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Chapter 15: Problem 9

In close analog to the half-lives of \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\), let's say two 80 elements have half lives of \(4.5\) billion years and 750 million years. If we start out having the same number of each (1:1 ratio), what will the ratio be after \(4.5\) billion years? Express as \(x: 1\), where \(x\) is the larger of the two.

Short Answer

Expert verified
Answer: The ratio between the remaining amounts of the two elements after 4.5 billion years is 1:32.

Step by step solution

01

Determine the number of half-lives

First, we should determine how many half-lives have passed for each element after 4.5 billion years. To find this, we will divide the total elapsed time by each element's half-life. For the first element (half-life of 4.5 billion years): Number of half-lives = 4.5 billion years / 4.5 billion years = 1 For the second element (half-life of 750 million years): Number of half-lives = 4.5 billion years / 750 million years = 6 So after 4.5 billion years, 1 half-life has passed for the first element, and 6 half-lives have passed for the second element.
02

Calculate the remaining amounts of each element

Now we need to calculate how much of each element remains after the respective half-lives. For the first element: After 1 half-life, the remaining amount is (1/2) * 1, since the initial ratio of the first element is 1. For the second element: After 6 half-lives, the remaining amount is (1/2)^6 * 1, since the initial ratio of the second element is 1.
03

Find the ratio

Now we need to find the ratio between the remaining amounts of the elements. Remaining amount of the first element = (1/2) * 1 = 1/2 Remaining amount of the second element = (1/2)^6 * 1 = 1/64 Since the ratio must be expressed in terms of the larger quantity to 1: Ratio = (1/64) : (1/2) The required ratio can be found by multiplying both sides of the ratio by 64: x:1 = 1:32 So the ratio between the remaining amounts of the two 80 elements after 4.5 billion years is 1:32.

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Most popular questions from this chapter

Based on the calculation that 18 TW would require an annual cube of seawater \(300 \mathrm{~m}\) on a side to provide enough deuterium, what is your personal share as one of 8 billion people on earth, in liters? Could you lift this yourself? One cubic meter is \(1,000 \mathrm{~L}\).

If a friend creates a nucleus whose half-life is 4 hours and gives it to you at noon, what is the probability that it will not have decayed by noon the following day?

A particular fission of \({ }^{235} \mathrm{U}+\mathrm{n}(\) total \(A=236)\) breaks up. One fragment has \(Z=54\) and \(N=86\), making it \({ }^{140} \mathrm{Xe} .\) If no extra neutrons are produced in this event, what must the other fragment be, so all numbers add up? Refer to a periodic table (e.g., Fig. B.1; p. 375 ) to learn which element has the corresponding \(Z\) value, and express the result in the notation \({ }^{\mathrm{A}} \mathrm{X}\).

A particular nuclide is found to have lost 3 neutrons and 1 proton after a decay chain. What combination of \(\alpha\) and \(\beta\) decays could account for this result?

Control rods in nuclear reactors tend to contain \({ }^{10} \mathrm{~B}\), which has a high neutron absorption cross section. \(^{81}\) What happens to this nucleus when it absorbs a neutron, and is the result stable? If not, track the decay chain until it lands on a stable nucleus.
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