Problem 5 - Physics for Scientists and Engineers with Modern Physics

Question

(II) What is the mass of a bare \(\alpha\) particle (without electrons) in MeV/\(c^{2} ?\)

Step-by-Step Solution

Answer

The mass of an \(\alpha\) particle is approximately 3755.674 MeV/ c^{2}.

Step 1: Understanding the Problem

We need to find the mass of an \(\alpha\) particle in terms of MeV/c^{2}. An \(\alpha\) particle is a helium nucleus, which consists of 2 protons and 2 neutrons.

Step 2: Mass of Protons

Each proton has a mass of approximately 938.272 MeV/c^{2}. Since an \(\alpha\) particle has 2 protons, the total mass of the protons is \(2 \times 938.272\, \text{MeV/}c^{2} = 1876.544\, \text{MeV/}c^{2}.\)

Step 3: Mass of Neutrons

Each neutron has a mass of approximately 939.565 MeV/c^{2}. Since an \(\alpha\) particle has 2 neutrons, the total mass of the neutrons is \(2 \times 939.565\, \text{MeV/}c^{2} = 1879.130\, \text{MeV/}c^{2}.\)

Step 4: Total Mass of the \(\alpha\) Particle

Adding the mass of the protons and neutrons together gives the total mass of the \(\alpha\) particle. \[\text{Total mass} = 1876.544\, \text{MeV/}c^{2} + 1879.130\, \text{MeV/}c^{2} = 3755.674\, \text{MeV/}c^{2}.\]

Step 5: Conclusion

The mass of a bare \(\alpha\) particle is therefore approximately 3755.674 MeV/c^{2}.

Key Concepts

Proton Mass

Protons are one of the basic building blocks of an atomic nucleus. Each proton is positively charged and contributes significantly to the mass of an atom. The mass of a proton is approximately 938.272 MeV/c². This unit, MeV/c², is often used in particle physics to denote energy-related mass.
Protons are a crucial component of an alpha particle. Since an alpha particle contains two protons, their combined mass can be calculated as \(2 \times 938.272 \, \text{MeV/c}^{2} = 1876.544 \, \text{MeV/c}^{2}\).
Understanding the proton's mass is vital because it helps in calculating the mass of composite particles, like alpha particles, which in turn helps in studies involving nuclear reactions and stability.

Neutron Mass

Neutrons are, alongside protons, the core constituents of atomic nuclei. Unlike protons, neutrons bear no electrical charge, and this neutrality is key to their role in atom stability. A neutron's mass is slightly higher than that of a proton, about 939.565 MeV/c².
An alpha particle contains two neutrons. The total mass of these neutrons is calculated by multiplying the mass of a single neutron by two: \(2 \times 939.565 \, \text{MeV/c}^{2} = 1879.130 \, \text{MeV/c}^{2}\).
The neutron's mass, similar to that of the proton, significantly contributes to the overall mass of the alpha particle. Recognizing the neutrons' mass helps us comprehend the balance between the forces at play within the nucleus, thereby aiding in the understanding of nuclear stability and reactions.

Helium Nucleus

The helium nucleus, also known as an alpha particle, plays an important role in nuclear physics. It consists of exactly two protons and two neutrons. When stripped of its electrons, this helium nucleus is referred to as a bare alpha particle.
The mass of an alpha particle is determined by summing the masses of its constituent protons and neutrons. The calculation is as follows:

• Total proton mass: 1876.544 MeV/c²
• Total neutron mass: 1879.130 MeV/c²
• Total mass of the alpha particle: \(1876.544 \, \text{MeV/c}^{2} + 1879.130 \, \text{MeV/c}^{2} = 3755.674 \, \text{MeV/c}^{2}\)

The computed mass of 3755.674 MeV/c² for the helium nucleus is crucial for understanding its energy dynamics and its applications in nuclear processes, like alpha decay. This knowledge assists in various scientific fields, including nuclear medicine and astrophysics.