-life is vital for determining how long a reaction needs to reach a certain stage, like the decomposition of a chemical, which is crucial in various scientific fields such as chemistry, biology, and even pharmacology.

Rate Constant Determination

The rate constant, \( k \), in first-order reaction kinetics offers insight into the speed of the reaction process. It is essential because it determines how quickly a reactant is transformed into products over a given time. For the decomposition reaction of a compound like \( \mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2} \), the rate constant can be calculated using the half-life formula rearranged as:\[ k = \frac{0.693}{t_{1/2}} \]Given a half-life, the rate constant is inversely related. A shorter half-life means a larger rate constant, indicating a faster reaction process. For instance, if the half-life is 30 minutes, substituting in the half-life equation gives:\[ k = \frac{0.693}{30 \, \text{min}} = 0.0231 \, \text{min}^{-1} \]This rate constant is crucial for calculating reaction kinetics, helping us understand not only the time required for concentration changes but also the underlying speed of chemical transformations.

Exponential Decay

Exponential decay is a key concept in first-order reactions, highlighting how quantities decrease rapidly at a rate proportional to their current value. The mathematical model describing first-order reaction rates leverages this principle to predict how a reactant concentration diminishes over time.
For first-order reactions, the concentration \([A]\) of the reactant at any time \(t\) can be expressed by the relation:\[ \ln \left( \frac{[A]_0}{[A]} \right) = kt \]where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(k\) is the rate constant. This shows that the concentration decreases exponentially with time, directly influenced by the rate constant.
In the given exercise, the exponential decay formula was used to find how long it takes for \(7.50 \, \text{mg} \) of \( \mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2} \) to reduce to \(0.25 \, \text{mg} \). By calculating the natural logarithm of the concentration ratio and dividing by the rate constant, one can determine the time required for such a change, emphasizing the core principle of exponential decay in first-order kinetics.