Log out Go to App
Go to App

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Which of the following is true about the fragments from \(\mathrm{a}^{235} \mathrm{U}\) fission event? a) any number of fragments ( 2 through 235 ) can be produced b) a small number of fragments will emerge ( 2 to 5 ) c) two nearly identical fragments will emerge d) two fragments of distinctly different size will emerge e) the fission is an alpha decay: a small piece having \(A=4\) is emitted

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?

If an atom were scaled up to be comparable to the extent of a mid-sized campus, how large would the nucleus be, and what sort of familiar object would be similar?

Cosmic rays impinging on our atmosphere generate radioactive \({ }^{14} \mathrm{C}\) from \({ }^{14} \mathrm{~N}\) nuclei. \(^{78}\) These \({ }^{14} \mathrm{C}\) atoms soon team up with oxygen to form \(\mathrm{CO}_{2}\), so that plants absorbing \(\mathrm{CO}_{2}\) from the air will have about one in a trillion of their carbon atoms in this form. Animals eating these plants \(^{79}\) will also have this fraction of carbon in their bodies, until they die and stop cycling carbon into their bodies. At this point, the fraction of carbon atoms in the form of \({ }^{14} \mathrm{C}\) in the body declines, with a half life of 5,715 years. If you dig up a human skull, and discover that only one-eighth of the usual one-trillionth of carbon atoms are \({ }^{14} \mathrm{C}\), how old do you deem the skull to be?

A large boulder whose mass is \(1,000 \mathrm{~kg}\) having a specific heat capacity of \(1,000 \mathrm{~J} / \mathrm{kg} /{ }^{\circ} \mathrm{C}\) is heated from \(0^{\circ} \mathrm{C}\) to a glowing \(1,800^{\circ} \mathrm{C}\). How much more massive is it, assuming no atoms have been added or subtracted?
See all solutions

Recommended explanations on Environmental Science Textbooks

Ecological Conservation

Read Explanation

Ecology Research

Read Explanation

Environmental Research

Read Explanation

Physical Environment

Read Explanation

Energy Resources

Read Explanation

Sustainability

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free