Ch.14 - Chemical Kinetics

Tro - Chemistry: A Molecular Approach 4th Edition

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Tro 4th Edition

Ch.14 - Chemical Kinetics

Problem 58a

Chapter 14, Problem 58a

The half-life for the radioactive decay of C-14 is 5730 years and is independent of the initial concentration. How long does it take for 25% of the C-14 atoms in a sample of C-14 to decay?

Verified step by step guidance

Identify that the problem involves first-order kinetics, as radioactive decay follows first-order kinetics.

Use the formula for first-order decay: \( N_t = N_0 e^{-kt} \), where \( N_t \) is the remaining quantity of substance, \( N_0 \) is the initial quantity, \( k \) is the rate constant, and \( t \) is time.

Recognize that 25% of the C-14 has decayed, meaning 75% remains. Therefore, \( N_t = 0.75N_0 \).

Use the relationship between half-life and rate constant for first-order reactions: \( k = \frac{0.693}{t_{1/2}} \), where \( t_{1/2} \) is the half-life.

Substitute \( k \) and \( N_t = 0.75N_0 \) into the first-order decay equation and solve for \( t \).

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Key Concepts

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

Half-life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For C-14, this period is 5730 years, meaning that after this time, only half of the original amount of C-14 remains. This concept is crucial for understanding the rate of decay and predicting how long it will take for a certain percentage of the substance to decay.

Radioactive Decay

Radioactive decay is a stochastic process by which unstable atomic nuclei lose energy by emitting radiation. This process occurs at a constant rate, characterized by the half-life, and is independent of external conditions such as temperature or pressure. Understanding this concept helps in calculating the remaining quantity of a radioactive isotope over time.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the amount of C-14 decreases exponentially over time, which means that after each half-life, the remaining quantity is halved. This concept is essential for determining how long it takes for a specific fraction of the substance to decay.

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Textbook Question

The diagram shows the energy of a reaction as the reaction progresses. Label each blank box in the diagram.

a. reactants b. products c. activation energy (E_a) d. enthalpy of reaction (ΔH_rxn)

Textbook Question

The half-life for the radioactive decay of C-14 is 5730 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 2255 years?

Textbook Question

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 10% of the U-238 atoms in a sample of U-238 to decay?

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Textbook Question

The activation energy of a reaction is 56.8 kJ/mol and the frequency factor is 1.5⨉10¹¹/ s. Calculate the rate constant of the reaction at 25 °C.

Textbook Question

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 1.5⨉10¹⁸ atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms does it contain today?

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?