/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 89 The half-life of \({ }^{27} \mat... [FREE SOLUTION] | 91影视

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Chapter 21: Problem 89

The half-life of \({ }^{27} \mathrm{Mg}\) is \(9.50 \mathrm{~min}\). (a) Initially there were \(4.20 \times 10^{12}{ }^{27} \mathrm{Mg}\) nuclei present. How many \({ }^{27}\) Mg nuclei are left 30.0 min later? (b) Calculate the \({ }^{27} \mathrm{Mg}\) activities \((\) in \(\mathrm{Ci})\) at \(t=0\) and \(t=30.0 \mathrm{~min}\) (c) What is the probability that any one \({ }^{27} \mathrm{Mg}\) nucleus decays during a 1 -s interval? What assumption is made in this calculation?

Short Answer

Expert verified
Part (a): The number of \( ^{27}Mg \) nuclei left is \( N = 4.20 \times 10^{12} e^{\frac{-ln 2}{570} \times 1800} \) nuclei. Part (b): The activity at \( t = 0 \) is \( A_0 = \frac{\lambda N_0}{3.70 \times 10^{10}} \) Ci, and the activity at \( t = 30 \) min is \( A = \frac{\lambda N}{3.70 \times 10^{10}} \) Ci. Part (c): The decay probability in a 1 sec interval is \( P = \lambda \times 1 \) sec. The assumption is that the decay probability is constant over that time interval.

Step by step solution

01

Calculation of Remaining Nuclei.

Using the exponential decay formula \( N = N_0 e^{-\lambda t} \), where \( \lambda = \frac{ln 2}{T} \). \( N_0 \) is the initial number of nuclei, \( N \) the final number of nuclei, \( t \) the elapsed time and \( T \) the half-life. With \( N_0 = 4.20 \times 10^{12} \) nuclei, \( T = 9.5 \) min = \( 570 \) seconds, and \( t = 30 \) min = \( 1800 \) seconds, we find \( N = N_0 e^{-\lambda t} = 4.20 \times 10^{12} e^{\frac{-ln 2}{570} \times 1800} \).
02

Calculation of Activity.

The activity \( A \) is given by \( A = \lambda N \). At \( t = 0 \), \( N = N_0 \), so \( A_0 = \lambda N_0 \). At \( t = 30 \) min, \( N = N \) from Step 1, so \( A = \lambda N \). 1 Ci = \( 3.70 \times 10^{10} \) decays/s, so to convert the activity in decays/s to Ci, divide by \( 3.70 \times 10^{10} \). Thus, \( A_0 = \frac{\lambda N_0}{3.70 \times 10^{10}} \), and \( A = \frac{\lambda N}{3.70 \times 10^{10}} \).
03

Calculation of Decay Probability.

The probability \( P \) that a nucleus decays in a given time interval \( \Delta t \) is \( P = \lambda \Delta t \). We find \( P = \lambda \times 1 \) sec. The underlying assumption is that the probability does not change during that time interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay
Exponential decay is a process where the quantity of a particular substance decreases at a rate proportional to its current amount. In the context of nuclear physics, it's used to describe how the number of radioactive nuclei in a sample reduces over time. The key feature of exponential decay is that it happens gradually and steadily, and the rate of decay depends on a constant known as the decay constant, denoted by \( \lambda \).

The mathematical expression for exponential decay is \( N = N_0 e^{-\lambda t} \), where:\
    \
  • \\N\\ is the number of remaining nuclei after time \(t\)\
    • \
  • \\N_0\\ is the initial number of nuclei\
    • \
  • \(\lambda\) is the decay constant\
    • \
  • \(t\) is the time elapsed\
    • \
\This formula is a cornerstone in understanding how substances like \( ^{27}Mg \) diminish over time, allowing us to predict the remaining amount after any given period.
Radioactive Decay
Radioactive decay is a natural, spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. Different radioactive isotopes, or radioisotopes, have varying levels of stability and thus decay at different rates. \( ^{27}Mg \) is a radioisotope with a specific half-life, which is the amount of time it takes for half of the sample to decay.

The half-life is crucial in radioactive decay because it's a constant value unique to each isotope and can be used to calculate decay activity, remaining radioactive material, and potential risks associated with the radioisotope. In nuclear chemistry and physics, understanding the half-life provides insight into the stability of the nucleus and the duration for which it will remain radioactive.
Decay Activity Calculation
Decay activity calculation involves determining the rate at which a radioactive substance undergoes decay. The activity, \(A\), quantifies the amount of decay occurring in a specified time and is measured in units like becquerels (Bq) or curies (Ci). Activity is directly proportional to the number of radioactive nuclei present; as the substance decays and the number of nuclei decreases, so does the activity.

To calculate the activity of a radioactive substance, you can use the formula \( A = \lambda N \), where:\
    \
  • \(\lambda\) is the decay constant\
    • \
  • \(N\) is the number of undecayed nuclei at the time of measurement\
    • \
\For instance, for \( ^{27}Mg \) at the starting point \(t = 0\), the activity \(A_0\) can be found using the initial number of nuclei \(N_0\). Whereas at a later time (such as 30 minutes) the remaining nuclei \(N\) would be used in the activity calculation, reflecting the decreased rate of decay as the number of undecayed nuclei lessens.

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Most popular questions from this chapter

To detect bombs that may be smuggled onto airplanes, the Federal Aviation Administration (FAA) will soon require all major airports in the United States to install thermal neutron analyzers. The thermal neutron analyzer will bombard baggage with low-energy neutrons, converting some of the nitrogen- 14 nuclei to nitrogen- \(15,\) with simultaneous emission of \(\gamma\) rays. Because nitrogen content is usually high in explosives, detection of a high dosage of \(\gamma\) rays will suggest that a bomb may be present. (a) Write an equation for the nuclear process. (b) Compare this technique with the conventional X-ray detection method.

Describe, with appropriate equations, nuclear processes that lead to the formation of the noble gases He, Ne, Ar, Kr, Xe, and Rn. (Hint: Helium is formed from radioactive decay, neon is formed from the positron emission of \({ }^{22} \mathrm{Na}\), the formation of \(\mathrm{Ar}\), Xe, and \(\mathrm{Rn}\) are discussed in the chapter, and \(\mathrm{Kr}\) is produced from the fission of \(\left.{ }^{235} \mathrm{U} .\right)\)

(a) Assume nuclei are spherical in shape, show that its radius \((r)\) is proportional to the cube root of mass number \((A) .\) (b) In general, the radius of a nucleus is given by \(r=r_{0} A^{\frac{1}{3}},\) where \(r_{0},\) the proportionality constant, is given by \(1.2 \times 10^{-15} \mathrm{~m}\). Calculate the volume of the \({ }^{238} \mathrm{U}\) nucleus.

Calculate the nuclear binding energy (in J) and the nuclear binding energy per nucleon of the following isotopes: (a) \({ }_{3}^{7} \mathrm{Li}(7.01600 \mathrm{amu})\) and (b) 35 17 Cl \((34.95952 \mathrm{amu})\)

The radioactive decay of \(\mathrm{T} 1-206\) to \(\mathrm{Pb}-206\) has a half- life of 4.20 min. Starting with \(5.00 \times 10^{22}\) atoms of \(\mathrm{Tl}-206,\) calculate the number of such atoms left after 42.0 min.
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