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Chapter 32: Problem 75

Identify the nucleus whose \(\beta^{-}\) decay produces the same nucleus as that produced by the \(\alpha\) decay of \(_{84 }^{214}\) Po.

Short Answer

Expert verified
The nucleus is \(_{81}^{210}Tl\).

Step by step solution

01

Understand the Problem

First, we need to identify the nucleus produced when \(\alpha\) decay is performed on \(_{84}^{214}Po\). Then, we need to find which nucleus undergoes \(\beta^{-}\) decay to produce the same nucleus.
02

Alpha Decay Results

During \(\alpha\) decay, a nucleus emits an \(\alpha\) particle, which consists of 2 protons and 2 neutrons. The atomic number decreases by 2 and the mass number decreases by 4. Thus, the decay of \(_{84}^{214}Po\) results in a new nucleus:\[ _{82}^{210}Pb \].
03

Identify the Beta-minus Decay Parent

\(\beta^{-}\) decay increases the atomic number by 1 while keeping the mass number the same because a neutron is transformed into a proton. We need to find a nucleus with an atomic number of 81 and mass number 210 that undergoes \(\beta^{-}\) decay to produce \(_{82}^{210}Pb\). This nucleus is \(_{81}^{210}Tl\).

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Key Concepts

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

Alpha Decay
Alpha decay is a type of nuclear decay where an unstable nucleus emits an alpha particle to become more stable. An alpha particle consists of 2 protons and 2 neutrons. So when a nucleus undergoes alpha decay, it loses these particles, resulting in a decrease in both its atomic and mass numbers:
  • The atomic number decreases by 2 because it loses 2 protons.
  • The mass number decreases by 4 because it loses a total of 4 nucleons (protons and neutrons).
For example, when Polonium-214 ( _{84}^{214}Po ) undergoes alpha decay, it converts into Lead-210 ( _{82}^{210}Pb ). This transformation is due to the loss of an alpha particle, which effectively reduces the atomic number to 82 and the mass number to 210.
Beta-Minus Decay
Beta-minus decay is another form of nuclear decay in which a neutron in the nucleus is converted into a proton. This process also releases a beta particle, which is an electron. Unlike alpha decay, beta-minus decay alters only the atomic number:
  • The atomic number goes up by 1 because a new proton is created.
  • The mass number remains the same as it involves only a proton replacing a neutron.
For instance, if a nucleus with an atomic number of 81 and mass number 210, like Thallium-210 ( _{81}^{210}Tl ), undergoes beta-minus decay, it produces Lead-210 ( _{82}^{210}Pb ). This change is due to the conversion of a neutron into a proton, increasing the atomic number but keeping the mass number stable.
Atomic Number
The atomic number is a very important aspect of an element. It tells you how many protons are in the nucleus of an atom. In fact, it is the atomic number that essentially defines what element an atom represents. For example, an atomic number of 84 belongs to Polonium because it has 84 protons in its nucleus.
  • It helps to identify the element in the periodic table.
  • In \(\alpha\) decay, the atomic number decreases by 2.
  • In \(\beta^{-}\) decay, the atomic number increases by 1.
Tracking the changes in the atomic number during nuclear decays is crucial to identifying the resulting elements and understanding the transformation process.
Mass Number
The mass number is another key feature of a nucleus, representing the total number of protons and neutrons it holds. Unlike the atomic number, the mass number can vary among isotopes of an element as they might have different numbers of neutrons.
  • It reveals the total nucleon count (protons + neutrons).
  • In \(\alpha\) decay, the mass number decreases by 4, since two protons and two neutrons are lost.
  • In \(\beta^{-}\) decay, the mass number remains the same as a neutron is simply converted into a proton.
Understanding the mass number's role in nuclear decay helps explain how isotopes of one element can transform into isotopes of another through radioactive processes.

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Most popular questions from this chapter

(a) Find the nuclear radius of \(_{ 15}^{30}\) P. (b) What mass number would be required for a nucleus to have twice the radius found in part (a)? (c) Verify your answer to part (b) with an explicit calculation.

Consider a nucleus that undergoes \(\beta\) decay. (a) Is the radius of the resulting daughter nucleus greater than, less than, or the same as that of the original nucleus? (b) Choose the best explanation from among the following: I. Capturing a \(\beta\) particle will cause the radius of a nucleus to increase. Therefore, the daughter nucleus has the greater radius. II. The original nucleus emits a \(\beta\) particle, and anytime a particle is emitted from a nucleus the result is a smaller radius. Therefore, the radius of the daughter nucleus is less than the radius of the original nucleus. III. When a nucleus emits a \(\beta\) particle a neutron is converted to a proton, but the number of nucleons is unchanged. As a result, the radius of the daughter nucleus is the same as that of the original nucleus.

A radioactive substance has a decay constant equal to \(8.9 \times 10^{-3} \mathrm{s}^{-1} .\) What is the half-life of this substance?

A Radioactive Tag A drug prepared for a patient is tagged with \(^{99}_{43} \mathrm{Tc},\) which has a half-life of \(6.05 \mathrm{h}\). (a) What is the decay constant of this isotope? (b) How many \(\$ 3\) Tc nuclei are required to give an activity of \(1.50 \mu\) Ci?

The number of radioactive nuclei in a particular sample decreases over a period of \(18 \mathrm{d}\) to one-sixteenth the original number. What is the half-life of these nuclei?
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