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Chapter 24: Problem 9

Write balanced nuclear equations for the following: (a) \(\beta^{-}\) decay of sodium- 26 (b) \(\beta^{-}\) decay of francium- 223 (c) Alpha decay of \({ }^{212} \mathrm{Bi}\) i

Short Answer

Expert verified
\( \beta^{-} \) decay of Na-26: \[ \mathrm{^{26}_{11}Na \rightarrow ^{26}_{12}Mg + e^{-} + \overline{u}} \]\( \beta^{-} \) decay of Fr-223: \[ \mathrm{^{223}_{87}Fr \rightarrow ^{223}_{88}Ra + e^{-} + \overline{u}} \] Alpha decay of \( \mathrm{^{212}Bi} \): \[ \mathrm{^{212}_{83}Bi \rightarrow ^{208}_{81}Tl + ^{4}_{2}He} \]

Step by step solution

01

Understand the Process of \( \beta^{-} \) Decay

\( \beta^{-} \) decay involves a neutron in the nucleus transforming into a proton while emitting a beta particle (an electron, \( e^{-} \)) and an antineutrino (\( \overline{u} \)). The atomic number increases by 1, while the mass number remains the same.
02

Write the \( \beta^{-} \) Decay Equation for Sodium-26

The symbol for sodium is Na. Sodium-26 decays by \( \beta^{-} \) decay. The resulting element will have an atomic number one unit higher than sodium (11), so it becomes magnesium (Mg). The mass number remains 26: \[ \mathrm{^{26}_{11}Na \rightarrow ^{26}_{12}Mg + e^{-} + \overline{u}} \]
03

Write the \( \beta^{-} \) Decay Equation for Francium-223

The symbol for francium is Fr. Francium-223 decays by \( \beta^{-} \) decay. The resulting element will have an atomic number one unit higher than francium (87), so it becomes radium (Ra). The mass number remains 223: \[ \mathrm{^{223}_{87}Fr \rightarrow ^{223}_{88}Ra + e^{-} + \overline{u}} \]
04

Understand Alpha Decay

Alpha decay involves the emission of an alpha particle, which is a helium nucleus (\( \alpha \) or \( \mathrm{^{4}_{2}He} \)). The atomic number decreases by 2, and the mass number decreases by 4.
05

Write the Alpha Decay Equation for \( \mathrm{^{212}Bi} \)

The symbol for bismuth is Bi. Bismuth-212 undergoes alpha decay. The resulting element will have an atomic number two units lower than bismuth (83), so it becomes thallium (Tl). The mass number decreases by 4, so it becomes 208: \[ \mathrm{^{212}_{83}Bi \rightarrow ^{208}_{81}Tl + ^{4}_{2}He} \]

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Key Concepts

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

Beta Decay
Beta decay is an important type of radioactive decay in which a neutron in the nucleus converts into a proton. During this conversion, a beta particle (an electron, denoted as \( e^{-} \) ) and an antineutrino (\( \overline{u} \) ) are emitted. The notable effects of beta decay are:
  • The atomic number of the element increases by 1, as the neutron becomes a proton, adding one positive charge to the nucleus.
  • The mass number remains unchanged because the total number of nucleons (protons and neutrons) stays the same.
For instance, when sodium-26 undergoes beta decay: \[ \mathrm{^{26}_{11}Na \rightarrow ^{26}_{12}Mg + e^{-} + \overline{u}} \] Sodium (Na) with atomic number 11 changes to magnesium (Mg) with atomic number 12, while maintaining its mass number at 26.
Alpha Decay
Alpha decay is another prevalent form of radioactive decay. In alpha decay, the nucleus emits an alpha particle, which is essentially a helium nucleus (\( \alpha \) or \( \mathrm{^{4}_{2}He} \) ). This form of decay has noticeable effects:
  • The atomic number of the original element decreases by 2 as two protons are removed.
  • The mass number decreases by 4 because the nucleus loses a total of four nucleons (two protons and two neutrons).
For example, in the alpha decay of bismuth-212: \[ \mathrm{^{212}_{83}Bi \rightarrow ^{208}_{81}Tl + ^{4}_{2}He} \] Bismuth (Bi), with atomic number 83, transitions to thallium (Tl), with atomic number 81, and the mass number drops from 212 to 208.
Nuclear Reactions
Nuclear reactions encompass a vast array of processes by which atomic nuclei interact and transform. These transformations can release or absorb significant amounts of energy. Key forms of nuclear reactions include:
  • Fusion: The combining of two light nuclei into a heavier nucleus, releasing energy. This is the process that powers the sun.
  • Fission: The splitting of a heavy nucleus into two lighter nuclei, along with a few neutrons and energy. This reaction is utilized in nuclear power plants.
  • Radioactive Decay: Spontaneous transformation of an unstable nucleus into a more stable one by emitting particles such as alpha particles, beta particles, or gamma rays. Beta and alpha decay are prime examples of this.
In the context of the given exercise, both beta decay and alpha decay are nuclear reactions. They involve changes in the atomic structure of the elements, leading to the emission of particles and the formation of new elements.

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Most popular questions from this chapter

A laboratory rat weighs \(265 \mathrm{~g}\) and absorbs \(1.77 \times 10^{10} \beta^{-}\) particles, each with an energy of \(2.20 \times 10^{-13} \mathrm{~J}\). (a) How many rads does the animal receive? (b) What is this dose in Gy? (c) If the RBE is \(0.75,\) what is the equivalent dose in \(\mathrm{Sv} ?\)

Uranium and radium are found in many rocky soils throughout the world. Both undergo radioactive decay, and one of the products is radon-222, the heaviest noble gas \(\left(t_{1 / 2}=\right.\) 3.82 days). Inhalation of this gas contributes to many lung cancers. According to the Environmental Protection Agency, the level of radioactivity from radon in homes should not exceed \(4.0 \mathrm{pCi} / \mathrm{L}\) of air. (a) What is the safe level of radon in Bq/L of air? (b) A home has a radon measurement of \(41.5 \mathrm{pCi} / \mathrm{L}\). The owner vents the basement in such a way that no more radon enters the living area. What is the activity of the radon remaining in the room air (in Bq/L) after 9.5 days? (c) How many more days does it take to reach the EPA recommended level?

The fraction of a radioactive isotope remaining at time \(t\) is \(\left(\frac{1}{2}\right)^{t / t_{12}},\) where \(t_{1 / 2}\) is the half-life. If the half-life of carbon-14 is 5730 yr, what fraction of carbon- 14 in a piece of charcoal remains after (a) \(10.0 \mathrm{yr} ;\) (b) \(10.0 \times 10^{3} \mathrm{yr} ;\) (c) \(10.0 \times 10^{4} \mathrm{yr} ?\) (d) Why is radiocarbon dating more reliable for the fraction remaining in part (b) than that in part (a) or in part (c)?

Write balanced nuclear equations for the following: (a) Alpha decay of \({ }^{234} \mathrm{U}\) (b) Electron capture by neptunium- 232 (c) Positron emission by \({ }_{7}^{12} \mathrm{~N}\)

The half-life of radium- 226 is \(1.60 \times 10^{3}\) yr. How many hours will it take for a 2.50 -g sample to decay to the point where \(0.185 \mathrm{~g}\) of the isotope remains?
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