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A scientist digs up a sample of arctic ice that is 458,000 years old. He takes it to his lab and finds that it contains 1.675 grams of krypton-81. If the half-life of krypton-81 is 229,000 years, how much krypton 81 was present when the ice first formed? Use the formula [tex]$N=N_0\left(\frac{1}{2}\right)^{\frac{t}{T}}$[/tex].
The ice originally contained $\square$ grams of krypton 81.
Answer (1)
Substitute the given values into the radioactive decay formula: 1.675 = N 0 ( 2 1 ) 229000 458000 .
Simplify the exponent: 229000 458000 = 2 .
Rewrite the equation: 1.675 = N 0 ( 2 1 ) 2 .
Solve for N 0 : N 0 = 1.675 × 4 = 6.7 .
Explanation
Understanding the Formula and Given Values We are given the formula for radioactive decay: N = N 0 ( 2 1 ) h t , where:
N is the current amount of krypton-81 (1.675 grams).
N 0 is the original amount of krypton-81 (what we want to find).
t is the time elapsed (458,000 years).
h is the half-life of krypton-81 (229,000 years).
Substituting the Values Now, let's plug in the given values into the formula:
1.675 = N 0 ( 2 1 ) 229000 458000
We need to solve for N 0 .
Simplifying the Exponent First, simplify the exponent:
229000 458000 = 2
So the equation becomes:
1.675 = N 0 ( 2 1 ) 2
Isolating N_0 Now, solve for N 0 :
1.675 = N 0 ( 4 1 )
N 0 = 4 1 1.675
N 0 = 1.675 × 4
Calculating the Original Amount Calculate the value of N 0 :
N 0 = 1.675 × 4 = 6.7
So, the original amount of krypton-81 was 6.7 grams.
Examples
Radioactive decay is used in carbon dating to determine the age of ancient artifacts. By measuring the amount of carbon-14 remaining in an artifact and knowing its half-life, scientists can estimate how old the artifact is. This technique is crucial in archaeology and paleontology for understanding the history of our planet and human civilization. Similarly, in medicine, radioactive isotopes are used in imaging techniques like PET scans to diagnose diseases. The decay of these isotopes helps create images of the body's internal organs and tissues, aiding in the detection of abnormalities.
