Problem 97 - Chemistry: The Central Science

Question

A first-order reaction \(\mathrm{A} \longrightarrow \mathrm{B}\) has the rate constant \(k=3.2 \times 10^{-3} \mathrm{~s}^{-1}\). If the initial concentration of \(\mathrm{A}\) is \(2.5 \times 10^{-2} M\), what is the rate of the reaction at \(t=660 \mathrm{~s}\) ?

Step-by-Step Solution

Answer

The rate of the reaction at \(t=660 s\) is approximately \(2.02 \times 10^{-5} M/s\).

Step 1: Write down the first-order rate law

For a first-order reaction, the rate law is given by: \[rate = k[A]\] Where, rate = rate of reaction, k = rate constant, and [A] = concentration of reactant A. We are given k and the initial concentration of A, and we need to find the rate at t=660 seconds.

Step 2: Use the integrated rate law for a first-order reaction to find the concentration of A at t=660 seconds

The integrated rate law for a first-order reaction is given by: \[\ln{\frac{[A]}{[A]_0}} = -kt\] Where, [A] = concentration of A at time t, [A]_0 = initial concentration of A, and t = time. In this problem, we have: [A]_0 = \(2.5 \times 10^{-2} M\), k = \(3.2 \times 10^{-3} s^{-1}\), and t = \(660 s\). We need to solve this equation for [A] to find the concentration of A at t=660 seconds.

Step 3: Calculate [A] at t=660 seconds

Substitute the given values of [A]_0, k, and t in the integrated rate law equation: \[\ln{\frac{[A]}{2.5 \times 10^{-2}}} = -(3.2 \times 10^{-3})(660)\] Now, solve for [A]: \[ [A] = 2.5 \times 10^{-2} \times e^{-(3.2 \times 10^{-3})(660)}\] \[ [A] \approx 6.32 \times 10^{-3} M\]

Step 4: Compute the rate of reaction at t=660 seconds

Now that we have the concentration of A at t=660 seconds, we can find the rate of reaction. Use the first-order rate law: \[rate = k[A]\] Substitute the given value of k and the calculated value of [A] in the equation: \[rate = (3.2 \times 10^{-3})(6.32 \times 10^{-3})\] \[rate \approx 2.02 \times 10^{-5} M/s\] Thus, the rate of the reaction at t=660 seconds is approximately \(2.02 \times 10^{-5} M/s\).

Key Concepts

Understanding the Rate Constant

The rate constant, denoted as \(k\), is a crucial component in the world of reaction kinetics, especially for first-order reactions. It is a measure that helps us understand how fast a reaction proceeds. In a first-order reaction like \(\text{A} \rightarrow \text{B}\), the rate constant determines how quickly reactant A is converted into product B.
Key features of the rate constant:

• Units: For first-order reactions, the unit is usually \(s^{-1}\), meaning the reaction rate changes with time.
• Dependence: The rate constant is constant at a given temperature but may change if the temperature changes.
• Value: In our example, \(k = 3.2 \times 10^{-3} \text{s}^{-1}\), indicates the reaction proceeds at a moderate pace.

If you are given a rate constant, you can predict how changes in concentration will affect the reaction speed. It's especially vital when calculating how much reactant remains or how fast a product forms over time.

The Integrated Rate Law for First-Order Reactions

The integrated rate law is a vital equation that links concentrations of reactants with time—providing a deeper understanding than the simple rate law. For a first-order reaction, the integrated rate law is expressed as:\[\ln{\frac{[A]}{[A]_0}} = -kt\]Here’s how it works:

• \([A]\): concentration of A at time \(t\).
• \([A]_0\): initial concentration of A.
• \(k\): rate constant, conveying the reaction speed.
• \(t\): time elapsed since the reaction began.

Using this equation, we can solve for \([A]\) at any desired time, such as the 660 seconds in our example. By solving the integrated rate law, you determine that \([A] = 6.32 \times 10^{-3} \text{M}\), the concentration of A left after 660 seconds. This equation shows the exponential relationship between concentration and time for first-order kinetics.

Exploring Reaction Kinetics

Reaction kinetics is the study of the speed or rate at which chemical reactions occur. For first-order reactions, understanding kinetics involves:

• Determining how reactant concentration changes over time using rate laws and integrated rate laws.
• Using rate constants to predict how quickly a reaction proceeds.

In our specific problem, reaction kinetics allows us to calculate that the rate of reaction at 660 seconds is \(2.02 \times 10^{-5} \text{M/s}\).

The reaction rate depends on both the rate constant and the current concentration of the reactant. Reaction kinetics helps chemists tailor reactions to desired speeds, crucial in industries where reaction time affects production.
By mastering the concepts of rate constants and integrated rate laws, you can predict and manipulate how quickly a reaction will reach completion. This skill is essential for chemists in both academic and industrial settings.