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Chapter 18: Problem 38

The half-life of thorium-234 is 24.10 days. How many days until only one-sixteenth of a 52.0 g sample of thorium-234 remains?

Short Answer

Expert verified
It will take 96.4 days for only one-sixteenth of a 52.0 g sample of thorium-234 to remain.

Step by step solution

01

Understand the Concept of Half-Life

First, understand that it takes one half-life for a substance to decay to half of its original value. Since we're looking for the time it takes for only one-sixteenth of the original 52.0 g thorium-234 sample to remain, we need to figure out how many half-lives it takes to get from the original amount to one-sixteenth of that amount.
02

Calculate Number of Half-Lives

Since each half-life halves the remaining amount of substance, to go from the whole to one-half is one half-life, from one-half to one-fourth is another half-life, from one-fourth to one-eighth is yet another half-life, and from one-eighth to one-sixteenth is still one more half-life. Add these up: one-half life, plus one-half life, plus one-half life, plus one-half life gives 4 half-lives to go from the whole to one-sixteenth of the original amount.
03

Determine the Total Time for the Decay

Now, calculate the total time for the decay process. Since each half-life of the thorium-234 is 24.10 days, and we have calculated that it takes 4 half-lives for the sample to decay to one-sixteenth of its original mass, we can calculate the total decay time. Multiply the half-life of the substance by the number of half-lives it takes for the sample to decay to the desired amount: 24.10 days/half-life * 4 half-lives = 96.4 days.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life
The concept of half-life is fundamental in nuclear chemistry and helps us understand the rate at which radioactive substances decay. The half-life of a substance is the time required for half of the substance's nuclei to decay. It's a constant rate, meaning no matter how much of the substance you have, it'll take the same amount of time for half of it to decay.

For example, if a radioactive material has a half-life of 10 days, then after 10 days you will only have half of the original material left. After another 10 days, you will have a quarter, and so on.
  • This property is crucial in calculating how long it will take for a sample to decay to a desired lower amount.
  • It also allows us to date ancient artifacts or determine the age of fossils by comparing the ratios of various isotopes.
Understanding half-life helps connect mathematical calculations with real-world phenomena in nuclear science.
Radioactive Decay
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. It's a spontaneous process and is a key aspect of nuclear chemistry.

There are different types of radioactive decay, including alpha decay, beta decay, and gamma decay, each involving the emission of different particles.
  • Alpha decay emits helium nuclei, which decreases the mass of the original atom.
  • Beta decay involves the transformation of a neutron to a proton or vice versa, resulting in the emission of electrons or positrons.
  • Gamma decay involves the release of energy without a change in the number or kind of particles in the nucleus.
In the context of the original exercise, we focus on the fact that with each step in radioactive decay, we move closer to reaching a more stable state, explaining why substances decay over time until they are no longer radioactive.
Thorium-234
Thorium-234 is an isotope of thorium which plays a key role in understanding radioactive decay processes. It is part of the uranium decay series, illustrating practical applications of nuclear chemistry.

Given its half-life of 24.10 days, thorium-234 provides a manageable timeframe for observing decay in a laboratory or educational setting.
  • It decays into protactinium-234 through beta decay, which then further decays in a sequence of events.
  • Its decay series ultimately leads to lead-206, a stable isotope, showcasing how radioactive substances can transform into stable ones over time.
Understanding thorium-234 and its behaviors helps illuminate broader concepts in nuclear science, such as radioactive decay chains and practical applications of the theory of half-life.

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Most popular questions from this chapter

Balance the following nuclear reactions. \begin{equation}a. \quad_{93}^{239} \mathrm{Np} \longrightarrow_{-1}^{0} e+?\end{equation} \begin{equation}\begin{array}{l}{\text { b. }_{4}^{9} \mathrm{Be}+_{2}^{4} \mathrm{He} \longrightarrow ?} \\ {\text { c. } _{15}^{32} \mathrm{P}+? \longrightarrow_{15}^{33} \mathrm{P}}\end{array}\end{equation} \begin{equation}d. _{92}^{236} \mathrm{U} \longrightarrow_{36}^{94} \mathrm{Kr}+?+3_{0}^{1} n\end{equation}

The radiation given off by iodine-131 in the form of beta particles is used to treat cancer of the thyroid gland.Write the nuclear equation to describe the decay of an iodine-131 nucleus.

What is a nucleon?

What is the relationship between mass defect and binding energy?

Calculating the Amount of Radioactive Material The graphing calculator can run a program that graphs the relationship between the amount of radioactive material and elapsed time. Given the half-life of the radioactive material and the initial amount of material in grams, you will graph the relationship between the amount of radioactive material and the elapsed time. Then, with the elapsed time, you will trace the graph to calculate the amount of radioactive material. Go to Appendix C. If you are using a TI-83 Plus, you can download the program RADIOACT and run the application as directed. If you are using another calculator, your teacher will provide you with key-strokes and data sets to use. After you have run the program, answer these questions. a. Determine the amount of neptunium-235 left after 2.0 years, given the half- life of neptunium-235 is 1.08 years and the initial amount was 8.00 g. b. Determine the amount of neptunium-235 left after 5.0 years, given the half- life of neptunium-235 is 1.08 years and the initial amount was 8.00 g. c. Determine the amount of uranium-232 left after 100 years, given the half- life of uranium-232 is 69 years and the initial amount was 10.0 g.
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