a. What is the half-life for the first-order decomposition of SO2Cl2 with a rate constant of 1.42 x 10^-4 s^-1? b. How long will it take for the concentration of SO2Cl2 to decrease to 25% of its initial concentration? c. If the initial concentration of SO2Cl2 is 1.00 M, how long will it take for the concentration to decrease to 0.78 M? d. If the initial concentration of SO2Cl2 is 0.150 M, what is the concentration of SO2Cl2 after 2.00 x 10^2 s? After 5.00 x 10^2 s?

Verified step by step guidance

1. Step 1: For part (a), use the formula for the half-life of a first-order reaction: \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant.
2. Step 2: For part (b), use the first-order integrated rate law: \( \ln \left( \frac{[A]_t}{[A]_0} \right) = -kt \). Set \( [A]_t = 0.25[A]_0 \) and solve for \( t \).
3. Step 3: For part (c), use the same first-order integrated rate law: \( \ln \left( \frac{[A]_t}{[A]_0} \right) = -kt \). Set \( [A]_0 = 1.00 \text{ M} \) and \( [A]_t = 0.78 \text{ M} \), then solve for \( t \).
4. Step 4: For part (d), use the first-order integrated rate law again: \( \ln \left( \frac{[A]_t}{[A]_0} \right) = -kt \). First, calculate \( [A]_t \) after \( 2.00 \times 10^2 \text{ s} \) with \( [A]_0 = 0.150 \text{ M} \).
5. Step 5: Repeat the calculation from Step 4 for \( 5.00 \times 10^2 \text{ s} \) to find the concentration \( [A]_t \) after this time period.

Key Concepts

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant decreases, the rate of the reaction also decreases. The mathematical representation of a first-order reaction is given by the equation: rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.

Half-Life

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For first-order reactions, the half-life is constant and can be calculated using the formula t1/2 = 0.693/k, where k is the rate constant. This property is crucial for predicting how long it will take for a substance to decompose or react under first-order kinetics.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of first-order reactions, the concentration of the reactant over time can be expressed with the equation [A] = [A]0 e^(-kt), where [A]0 is the initial concentration, k is the rate constant, and t is time. This concept is essential for calculating the concentration of a reactant at any given time during the reaction.