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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.

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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’s 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’s 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.

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