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Chapter 40: Problem 34

The half-life of a sample of \(10^{11}\) atoms that decay by alpha emission is \(10 \mathrm{~min} .\) How many alpha particles are emitted between the time interval 100 min and 200 min?

Short Answer

Expert verified
Answer: Approximately \(8.95 \times 10^9\) alpha particles are emitted between the time interval 100 and 200 minutes.

Step by step solution

01

Write down the radioactive decay formula

The radioactive decay formula is the following: \(N(t) = N_0 e^{-\lambda t}\) Here, \(N(t)\) is the number of remaining atoms at time t, \(N_0\) is the initial number of atoms, \(\lambda\) is the decay constant, and \(t\) is the time.
02

Find the decay constant using the half-life

The decay constant \(\lambda\) can be found using the following half-life formula: \(\lambda = \frac{\ln(2)}{T_{1/2}}\) Given that the half-life (\(T_{1/2}\)) is 10 minutes, we can calculate the decay constant: \(\lambda = \frac{\ln(2)}{10} = 0.0693\)
03

Find the number of remaining atoms at t = 100 minutes

Now we will use the radioactive decay formula with the initial number of atoms (\(N_0 = 10^{11}\) atoms), the decay constant (\(\lambda = 0.0693\)), and the time interval t = 100 minutes: \(N(100) = 10^{11}e^{-(0.0693)(100)} = 10^{11}e^{-6.93} \approx 9.93 \times 10^9\)
04

Find the number of remaining atoms at t = 200 minutes

Similarly, we will find the number of remaining atoms at t = 200 minutes using the radioactive decay formula: \(N(200) = 10^{11}e^{-(0.0693)(200)} = 10^{11}e^{-13.86} \approx 9.86 \times 10^8\)
05

Find the number of alpha particles emitted between t = 100 and t = 200 minutes

To find the number of alpha particles emitted during the time interval, we will find the difference between the remaining atoms at the beginning and end of the interval: Number of alpha particles emitted = \(N(100) - N(200) = 9.93 \times 10^9 - 9.86 \times 10^8 = (9.93 - 0.986) \times 10^9 \approx 8.95 \times 10^9\) Therefore, approximately \(8.95 \times 10^9\) alpha particles are emitted between the time interval 100 and 200 minutes.

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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?

The radon isotope \({ }^{222} \mathrm{Rn}\), which has a half-life of 3.825 days, is used for medical purposes such as radiotherapy. How long does it take until \({ }^{222} \mathrm{Rn}\) decays to \(10.00 \%\) of its initial quantity?

When a target nucleus is bombarded by an appropriate beam of particles, it is possible to produce a) a less massive nucleus, but not a more massive one. b) a more massive nucleus, but not a less massive one. c) a nucleus with smaller charge number, but not one with a greater charge number. d) a nucleus with greater charge number, but not one with a smaller charge number. e) a nucleus with either greater or smaller charge number.

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

Which of the following quantities is conserved during a nuclear reaction, and how? a) charge d) linear momentum b) the number of nucleons, A e) angular momentum c) mass-energy
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