Problem 49 - University Physics with Modern Physics

Question

Use conservation of mass-energy to show that the energy released in alpha decay is positive whenever the mass of the original neutral atom is greater than the sum of the masses of the final neutral atom and the neutral \(^{4}\) He atom. (Hint: Let the parent nucleus have atomic number \(Z\) and nucleon number \(A .\) First write the reaction in terms of the nuclei and particles involved, and then add \(Z\) electron masses to both sides of the reaction and allot them as needed to arrive at neutral atoms.)

Step-by-Step Solution

Answer

Energy is released when the parent atom's mass exceeds the sum of the daughter atom's and the alpha particle's masses.

Step 1: Understand Alpha Decay

In alpha decay, a nucleus of the parent atom emits an alpha particle (which is essentially a helium nucleus, consisting of 2 protons and 2 neutrons) and transforms into a different nucleus. The general nuclear reaction can be expressed as: \[ \text{Parent Nucleus} \rightarrow \text{Daughter Nucleus} + \alpha\text{-particle} \] The alpha particle is denoted as \(^{4} \text{He} \).

Step 2: Express the Reaction with Atoms and Electrons

For a neutral parent atom with atomic number \(Z\) and mass number \(A\), the initial mass is the mass of the neutral parent atom, \(M(\text{Parent})\). The decay reaction considering the electrons becomes: \[ M(\text{Parent}) \rightarrow M(\text{Daughter}) + M(^{4} \text{He}) + \Delta E \] where \(M(\text{Daughter})\) is the mass of the neutral daughter atom, \(^{4} \text{He}\) is the alpha particle, and \(\Delta E\) is the energy released.

Step 3: Apply Conservation of Mass-Energy

According to conservation of mass-energy, the mass difference before and after the reaction is converted into energy. Therefore, \[ M(\text{Parent}) = M(\text{Daughter}) + M(^{4} \text{He}) + \Delta E/c^2 \] where \(c\) is the speed of light. Solving for the energy released, the equation becomes: \[ \Delta E = (M(\text{Parent}) - M(\text{Daughter}) - M(^{4} \text{He}))c^2 \]

Step 4: Calculate Energy Release Condition

For \(\Delta E\) to be positive, implying that energy is indeed released during the decay, the condition \(M(\text{Parent}) > M(\text{Daughter}) + M(^{4} \text{He})\) must be satisfied. Therefore, when the mass of the original neutral atom is greater than the sum of the masses of the final neutral atom and the alpha particle, energy is released.

Key Concepts

Alpha Decay

Alpha decay is a type of nuclear reaction where an unstable nucleus transforms by emitting an alpha particle.
The alpha particle consists of 2 protons and 2 neutrons, identical to a helium nucleus. This process results in the original atom transforming into a new element, known as the daughter nucleus.
In this decay:

• The atomic number decreases by 2.
• The mass number decreases by 4.

This change is due to the loss of the alpha particle, leading to a lighter and more stable configuration. Alpha decay is commonly observed in heavy elements, such as uranium and radium, as they seek stability by shedding excess particles.

Nuclear Reaction

Nuclear reactions involve changes within an atom's nucleus, where protons and neutrons are rearranged.
These changes release or absorb energy. In the case of alpha decay, the nuclear reaction can be expressed as: \[ \text{Parent Nucleus} \rightarrow \text{Daughter Nucleus} + \alpha\text{-particle} \] In nuclear reactions such as this, the conservation laws of physics apply:

• Conservation of mass-energy: The total mass-energy remains constant.
• Conservation of charge: The total charge before and after the reaction is the same.
• Conservation of nucleon number: The total number of nucleons (protons plus neutrons) is unchanged.

These laws ensure that the "ingredients" of the reaction are balanced, encompassing both matter and energy forms.

Mass-Energy Equivalence

Mass-energy equivalence, one of the cornerstones of Einstein's theory of relativity, is encapsulated in the famous equation:\[ E = mc^2 \]This equation asserts that mass and energy are interchangeable; they are different forms of the same entity. In nuclear reactions like alpha decay, part of the mass is converted to energy, as regulated by this principle.
When the parent atom decays by emitting an alpha particle:

• The loss in mass becomes a gain in energy.
• This energy, denoted as \(\Delta E\), is calculated using the mass difference before and after the decay.
• High energy particles or radiation is often the result of mass-energy conversion.

Thus, mass-energy equivalence provides the framework to understand why energy is released when heavier nuclei decay.

Neutral Atoms

Neutral atoms have no overall charge because they possess an equal number of protons and electrons. Their overall electrical neutrality allows for stable and predictable behavior in chemical and nuclear reactions.
In alpha decay, the neutrality of atoms is critical when expressing the mass-energy balance:

• Mass calculations begin with neutral parent atoms and end with neutral daughter atoms.
• Electrons accompany both the parent and daughter atoms to ensure they are compared on the same basis.
• The reaction balances include all particles involved to maintain overall charge neutrality.

Including electrons helps provide an accurate theoretical calculation of mass differences, critical for deriving the energy released. Overall, analyzing reactions with neutral atoms simplifies the comparison between reactants and products in nuclear decay.