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Chapter 43: Problem 29

The radioactive nuclide \(^1$$^9$$^9\)Pt has a half-life of 30.8 minutes. A sample is prepared that has an initial activity of 7.56 \(\times\) 10\(^1$$^1\) Bq. (a) How many \(^1$$^9$$^9\)Pt nuclei are initially present in the sample? (b) How many are present after 30.8 minutes? What is the activity at this time? (c) Repeat part (b) for a time 92.4 minutes after the sample is first prepared.

Short Answer

Expert verified
Initially, there are approximately 3.36 脳 10鹿鲁 nuclei. After 30.8 minutes, there are 1.68 脳 10鹿鲁 nuclei and 3.78 脳 10鹿鹿 Bq. After 92.4 minutes, there are 4.20 脳 10鹿虏 nuclei and 9.45 脳 10鹿鈦� Bq.

Step by step solution

01

Calculate the decay constant

The decay constant, \( \lambda \), can be calculated using the half-life formula: \( \lambda = \frac{\ln(2)}{T_{1/2}} \). Given that the half-life \( T_{1/2} \) is 30.8 minutes, we find:\[\lambda = \frac{\ln(2)}{30.8 \text{ min}} = \frac{0.693}{30.8 \text{ min}}\approx 0.0225 \text{ min}^{-1}\]
02

Calculate the initial number of nuclei

The activity \( A \) of a radioactive sample is related to the number of radioactive nuclei \( N \) by the equation \( A = \lambda N \). We are given that the initial activity \( A_0 \) is 7.56 \times 10^{11} \text{ Bq}. Solving for \( N_0 \):\[N_0 = \frac{A_0}{\lambda} = \frac{7.56 \times 10^{11}}{0.0225}\approx 3.36 \times 10^{13} \text{ nuclei}\]
03

Determine the number and activity of nuclei after 30.8 minutes

After one half-life (30.8 minutes), half of the nuclei will have decayed. Thus, the number of nuclei \( N \) is:\[N = \frac{N_0}{2} = \frac{3.36 \times 10^{13}}{2}= 1.68 \times 10^{13} \text{ nuclei}\]The activity after one half-life is also halved:\[A = \frac{A_0}{2} = \frac{7.56 \times 10^{11}}{2}\approx 3.78 \times 10^{11} \text{ Bq}\]
04

Determine the number and activity of nuclei after 92.4 minutes

92.4 minutes corresponds to three half-lives (\(92.4 \div 30.8 = 3\)). After three half-lives, the number of nuclei is:\[N = \frac{N_0}{2^3} = \frac{3.36 \times 10^{13}}{8}= 4.20 \times 10^{12} \text{ nuclei}\]The activity is:\[A = \frac{A_0}{2^3} = \frac{7.56 \times 10^{11}}{8}\approx 9.45 \times 10^{10} \text{ Bq}\]

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Key Concepts

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

Half-life
Half-life is the time it takes for half of the radioactive nuclide in a sample to decay. It is a crucial concept in understanding how quickly a radioactive substance breaks down over time. In this context, the nuclide \(^1$$^9$$^9\)Pt has a half-life of 30.8 minutes. This means that after 30.8 minutes, only half of the original sample will remain. This is consistent for each subsequent half-life period.
  • After one half-life, 50% of the nuclides remain.
  • After two half-lives, 25% remain.
  • After three half-lives, only 12.5% of the initial amount remains.
Understanding half-life helps predict the amount of substance that remains over time.
Decay constant
The decay constant, represented by the symbol \( \lambda \), is a measure of the probability of a decay of a radioactive nuclide per unit time. It links closely with the concept of half-life.
You can calculate the decay constant with the formula: \( \lambda = \frac{\ln(2)}{T_{1/2}} \), where \( T_{1/2} \) is the half-life. For the nuclide \(^1$$^9$$^9\)Pt with a half-life of 30.8 minutes, this calculates to be approximately 0.0225 min\(^{-1}\).
  • The decay constant gives you a rate for exponential decay.
  • A larger \( \lambda \) indicates a faster decay rate and a shorter half-life.
The decay constant is essential for calculating the remaining number of nuclei and predicting radioactive decay outcomes.
Activity of radioactive samples
The activity of a radioactive sample refers to the number of decays that occur in the sample per second. It is measured in becquerels (Bq). In this exercise, the initial activity of the \(^1$$^9$$^9\)Pt sample is 7.56 \( \times \) 10\(^{11}\) Bq.
The activity \( A \) of a sample changes as nuclei decay, related to the number of radioactive nuclei \( N \) present and the decay constant \( \lambda \) by the formula: \( A = \lambda N \). Over time, as half of the nuclei decay every half-life period, the activity also reduces by half.
  • After one half-life, activity decreases to half of its initial value.
  • This makes it easier to track how active the sample remains over time.
Calculating activity helps in determining the potential radiation exposure from the sample.
Radioactive nuclides
Radioactive nuclides, or radionuclides, are atoms with unstable nuclei that release energy in the form of radiation as they transform into more stable forms. This transformation occurs at a predictable rate defined by its half-life and decay constant.
The nuclide \(^1$$^9$$^9\)Pt is an example of a radioactive nuclide, experiencing decay over time as it releases radiation. Observing radioactive nuclides:
  • Plays a vital role in nuclear medicine, carbon dating, and energy production.
  • Shows how matter and energy can transform in the universe.
Comprehending their behavior is crucial for applications involving radioactivity and safety measures.

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Most popular questions from this chapter

Measurements indicate that 27.83% of all rubidium atoms currently on the earth are the radioactive \(^8$$^7\)Rb isotope. The rest are the stable \(^8$$^5\)Rb isotope. The half-life of \(^8$$^7\)Rb is 4.75 \(\times\) 10\(^1$$^0\) y. Assuming that no rubidium atoms have been formed since, what percentage of rubidium atoms were \(^8$$^7\)Rb when our solar system was formed 4.6 \(\times\) 10\(^9\) y ago?

A nuclear chemist receives an accidental radiation dose of 5.0 Gy from slow neutrons (RBE \(=\) 4.0). What does she receive in rad, rem, and J/kg?

Radioactive isotopes used in cancer therapy have a "shelf-life," like pharmaceuticals used in chemotherapy. Just after it has been manufactured in a nuclear reactor, the activity of a sample of \(^6$$^0\)Co is 5000 Ci. When its activity falls below 3500 Ci, it is considered too weak a source to use in treatment. You work in the radiology department of a large hospital. One of these \(^6$$^0\)Co sources in your inventory was manufactured on October 6, 2011. It is now April 6, 2014. Is the source still usable? The half-life of \(^6$$^0\)Co is 5.271 years.

At the beginning of Section 43.7 the equation of a fission process is given in which \(^2$$^3$$^5\)U is struck by a neutron and undergoes fission to produce \(^1$$^4$$^4\)Ba, \(^8$$^9\)Kr, and three neutrons. The measured masses of these isotopes are 235.043930 u (\(^2$$^3$$^5\)U), 143.922953 u (\(^1$$^4$$^4\)Ba), 88.917631 u (\(^8$$^9\)Kr), and 1.0086649 u (neutron). (a) Calculate the energy (in MeV) released by each fission reaction. (b) Calculate the energy released per gram of \(^2$$^3$$^5\)U, in MeV/g.

Thorium \(^{230}_{90}Th\) decays to radium \(^{226}_{88}Ra\) by a emission. The masses of the neutral atoms are 230.033134 u for \(^{230}_{90}Th\) and 226.025410 u for \(^{226}_{88}Ra\). If the parent thorium nucleus is at rest, what is the kinetic energy of the emitted \(\alpha\) particle? (Be sure to account for the recoil of the daughter nucleus.)
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