Ch.15 - Chemical Kinetics

Tro - Chemistry: A Molecular Approach 6th Edition

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Tro 6th Edition

Ch.15 - Chemical Kinetics

Problem 63b

Chapter 15, Problem 63b

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. If a sample of U-238 initially contained 3.2⨉10¹⁸ atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms does it contain today?

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Identify the given values: the half-life of U-238 is 4.5 billion years, the initial number of atoms is 3.2\(\times\)10^{18}, and the time elapsed is 13.8 billion years.

Use the formula for radioactive decay: N(t) = N_0 \(\times\) (1/2)^{t/t_{1/2}}, where N(t) is the number of atoms remaining, N_0 is the initial number of atoms, t is the time elapsed, and t_{1/2} is the half-life.

Substitute the given values into the formula: N(t) = 3.2\(\times\)10^{18} \(\times\) (1/2)^{13.8/4.5}.

Calculate the exponent: 13.8/4.5 to determine how many half-lives have passed.

Evaluate the expression to find the remaining number of U-238 atoms, N(t).

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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 an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. For U-238, this half-life is 4.5 billion years, meaning that after this period, half of the original amount of U-238 will have transformed into other elements.

Half-Life

Half-life is a fundamental concept in nuclear chemistry that quantifies the time it takes for half of a radioactive substance to decay. It is a constant for each isotope and does not depend on the initial quantity of the substance. In the case of U-238, knowing its half-life allows us to calculate how much of the original sample remains after a given period, such as the 13.8 billion years since the universe's formation.

Exponential Decay

Exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. This means that as time progresses, the amount of the substance decreases rapidly at first and then slows down. The relationship can be mathematically expressed using the formula N(t) = N0 * (1/2)^(t/T), where N0 is the initial quantity, t is the elapsed time, and T is the half-life. This concept is crucial for calculating the remaining U-238 atoms after 13.8 billion years.

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The half-life for the radioactive decay of C-14 is 5715 years and is independent of the initial concentration. If a sample of C-14 initially contains 1.5 mmol of C-14, how many millimoles are left after 2725 years?

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The half-life for the radioactive decay of C-14 is 5715 years and is independent of the initial concentration. How long does it take for 25.00% of the C-14 atoms in a sample of C-14 to decay?

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The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. How long will it take for 20% of the U-238 atoms in a sample of U-238 to decay?

Textbook Question

The decomposition of XY is second order in XY and has a rate constant of 7.02⨉10^-3 M^-1• s^-1 at a certain temperature. a. What is the half-life for this reaction at an initial concentration of 0.100 M?

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. If a sample of U-238 initially contained 3.2⨉1018 atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms does it contain today?

Verified video answer for a similar problem:

Key Concepts

Radioactive Decay

Half-Life

Exponential Decay

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. If a sample of U-238 initially contained 3.2⨉10¹⁸ atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms does it contain today?