Problem 27 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A \({ }_{92}^{238} \mathrm{U}\) nucleus emits an \(\alpha\) particle with kinetic energy \(=4.20 \mathrm{MeV}\). (a) What is the daughter nucleus, and (b) what is the approximate atomic mass (in u) of the daughter atom? Ignore recoil of the daughter nucleus.

Step-by-Step Solution

Answer

The daughter nucleus is Th-234 with an approximate atomic mass of 234 u.

Step 1: Understand Alpha Decay

In alpha decay, a nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons ( 2He^4 ). This decreases the original nucleus's mass number by 4 and atomic number by 2.

Step 2: Determine Daughter Nucleus

Subtract the mass number and atomic number of the emitted alpha particle from the original nucleus of uranium-238: 1. Mass number: 238 - 4 = 234 2. Atomic number: 92 - 2 = 90 The new nucleus is Z_{90}^{234} , which is Thorium (Th).

Step 3: Estimate Atomic Mass

The atomic mass of the daughter nucleus is approximately the mass number of the daughter nucleus (234 u), since the mass of electrons is negligible compared to protons and neutrons, and we ignore recoil.

Key Concepts

Thorium

Thorium is a naturally occurring radioactive element that is part of the actinide series on the periodic table. It is represented by the symbol \( \text{Th} \) and has an atomic number of 90. This metal is silvery and tarnishes to a black oxide when exposed to air. When a uranium nucleus undergoes alpha decay, it sheds 2 protons and 2 neutrons in the form of an alpha particle. This transforms it into a nucleus with 2 fewer protons and 4 less in mass number, leading to the formation of thorium. Hence, when \( \text{U} \) decays, thorium is often one of the resulting products. Here's why:

• The alpha particle consists of 2 protons and 2 neutrons, equivalent to \( \text{He}^{4}_{2} \).
• In our example, \( _{92}^{238} \text{U} \rightarrow _{90}^{234} \text{Th} + \text{He}^{4}_{2} \).

Thorium-234 is the specific isotope produced after uranium-238 undergoes alpha decay.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In nuclear reactions, such as alpha decay, part of the energy released is transferred to the kinetic energy of the emitted particles, like alpha particles.The kinetic energy of the alpha particle in the provided example is 4.20 MeV (Mega electron Volt). This is a significant amount of energy that indicates the speed and velocity of the emitted particle.Key points to remember about kinetic energy in alpha decay:

• It is conserved in nuclear reactions; however, it is distributed among the decay products.
• Most of the kinetic energy is carried away by the lighter particle, which is the alpha particle in this case.

For reference, 1 MeV = \( 1.60218 \times 10^{-13} \) joules. This transformation is essential when dealing with energy calculations in nuclear physics.

Nuclear Physics

Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. It explores how nuclear processes, such as radioactive decay, nuclear fission, and nuclear fusion, occur and their respective governing principles. Alpha decay, as highlighted here, is a common form of radioactive decay within nuclear physics. Here's why it's significant:

• It helps scientists understand the stability of a nucleus and the forces at play within it.
• It reveals energy transformation processes and how different elements are formed.

In alpha decay, understanding how a heavy nucleus like uranium-238 decays into thorium-234 helps illustrate concepts like nuclear stability and the balance of forces between nuclear particles.

Atomic Mass

The atomic mass of an element is the mass of an atom, typically expressed in atomic mass units (u). It is approximately equal to the mass number, which is the total number of protons and neutrons in an atom's nucleus. For nuclei undergoing alpha decay, the atomic mass slightly decreases due to the emission of the alpha particle, whose mass is roughly 4 u. Considerations in the provided example:

• The new atomic mass is estimated by subtracting the alpha particle's mass from the original nucleus, resulting in 234 u for the daughter nucleus, thorium.
• Electrons have negligible mass relative to protons and neutrons, so they don't significantly impact the atomic mass.

Thus, the atomic mass aids in calculating the mass defects and binding energy which are pivotal in nuclear stability and energy release during decay processes.