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Chapter 10: Problem 68

Find the mass of \(^{239} \mathrm{Pu}\) that has an activity of \(1.00 \mu \mathrm{Ci}\).

Short Answer

Expert verified
The mass of Plutonium-239 (\(^{239} \mathrm{Pu}\)) that has an activity of 1.00 microCurie is approximately 0.014 g

Step by step solution

01

Convert the activity in Curie to dps

1 Ci is equivalent to \(3.7 \times 10^{10}\) dps. Therefore, an activity of \(1.00 \mu \mathrm{Ci}\) is equivalent to \(1.00 \mu \mathrm{Ci} \times \frac{3.7 \times 10^{10} \mathrm{dps}}{1 \mathrm{Ci}} \times \frac{1 \mathrm{Ci}}{10^6 \mu \mathrm{Ci}} = 3.7 \times 10^4 \mathrm{dps}\)
02

Convert the activity in dps to number of atoms.

Activity is the rate of decay of a radioactive substance, which is the number of decays per second (dps). If we know the number of decays in one second, we can calculate the total number of atoms in our sample because each atom would decay once. The number of atoms is therefore equal to the activity in dps, which is \(3.7 \times 10^4 \) atoms.
03

Convert the number of atoms to moles using Avogadro's number

A mole of any substance contains \(6.022 \times 10^{23}\) entities (atoms, molecules, etc). Therefore, \(3.7 \times 10^4 \) atoms would be equal to \(\frac{3.7 \times 10^4 \mathrm{atoms}}{6.022 \times 10^{23} \mathrm{atoms/mol}}\) moles.
04

Convert the number of moles to mass using the atomic mass of Pu-239

The atomic mass of Pu-239 is 239.0521634 g/mol. Therefore, the mass of \(3.7 \times 10^4 \mathrm{atoms}\) would be equal to \(\frac{3.7 \times 10^4 \mathrm{atoms}}{6.022 \times 10^{23} \mathrm{atoms/mol}} \times 239.0521634 \mathrm{g/mol}\) g

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Key Concepts

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

Activity Measurement
In radioactive decay, activity measurement is crucial. It tells us how fast a radioactive substance is disintegrating. Activity is often measured in decays per second (dps), which signifies how many atoms decay each second. A more common unit is the Curie (Ci), which equals approximately \(3.7 \times 10^{10}\) dps.

For smaller quantities, microcurie (\(\mu\)Ci) is used. It's just one-millionth of a curie, designed for dealing with less radioactive samples. Measuring activity in microcurie helps scientists work conveniently with more manageable numbers.
  • 1 Ci = \(3.7 \times 10^{10}\) dps
  • 1 \(\mu\)Ci = \(10^{-6}\) Ci
Converting between these units helps us understand the scale of radioactive decay happening in a sample.
Avogadro's Number
Avogadro's number is a constant used to count the number of particles in a mole of any substance. The value is \(6.022 \times 10^{23}\). This seemingly enormous figure helps simplify the handling of extremely large numbers of atoms or molecules by grouping them into moles.

When dealing with radioactive materials, knowing how many atoms you have can be essential. Using Avogadro's number allows conversion from the actual number of atoms to moles, making calculations more feasible:
  • 1 mole = \(6.022 \times 10^{23}\) atoms
In our calculation, using this number turned the large quantity of \(3.7 \times 10^4\) atoms into a manageable number of moles.
Atomic Mass
Atomic mass provides insight into the mass of a single atom in atomic mass units (amu). For calculations, it's generally expressed in grams per mole (g/mol).

For plutonium-239 (\(^{239} \mathrm{Pu}\)), the atomic mass is 239.0521634 g/mol. This means one mole of plutonium-239 weighs about 239 grams.

In practical chemistry, converting moles to grams using atomic mass is common. This conversion helps us find how much material we actually have in terms of mass. It's especially useful in scenarios where reaction ratios are important, or, as in our case, when calculating the mass of a radioactive sample.
  • 1 mole of \(^{239} \mathrm{Pu}\) = 239.0521634 g
Curie to DPS Conversion
The conversion between Curie (Ci) and decays per second (dps) is foundational in understanding radioactive activity. As mentioned, 1 Curie equals \(3.7 \times 10^{10}\) dps. This unit conversion is key to expressing radioactive decay rates in a more standard scientific format.

In practical terms, this means if you have a sample with a known activity in curies, you can easily convert it to dps using this factor. This helps in comparing and calculating decay rates irrespective of the original units.
  • \(1 \mu\)Ci = \(3.7 \times 10^4\) dps (since \(1 \mu\)Ci is \(10^{-6}\) of a Ci)
This conversion is particularly vital when the activity needs to be linked with the number of atoms and further calculations.

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Most popular questions from this chapter

(a) Calculate the energy released in the \(\alpha\) decay of \(^{238} \mathrm{U} .\) (b) What fraction of the mass of a single \(^{238} \mathrm{U}\) is destroyed in the decay? The mass of \(^{234} \mathrm{Th}\) is 234.043593 u. (c) Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium. Why is this?

If two nuclei are to fuse in a nuclear reaction, they must be moving fast enough so that the repulsive Coulomb force between them does not prevent them for getting within \(R \approx 10^{-14} \mathrm{m}\) of one another. At this distance or nearer, the attractive nuclear force can overcome the Coulomb force, and the nuclei are able to fuse. (a) Find a simple formula that can be used to estimate the minimum kinetic energy the nuclei must have if they are to fuse. To keep the calculation simple, assume the two nuclei are identical and moving toward one another with the same speed \(v\). (b) Use this minimum kinetic energy to estimate the minimum temperature a gas of the nuclei must have before a significant number of them will undergo fusion. Calculate this minimum temperature first for hydrogen and then for helium. (Hint: For fusion to occur, the minimum kinetic energy when the nuclei are far apart must be equal to the Coulomb potential energy when they are a distance \(R\) apart.)

If a 1.50 -cm-thick piece of lead can absorb \(90.0 \%\) of the rays from a radioactive source, how many centimeters of lead are needed to absorb all but \(0.100 \%\) of the rays?

The \(^{210}\) Po source used in a physics laboratory is labeled as having an activity of \(1.0 \mu \mathrm{Ci}\) on the date it was prepared. A student measures the radioactivity of this source with a Geiger counter and observes 1500 counts per minute. She notices that the source was prepared 120 days before her lab. What fraction of the decays is she observing with her apparatus?

Engineers are frequently called on to inspect and, if necessary, repair equipment in nuclear power plants. Suppose that the city lights go out. After inspecting the nuclear reactor, you find a leaky pipe that leads from the steam generator to turbine chamber. (a) How do the pressure readings for the turbine chamber and steam condenser compare? (b) Why is the nuclear reactor not generating electricity?
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