/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 Suppose you find a piece of anci... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 51

Suppose you find a piece of ancient pottery and take it to the laboratory of a physicist friend. He finds that the glaze contains radium, a radioactive element that decays to radon and has a half-life of 1,620 years. He tells you that there could not have been any radon in the glaze when the pottery was being fired, but that it now contains three atoms of radon for each atom of radium. How old is the pottery?

Short Answer

Expert verified
The pottery is 3,240 years old.

Step by step solution

01

Define the half-life

First, note that the half-life of radium is 1,620 years.
02

Define the ratio and understand decay process

There are three atoms of radon for every one atom of radium. This means that for every remaining atom of radium, three atoms have decayed over time.
03

Use decay formula

Use the decay formula \[ N(t) = N_0 \times (0.5)^{t/T} \] where \(N(t)\) is the remaining quantity after time \(t\), \(N_0\) is the initial quantity, and \(T\) is the half-life.
04

Set up equation based on decay

Since radon is the decay product and we have 3 atoms of radon for every atom of radium, we can set it up as follows: \[ 1 = 4 \times (0.5)^{t/1620} \] This is because originally there were 4 atoms of radium for each 1 atom now existing (1 current and 3 decayed).
05

Solve for time (t)

Taking the logarithm of both sides, we get: \[ \frac{t}{1620} \times \text{log}(0.5) = \text{log}(0.25) \] Next, solve for \(t\): \[ t = \frac{\text{log}(0.25)}{\text{log}(0.5)} \times 1620 \] Since \( \text{log}(0.25) = 2 \times \text{log}(0.5) \), this simplifies to: \[ t = \frac{2 \times \text{log}(0.5)}{\text{log}(0.5)} \times 1620 = 2 \times 1620 = 3240 \]
06

Finalize the result

The pottery is approximately 3,240 years old.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Half-Life
The half-life of a radioactive element is the time it takes for half of the radioactive atoms in a sample to decay.
In our problem, radium has a half-life of 1,620 years. This means that if you start with 100 atoms of radium, after 1,620 years, only 50 will remain.
This process continues: after another 1,620 years, only 25 of the original 100 atoms will remain.
  • Half-life helps us determine how quickly a substance decays.
  • Knowing the half-life lets us calculate the age of ancient objects containing radioactive materials.
  • It鈥檚 a key concept in radiometric dating methods.
Radioactive Decay
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. In this problem, radium decays into radon.
This decay process follows an exponential pattern, meaning it decreases rapidly at first and then more slowly over time.
  • Radium鈥檚 transformation to radon involves the loss of particles to form a different element.
  • Over time, the amount of radium decreases while the amount of radon increases until the radium is all gone.
  • The decay is predictable and can be calculated using its half-life and decay formula.
Logarithmic Functions
Logarithmic functions are used to solve equations involving exponential decay or growth. In our example, we use the logarithm to find the age of the pottery.
By taking the logarithm of both sides of our decay formula, we can isolate the variable representing time.
  • Logarithms help us convert multiplication into addition, making it easier to solve exponential equations.
  • The property \( \text{log}(a^b) = b \times \text{log}(a) \) is essential in solving for time in decay problems.
  • Understanding these properties allows us to compute the age of objects based on radioactive decay data.
Ancient Pottery Dating
Dating ancient pottery using radioactive elements like radium gives us a window into the past. This method relies on the predictable decay of radioactive elements.
In our case, the presence of radon in the glaze, which was initially non-existent, indicates the passage of time.
  • By measuring the amount of decay products, we can estimate the age of the pottery.
  • This technique, known as radiometric dating, is widely used for archaeology and geology.
  • Knowing the initial conditions and half-life of the decaying element is crucial for accurate dating.

Using the half-life of radium (1,620 years) and observing that the pottery now contains three atoms of radon per one atom of radium, we solve for the time elapsed.
This timespan of decay indicates the pottery is approximately 3,240 years old.
Such precise dating provides significant historical context and understanding of ancient civilizations.

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

Although oceans cover 72 percent of Earth's surface, they represent but a tiny fraction of its mass. Earth's mass is \(6.0 \times 10^{24} \mathrm{kg},\) and its oceans have a total mass of \(1.5 \times 10^{21} \mathrm{kg}\) a. What fraction of Earth's total mass do the oceans represent? b. Does surface and atmospheric water represent Earth's total mass of water? Explain.

The Google Lunar X Prize (http://googlelunarxprize.org) goes to the first privately funded team to send a robot to the Moon. The winning robot must travel some distance on the Moon's surface, and send back pictures, by December 2015 On the website, click on "The Teams" and read about a few of the teams. For each team, consider these questions: What kind of people and companies are on the team? What is their plan to go to the Moon? Aside from this prize, why do they want to go to the Moon; what commercial opportunities on the Moon do they anticipate?

If crater \(\mathrm{A}\) is inside crater \(\mathrm{B}\), we know that a. crater A was formed before crater B. b. crater B was formed before crater A. c. both craters were formed at about the same time. d. crater B formed crater A. e. crater A formed crater B.

Scientists know the history of Earth's magnetic field because a. the magnetic field hasn't changed since the formation of Earth. b. they see how it's changing today, and project backward in time. c. the magnetic field becomes frozen into rocks, and plate tectonics spreads those rocks out. d. they compare the magnetic fields on other planets to Earth's. e. there are written documents of magnetic field measurements since the beginning of Earth.

Different radioisotopes have different half-lives. For example, the half-life of carbon-14 is 5,700 years, the half-life of uranium-235 is 704 million years, the half-life of potassium-40 is 1.3 billion years, and the half-life of rubidium-87 is 49 billion years. a. Why wouldn't you use an isotope with a half-life similar to that of carbon-14 to determine the age of the Solar System? b. The age of the universe is approximately 14 billion years. Does that mean that no rubidium- 87 has decayed yet?
See all solutions

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Read Explanation

Electrostatics

Read Explanation

Energy Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Mechanics and Materials

Read Explanation

Geometrical and Physical Optics

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.