31.10

Concepts & Calculations

Radioactive decay obeys the conservation laws of physics, and we have studied five of these laws, as the Concepts-at-a-Glance chart in Figure 31-6 illustrates. Two of them are particularly important in understanding the types of radioactivity that occur and the nuclear changes that accompany them. These are the conservation of electric charge and the conservation of nucleon number. Example 13 emphasizes their importance and reviews how they are applied.

| Concepts & Calculations Example 13 | | | Electric Charge and Nucleon Number

Thorium produces a daughter nucleus that is radioactive. The daughter, in turn, produces its own radioactive daughter, and so on. This process continues until bismuth is reached. What are the total number of particles and the total number of particles that are generated in this series of radioactive decays?
Concept Questions and Answers How many of the 90 protons in the thorium nucleus are carried off by the particles?

> Answer Each particle is a helium nucleus and carries off two protons. Therefore, the total number of protons carried off by the particles is (2).

How many protons are left behind when the particles are emitted?

> Answer Each particle is an electron and is emitted when a neutron in the nucleus decays into a proton and an electron. Therefore, the total number of protons left behind by the particles is .

How many of the 228 nucleons in the thorium nucleus are carried off by the particles?

> Answer Each particle is a helium nucleus and carries off four nucleons. Therefore, the total number of nucleons carried off by the particles is (4).

Does the departure of a particle alter the number of nucleons?

> Answer No. Each particle is an electron and is emitted when a neutron in the nucleus decays into a proton and an electron. In effect, a neutron is replaced by a proton. But since each is a nucleon, the number of nucleons is not changed.

Solution The overall decay process can be written as

Since electric charge must be conserved, we know that the charge on the left side of this reaction must be equal to the total charge on the right side, the result being

Since the nucleon number must be conserved also, we know that the 228 nucleons on the left must equal the total number of nucleons on the right, so that

Solving this result for the number of particles gives . Substituting into the conservation-of-charge equation, we find that the number of particles is .
Energy in the form of heat is needed to raise the temperature of an object, as Section 12.7 discusses. One source of the heat can be radioactive decay, as the next example discusses.

| Concepts & Calculations Example 14 | | | The Energy from Radioactive Decay

A one-gram sample of thorium contains atoms and undergoes decay with a half-life of 1.913 yr . Each disintegration releases an energy of 5.52 MeV . Assuming that all of the energy is used to heat a 3.8-kg sample of water, find the change in temperature of the water that occurs in one hour.
Concept Questions and Answers How much heat Q is needed to raise the temperature of a mass m of water by degrees?

> Answer According to Equation 12.4, the heat needed is , where the specific heat capacity of water is .

The energy released by each disintegration is E. What is the total energy released by a number n of disintegrations?

> Answer The total energy released is just the number of disintegrations times the energy for each one, or .

What is the number n of disintegrations that occur during a time t?

> Answer The number of radioactive nuclei remaining after a time t is given by Equations 31.5 and 31.6 as

> where is the number of radioactive nuclei present at and is the half-life for the decay. Therefore, the number n of disintegrations that occur during the time t is or

> | (31.7)

Solution According to Equation 12.4, the heat needed to change the temperature of the water is . This heat is provided by the total energy released in one hour of radioactive decay, or . Setting , we obtain , which can be solved for the change in temperature :

Using Equation 31.7 for the number of disintegrations n that occur in the time t, we obtain

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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