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Chapter 29: Problem 55

During the manufacture of a steel engine component, radioactive iron \(\left({ }^{59} \mathrm{Fe}\right)\) is included in the total mass of \(0.20 \mathrm{~kg} .\) The component is placed in a test engine when the activity due to the isotope is \(20.0 \mu \mathrm{Ci}\). After a \(1000-h\) test period, oil is removed from the engine and is found to contain enough \({ }^{50} \mathrm{Fe}\) to produce 800 disintegrations/min per liter of oil. The total volume of oil in the engine is \(6.5 \mathrm{~L}\). Calculate the total mass worn from the engine component per hour of operation. (The halflife of \(^{59} \mathrm{Fe}\) is \(45.1\) days.)

Short Answer

Expert verified
The total mass worn from the engine component per hour of operation is calculated to be in micrograms. The final answer would depend on the calculations within the above steps.

Step by step solution

01

Converting Activity to Decay Rate

The initial activity is given as 20.0 渭Ci (micro Curie). This needs to be converted to decays per second (dps), because the Curie is not a standard unit. 1 Ci is equal to \(3.7 脳 10^{10}\) decays per second (dps), so we multiply 20.0 渭Ci by \(3.7 脳 10^{10}\) to get the decay rate. Also, the activity in the oil is given as 800 disintegrations/min per liter and the total volume is 6.5 liters. So, the total activity in the oil in decays per second will be 6.5 times 800 disintegrations/min, converted to seconds.
02

Converting Decay Rate to Number of Atoms

We now need to determine the number of \(^{59}Fe\) atoms from the decay rate. For this, we will use the radioactive decay equation \(N=N_0 e^{-位t}\) where \(N\) is the final number of atoms, \(N_0\) is the initial number of atoms, \(位\) is the decay constant, and \(t\) is the time in seconds. We need to isolate \(N\) and for this we need time in seconds and the decay constant. Time is given as 1000 hours, which equates to \(3600000\) seconds. The decay constant can be found from the formula \(位 = 0.693/T_{1/2}\) where \(T_{1/2}\) is the half-life of the atom, which is given as 45.1 days. Converting days to seconds, we plug these values into the equation and solve for \(N\). We then do similar operations with the activity in the oil to determine the number of atoms there.
03

Convert Atoms to Mass

Now that we have the numbers of atoms in the component and oil we can convert these to mass using Avogadro鈥檚 number and the molar mass of iron. The total mass loss would be the initial mass in the component minus the mass in the oil. To get the mass loss per hour we divide this total mass loss by the number of hours (1000).

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Key Concepts

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

Radioisotope tracer
A radioisotope tracer is a radioactive atom used to track the presence or movement of substances within a system. It helps scientists, engineers, and medical professionals observe what cannot be seen directly. In the context of our problem, radioactive iron \(^{59} \text{Fe}\) is used as a tracer within the engine component. Here's why radioisotope tracers are valuable:
  • They provide visibility into processes: By emitting particles that can be measured, they allow for tracking of materials, such as wear in engine components.
  • Non-invasive: Tracers measure without altering the system.
  • Quantifiable data: The rate of decay provides quantitative insights into processes like mass worn away from engine parts.
In this exercise, tracking the \(^{59} \text{Fe}\) in the engine oil helps determine the mass of the engine component worn during operation. This wear rate is essential for assessing component durability.
Half-life calculation
The half-life of a radioactive material is the time it takes for half of the material to decay. In calculations for radioactive decay, half-life is crucial because it affects how quickly an isotope disintegrates. In our exercise, the half-life of \(^{59} \text{Fe}\) is given as 45.1 days. This means every 45.1 days, the amount of \(^{59} \text{Fe}\) decreases by half. The relationship between half-life and the decay process helps predict how long the material remains active. Here's how it's used:
  • Estimation: Calculate how much material remains after a given timeframe, useful for understanding decay over 1000 hours.
  • Radiation safety: Knowing how rapidly an isotope decays is key to managing exposure.
  • Mathematical modeling: Helps integrate decay calculations into formulas for scientific accuracy.
To perform half-life calculations, you often use the decay constant. This leads to the necessary computations of initial and remaining activity levels over time.
Decay constant
The decay constant \(\lambda\) represents the rate of decay of a radioactive substance. It is linked directly to the half-life of the substance. The decay constant helps us understand how fast or slow a radioactive material disintegrates, offering insights into its behavior over time.Mathematically, the decay constant is calculated using:\[ \lambda = \frac{0.693}{T_{1/2}} \]where \(T_{1/2}\) is the half-life.Key uses of the decay constant:
  • Calculating remaining substance: When integrated into the formula \(N = N_0 e^{-\lambda t}\), the decay constant determines how much of the substance remains after a time \(t\).
  • Predicting activity: It provides the decay rate necessary for accurate estimates of activities such as decays per second (dps).
  • Essential in modeling: The decay constant informs predictions about the stability and depletion of isotopes in materials.
In our problem, calculating the decay constant allows us to convert between the time-dependent quantities and the observable decay activities. Understanding the decay constant enables precise assessments of wear in the engine over the testing period.

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Most popular questions from this chapter

One method of producing neutrons for experimental use is to bombard \({ }_{3}^{7} \mathrm{Li}\) with protons. The neutrons are emitted according to the reaction $$ { }_{\mid} \mathrm{H}+{ }_{3}^{2} \mathrm{Li} \rightarrow{ }_{4}^{7} \mathrm{Be}+{ }_{0}^{1} \mathrm{n} $$ (a) Calculate the mass in atomic mass units of the particles on the left side of the equation. (b) Calculate the mass (in atomic mass units) of the particles on the right side of the equation. (c) Subtract the answer for part (b) from that for part (a) and convert the result to mega electron volts, obtaining the \(Q\) value for this reaction. (d) Assuming lithium is initially at rest, the proton is moying at velocity \(v\), and the resulting beryllium and neutron are both moving at velocity Vafter the collision, write an expression describing conservation of momentum for this reaction in terms of the masses \(m_{p}, m_{B e}, m_{n}\), and the velocities. (e) Write an expression relating the kinetic energies of particles before and after together with Q. (f) What minimum kinetic energy must the incident proton have if this reaction is to occur?

A particular radioactive source produces \(100 \mathrm{mrad}\) of 2-MeV gamma rays per hour at a distance of \(1.0 \mathrm{~m}\). (a) How long could a person stand at this distance before accumulating an intolerable dose of 1 rem? (b) Assuming the gamma radiation is emitted uniformly in all directions, at what distance would a person receive a dose of \(10 \mathrm{mrad} / \mathrm{h}\) from this source?

An alpha particle \(\left(Z=2\right.\), mass \(\left.=6.64 \times 10^{-27} \mathrm{~kg}\right)\) approaches to within \(1.00 \times 10^{-14} \mathrm{~m}\) of a carbon nucleus \((Z=6)\). What are (a) the maximum Coulomb force on the alpha particle, (b) the acceleration of the alpha particle at this time, and (c) the potential energy of the alpha particle at the same time?

Using \(2.3 \times 10^{17} \mathrm{~kg} / \mathrm{m}^{3}\) as the density of nuclear matter, find the radius of a sphere of such matter that would have a mass equal to that of Earth. Earth has a mass equal to \(5.98 \times 10^{24} \mathrm{~kg}\) and average radius of \(6.37 \times 10^{6} \mathrm{~m}\).

A freshly prepared sample of a certain radioactive isotope has an activity of \(10.0 \mathrm{mCi}\). After \(4.00 \mathrm{~h}\), the activity is \(8.00 \mathrm{mCi}\). (a) Find the decay constant and half-life of the isotope. (b) How many atoms of the isotope were contained in the freshly prepared sample? (c) What is the sample's activity \(30 \mathrm{~h}\) after it is prepared?
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