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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

Radiation therapy is one of the modalities for cancer treatment. Based on the approximate mass of a tumor, oncologists can calculate the radiation dose necessary to treat their patients. Suppose a patient has a 50.0 -g tumor and needs to receive 0.180 J of energy to kill the cancer cells. What rad (radiation absorbed dose) should the patient receive?

What is the nuclear configuration of the daughter nucleus associated with the alpha decay of \(\mathrm{Hf}(A=157,\) \(Z=72\) )?

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Pu decays with a half-life of 24,100 yr via a 5.25 \(\mathrm{MeV}\) alpha particle. If you have a \(1.00 \mathrm{~kg}\) spherical sample of \({ }^{239} \mathrm{Pu},\) find the initial activity in \(\mathrm{Bq} .\)

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