A first-order reaction A → B has the rate constant k = 3.2 * 10^-3 s^-1. If the initial concentration of A is 2.5 * 10^-2 M, what is the rate of the reaction at t = 660 s?

Verified step by step guidance

Step 1: Understand that for a first-order reaction, the rate of the reaction is given by the formula: rate = k[A], where k is the rate constant and [A] is the concentration of A at time t.

Step 2: Use the integrated rate law for a first-order reaction to find the concentration of A at time t. The integrated rate law is: ln([A]_t/[A]_0) = -kt, where [A]_t is the concentration at time t, [A]_0 is the initial concentration, and t is the time.

Step 3: Substitute the given values into the integrated rate law: ln([A]_t/2.5 \times 10^{-2}) = -(3.2 \times 10^{-3}) \times 660.

Step 4: Solve the equation from Step 3 to find [A]_t, the concentration of A at time t = 660 s.

Step 5: Substitute the value of [A]_t obtained in Step 4 into the rate equation from Step 1 to calculate the rate of the reaction at t = 660 s.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 rate law for a first-order reaction can be expressed as rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.

Rate Constant (k)

The rate constant (k) is a proportionality factor in the rate law that is specific to a given reaction at a specific temperature. It indicates how quickly a reaction proceeds; a larger k value signifies a faster reaction. For first-order reactions, the units of k are typically s^-1, reflecting the dependence of the reaction rate on the concentration of the reactant.

Integrated Rate Law for First-Order Reactions

The integrated rate law for a first-order reaction relates the concentration of the reactant at any time (t) to its initial concentration and the rate constant. It is expressed as ln([A]0/[A]) = kt, where [A]0 is the initial concentration, [A] is the concentration at time t, and k is the rate constant. This equation allows for the calculation of concentration changes over time, which is essential for determining the rate of the reaction at a specific time.

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First-Order Reactions

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The reaction 2 NO₂ → 2 NO + O₂ has the rate constant k = 0.63 M^-1s^-1. (a) Based on the units for k, is the reaction first or second order in NO₂?

Textbook Question

The reaction 2 NO₂ → 2 NO + O₂ has the rate constant k = 0.63 M^-1s^-1.

(b) If the initial concentration of NO₂ is 0.100 M, how would you determine how long it would take for the concentration to decrease to 0.025 M?

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Consider two reactions. Reaction (1) has a constant halflife, whereas reaction (2) has a half-life that gets longer as the reaction proceeds. What can you conclude about the rate laws of these reactions from these observations?

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Americium-241 is used in smoke detectors. It has a first-order rate constant for radioactive decay of k = 1.6 * 10-3 yr-1. By contrast, iodine-125, which is used to test for thyroid functioning, has a rate constant for radioactive decay of k = 0.011 day-1. (a) What are the half-lives of these two isotopes? (b) Which one decays at a faster rate?

Textbook Question

Americium-241 is used in smoke detectors. It has a first-order rate constant for radioactive decay of k = 1.6 * 10-3 yr-1. By contrast, iodine-125, which is used to test for thyroid functioning, has a rate constant for radioactive decay of k = 0.011 day-1. (c) How much of a 1.00-mg sample of each isotope remains after three half-lives? (d) How much of a 1.00-mg sample of each isotope remains after 4 days?