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Chapter 40: Problem 66

\(10^{30}\) Atoms of a radioactive sample remain after 10 half-lives. How many atoms remain after 20 half-lives?

Short Answer

Expert verified
Answer: After 20 half-lives, there will be N_{20} = 10^{30} * 2^{-10} atoms remaining.

Step by step solution

01

Understand half-life

A half-life is the time it takes for half of a radioactive sample to decay. After each half-life, the number of remaining radioactive atoms is halved.
02

Find the initial number of atoms

We are given that 10^{30} atoms remain after 10 half-lives. Relating to the half-life concept, we can find the initial number of atoms (N_0) by multiplying the remaining atoms by 2, 10 times since it has gone through 10 half lives: N_0 = 10^{30} * 2^{10}
03

Calculate number of atoms after 20 half-lives

Knowing the initial number of atoms, we can now find the number of atoms remaining after 20 half-lives. After each half-life, the number of remaining atoms is halved. Thus, after 20 half-lives, the remaining atoms can be calculated by dividing the initial number of atoms by 2, 20 times: N_{20} = N_{0} * 2^{-20}
04

Substitute and solve

Replace N_0 with the expression we found in step 2 and solve for N_{20}: N_{20} = (10^{30} * 2^{10}) * 2^{-20} Simplify the expression: N_{20} = 10^{30} * 2^{-10}
05

Final answer

After 20 half-lives, there will be: N_{20} = 10^{30} * 2^{-10} atoms remaining.

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Most popular questions from this chapter

The Sun radiates energy at the rate of \(3.85 \cdot 10^{26} \mathrm{~W}\) a) At what rate, in \(\mathrm{kg} / \mathrm{s}\), is the Sun's mass converted into energy? b) Why is this result different from the rate calculated in Example \(40.6,6.02 \cdot 10^{11}\) kg protons being converted into helium each second? c) Assuming that the current mass of the Sun is \(1.99 \cdot 10^{30} \mathrm{~kg}\) and that it radiated at the same rate for its entire lifetime of \(4.50 \cdot 10^{9} \mathrm{yr}\), what percentage of the Sun's mass was converted into energy during its entire lifetime?

\(^{214} \mathrm{Pb}\) has a half-life of \(26.8 \mathrm{~min}\). How many minutes must elapse for \(90.0 \%\) of a given sample of \({ }^{214} \mathrm{~Pb}\) atoms to decay?

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A nuclear reaction of the kind \({ }_{2}^{3} \mathrm{He}+{ }_{6}^{12} \mathrm{C} \rightarrow \mathrm{X}+\alpha\) is called a pick-up nuclear reaction. a) Why is it called a pick-up reaction, that is, what is picked up, what picked it up, and where did it come from? b) What is the resulting nucleus X? c) What is the \(\mathrm{Q}\) -value of this reaction? d) Is this reaction endothermic or exothermic?

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