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Chapter 30: Problem 24

Identify the unknown isotope \(X\) in the following decays. a. \({ }^{234} \mathrm{U} \rightarrow \mathrm{X}+\alpha\) b. \({ }^{32} \mathrm{P} \rightarrow \mathrm{X}+\mathrm{e}^{-}\) c. \(\mathrm{X} \rightarrow{ }^{30} \mathrm{Si}+\mathrm{e}^{+}\) d. \({ }^{24} \mathrm{Mg} \rightarrow \mathrm{X}+\gamma\)

Short Answer

Expert verified
The unknown isotopes for the given decays are; \n a. Thorium (Th) \n b. Sulfur (S) \n c. Aluminum (Al) \n d. Magnesium (Mg)

Step by step solution

01

Identify the type of decay and its effect on atomic number and atomic mass

a. For the decay \({ }^{234} \mathrm{U} \rightarrow \mathrm{X}+\alpha\), it's an alpha decay. Alpha decay decreases the atomic number by 2 and the atomic mass by 4. \n b. For the decay \({ }^{32} \mathrm{P} \rightarrow \mathrm{X}+\mathrm{e}^{-}\), it's a beta decay. Beta decay increases the atomic number by 1. \n c. For the decay \(\mathrm{X} \rightarrow{ }^{30} \mathrm{Si}+\mathrm{e}^{+}\), it's a positron emission, which decreases the atomic number by 1. \n d. For the decay \({ }^{24} \mathrm{Mg} \rightarrow \mathrm{X}+\gamma\), it's a gamma decay. Gamma decay does not change the atomic number or atomic mass.
02

Calculate the atomic number and atomic mass of the unknown isotope

a. For the alpha decay, we subtract 2 from the atomic number of Uranium (92) and 4 from the atomic mass to get the atomic number and atomic mass of the unknown isotope. \n b. For the beta decay, we add 1 to the atomic number of Phosphorus (15) to get the atomic number of the unknown isotope. The atomic mass remains the same. \n c. For the positron emission, we add 1 to the atomic number of Silicon (14) to get the atomic number of the unknown isotope. The atomic mass remains the same. \n d. For the gamma decay, the atomic number and atomic mass remains the same as Magnesium (12 and 24 respectively).
03

Identify the unknown isotope

Using the calculated atomic numbers and atomic masses, lookup the element in the periodic table. \n a. For the alpha decay, we find that the unknown isotope is Thorium (Th), with atomic number 90 and atomic mass 230. \n b. For the beta decay, we find that the unknown isotope is Sulfur (S), with atomic number 16 and atomic mass 32. \n c. For the positron emission, we find that the unknown isotope is Aluminum (Al), with atomic number 13 and atomic mass 30. \n d. For the gamma decay, we find that the unknown isotope is still Magnesium (Mg), with atomic number 12 and atomic mass 24.

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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 loses an alpha particle. An alpha particle is composed of 2 protons and 2 neutrons, which is essentially a helium nucleus. When an element undergoes alpha decay, it loses these 4 particles from its nucleus. This leads to two significant changes:

  • The atomic number decreases by 2.
  • The atomic mass decreases by 4.
For example, in the decay of Uranium-234 to Thorium-230, the Uranium nucleus releases an alpha particle, resulting in the formation of a new element, Thorium. Its atomic number drops from 92 to 90, and its atomic mass goes from 234 to 230. It is quite a straightforward decay process, yet an essential concept in understanding how heavier elements transform into others.
Beta Decay
Beta decay is another form of nuclear decay but operates differently than alpha decay. In beta decay, a neutron in the nucleus is transformed into a proton, emitting an electron (beta particle) and an antineutrino. The important changes that occur during beta decay include:

  • An increase in the atomic number by 1.
  • The atomic mass remains unchanged.
In the specific example of Phosphorus-32 undergoing beta decay, the resulting element is Sulfur-32. Phosphorus' atomic number increases from 15 to 16, turning it into sulfur. However, the atomic mass stays the same because the nucleon count does not change. This process gradually transforms a neutron-rich nucleus into a more stable form.
Gamma Decay
Gamma decay is a much different decay process than both alpha and beta decay. It occurs when a nucleus emits gamma radiation, which consists of photons 鈥� high-energy light waves. Gamma decay usually happens after other types of decay, serving to release excess energy without changing the atomic number or mass. The key characteristics include:

  • No change in atomic number.
  • No change in atomic mass.
In the case of Magnesium-24 emitting gamma radiation, there is no transformation into another element or isotope. Instead, the magnesium nucleus loses energy and becomes more stable. Gamma decay is critical for understanding how nuclei release energy without altering their identity.
Periodic Table Identification
Using the periodic table is a vital skill for chemistry and nuclear physics. It helps identify elements based on their atomic number and mass number. Here are some quick steps to identify isotopes after nuclear decay:

  • Start by determining the new atomic number and atomic mass after a decay process.
  • Locate the new element on the periodic table using the updated atomic number.
  • The position of this element in the periodic table tells you the identity of the new isotope.
For instance, after alpha decay of Uranium-234, where it becomes Thorium with an atomic number of 90, we refer to the periodic table to confirm this element as Thorium. Periodic table identification requires a careful look at information derived from different nuclear decay processes and gives insight into the resulting elements involved.

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Most popular questions from this chapter

The barium isotope \({ }^{133}\) Ba has a half-life of 10.5 years. A sample begins with \(1.0 \times 10^{10133} \mathrm{Ba}\) atoms. How many are left after. (a) 2 years, (b) 20 years, and (c) 200 years?

\({ }^{235} \mathrm{U}\) decays to \({ }^{207} \mathrm{Pb}\) via the decay series shown in Figure \(30.16 .\) The first decay in the chain, that of \({ }^{235} \mathrm{U},\) has a halflife of \(7.0 \times 10^{8}\) years. The subsequent decays are much more rapid, so we can take this as the half-life for the complete decay of \({ }^{235} \mathrm{U}\) to \({ }^{207} \mathrm{Pb}\). Certain minerals exclude lead but not uranium from their crystal structure, so when the minerals form they have no lead, only uranium. As time goes on, the uranium decays to lead, so measuring the ratio of lead atoms to uranium atoms allows investigators to determine the ages of the minerals. If a sample of a mineral contains 3 atoms of \({ }^{207} \mathrm{Pb}\) for every 1 atom of \({ }^{235} \mathrm{U},\) how many years ago was it formed?

\({ }^{15} \mathrm{O}\) and \({ }^{131} \mathrm{I}\) are isotopes used in medical imaging. \({ }^{15} \mathrm{O}\) is a beta-plus emitter, \({ }^{131}\) I a beta- minus emitter. What are the daughter nuclei of the two decays?

The cadmium isotope \({ }^{109} \mathrm{Cd}\) has a half-life of 462 days. A sample begins with \(1.0 \times 10^{12} 109 \mathrm{Cd}\) atoms. How many are left after (a) 50 days, (b) 500 days, and (c) 5000 days?

The Chernobyl reactor accident in what is now Ukraine was the worst nuclear disaster of all time. Fission products from the reactor core spread over a wide area. The primary radiation exposure to people in western Europe was due to the short-lived (half-life 8.0 days isotope \({ }^{131} \mathrm{I},\) which fell across the landscape and was ingested by grazing cows that concentrated the isotope in their milk. Farmers couldn't sell the contaminated milk, so many opted to use the milk to make cheese, aging it until the radioactivity decayed to acceptable levels. How much time must elapse for the activity of a block of cheese containing \({ }^{131} \mathrm{I}\) to drop to \(1.0 \%\) of its initial value?
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