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?

Verified step by step guidance

Step 1: Understand the concept of half-life. In each half-life, half of the remaining radioactive isotope decays. So, after one half-life, half of the original amount remains. After two half-lives, half of the remaining half decays, leaving 1/4 of the original amount. After three half-lives, half of the remaining 1/4 decays, leaving 1/8 of the original amount.

Step 2: Apply this concept to the given problem. We start with a 1.00-mg sample of each isotope. After three half-lives, 1/8 of this original amount remains.

Step 3: Calculate the remaining amount for each isotope. For Americium-241, multiply the original amount (1.00 mg) by 1/8 to find the remaining amount.

Step 4: Repeat the calculation for Iodine-125. Again, multiply the original amount (1.00 mg) by 1/8 to find the remaining amount.

Step 5: Note that the rate constants given in the problem are not needed to solve this part of the problem. The amount remaining after a certain number of half-lives is independent of the specific rate of decay.

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Key Concepts

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

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay occurs at a characteristic rate for each isotope, defined by its decay constant (k). The rate of decay is exponential, meaning that the quantity of the substance decreases by a consistent fraction over equal time intervals.

Half-Life

The half-life of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. This concept is crucial for understanding how much of a substance remains after a certain period. For any isotope, after one half-life, 50% of the original amount remains; after two half-lives, 25% remains, and so on.

First-Order Kinetics

First-order kinetics refers to a reaction or decay process where the rate is directly proportional to the concentration of one reactant. In the context of radioactive decay, this means that the decay rate of an isotope is constant and independent of the amount present. The relationship can be described mathematically, allowing for the calculation of remaining quantities after a given time.

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First-Order Reactions

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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. (a) What are the half-lives of these two isotopes? (b) Which one decays at a faster rate?

Textbook Question

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorbance of a colored reactant at 520 nm. The reaction occurs in a 1.00-cm sample cell, and the only colored species in the reaction has an extinction coefficient of 5.60 × 10³ M^-1 cm^-1 at 520 nm.

(a) Calculate the initial concentration of the colored reactant if the absorbance is 0.605 at the beginning of the reaction.

Textbook Question

(d) How long does it take for the absorbance to fall to 0.100?