Chapter 18: Problem 2513
if \(_{92} \mathrm{U}^{238}\) undergoes successively \(8 \alpha\) decays and \(6 \beta\) decays then resulting nucleus is (A) \(\mathrm{Pb}^{206}\) (B) \(\mathrm{Pb}^{208}\) (C) \(\mathrm{Pb}^{214}\) (D) None of these
Short Answer
Expert verified
The resulting nucleus after 8 alpha decays and 6 beta decays of \(_{92} \mathrm{U}^{238}\) is (A) \(\mathrm{Pb}^{206}\).
Step by step solution
01
Understanding Alpha decay and Beta decay
In an Alpha decay, an atom emits an alpha particle which consists of 2 protons and 2 neutrons. As a result, the atomic number (number of protons) of the decaying nucleus decreases by 2 and the mass number (number of protons + number of neutrons) decreases by 4.
In a Beta decay (specifically, beta-minus decay), a neutron is converted into a proton, with the emission of an electron. The atomic number increases by 1, while the mass number remains the same.
02
Start with the given isotope
The initial isotope given is \(_{92} \mathrm{U}^{238}\). Initially, the atomic number (Z) is 92 and the mass number (A) is 238.
03
Calculate the result after 8 Alpha decays
For 8 alpha decays, there will be a loss of \(8 * 2 =16\) protons and \(8 * 4 =32\) nucleons (protons and neutrons combined). After these decays, the atomic number will be \(92 - 16 = 76\) and the mass number will be \(238 - 32 = 206\).
Now, the nucleus is represented by \(_{76} \mathrm{X}^{206}\), where X is the symbol of the resulting element with atomic number 76.
04
Calculate the result after 6 Beta decays
For 6 beta decays, the mass number remains the same while the atomic number increases by 6. The updated atomic number will be \(76 + 6 = 82\). The nucleus after all decays will be \(_{82} \mathrm{Y}^{206}\), where Y is the symbol of the resulting element with atomic number 82.
05
Identify the resulting element
Based on the periodic table, the element with atomic number 82 is Lead (Pb). Thus, the resulting nucleus after 8 alpha and 6 beta decays will be \(_{82} \mathrm{Pb}^{206}\).
The correct answer is: (A) \(\mathrm{Pb}^{206}\).
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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 atom emits an alpha particle. An alpha particle is composed of two protons and two neutrons, essentially a helium nucleus. This emission causes significant changes in the decaying atom.
During alpha decay:
During alpha decay:
- The atomic number decreases by 2 because two protons are lost. This changes the element into a different one on the periodic table.
- The mass number decreases by 4 as both the protons and neutrons are counted among the nucleons.
Beta Decay
In beta decay, an atom transforms by converting a neutron into a proton while releasing a beta particle. In beta-minus decay, the particle released is an electron.
This process affects the atom in the following ways:
This process affects the atom in the following ways:
- The atomic number increases by 1, reflecting the additional proton created.
- The mass number remains unchanged since the number of nucleons (protons and neutrons together) stays the same.
Atomic Number
The atomic number, frequently symbolized as Z, represents the number of protons in the nucleus of an atom. This number is crucial because it determines the identity of an element on the periodic table. Each element has a unique atomic number.
- For example, hydrogen has an atomic number of 1, while helium's atomic number is 2.
- Changes in the atomic number mean the substance is transforming into a different element. Such changes occur during nuclear reactions like alpha or beta decay.
Mass Number
The mass number, denoted as A, refers to the total count of protons and neutrons in an atom's nucleus. Together, these particles constitute the atom's mass since electrons have negligible mass.
Here are some key points about mass number:
Here are some key points about mass number:
- It is essential for identifying isotopes of an element, which have the same number of protons but different numbers of neutrons.
- The mass number is crucially affected during nuclear decay. For instance, in alpha decay, it decreases by 4.