Problem 60 - Master Resource Book in JEE Main Physics

Question

\(M_{x}\) and \(M_{y}\) denote the atomic masses of the parent and the daughter nuclei respectively in a radioactive decay. The \(Q\)-value of a \(\beta^{-}\)decay is \(Q_{1}\) and that for a \(\beta^{+}\) decay is \(Q_{2}\). If \(m_{e}\) denotes the mass of an electron, then which of the following statements is correct? [NCERT Exemplar] (a) \(Q_{1}=\left(M_{x}-M_{y}\right) c^{2}\) and \(Q_{2}=\left(M_{x}-M_{y}-2 m_{e}\right) c^{2}\) (b) \(Q_{1}=\left(M_{x}-M_{y}\right) c^{2}\) and \(Q_{2}=\left(M_{x}-M_{y}\right) c^{2}\) (c) \(Q_{1}=\left(M_{x}-M_{y}-2 m_{e}\right) c^{2}\) and \(Q_{2}=\left(M_{x}-M_{y}+2 m_{e}\right) c^{2}\) (d) \(Q_{1}=\left(M_{x}-M_{y}+2 m_{e}\right) c^{2}\) and \(Q_{2}=\left(M_{x}-M_{y}+2 m_{e}\right) c^{2}\)

Step-by-Step Solution

Answer

Option (a) is correct.

Step 1: Understand the Question

We need to determine how the Q-values for β-decay are related to the masses of parent and daughter nuclei and the mass of an electron. This involves checking the mass-energy equivalence principles used in radioactive decay.

Step 2: Review β-decay

In β⁻ decay, a neutron is converted into a proton, with the emission of an electron and an antineutrino. In β⁺ decay, a proton is converted into a neutron, with the emission of a positron and a neutrino. The Q-value represents the energy released during these decays.

Step 3: Apply Mass-Energy Conservation for β⁻ Decay

For β⁻ decay, the Q-value, denoted as Q₁, is given by the difference in atomic masses of the parent and daughter nuclei. Since an electron is emitted, its mass affects the net energy outcome.Thus, the formula becomes: \[ Q₁ = (M_{x} - M_{y})c² \]This reflects the mass-energy equivalence without accounting for the mass of the emitted electron independently, as it's already part of the energy balance.

Step 4: Apply Mass-Energy Conservation for β⁺ Decay

For β⁺ decay, an additional factor is the appearance of a positron. Thus, the Q-value, Q₂, considers both the mass difference and the additional mass of two electrons (a positron and an electron cancel in mass terms) which aren't part of the atomic masses.Hence, \[ Q₂ = (M_{x} - M_{y} - 2m_{e})c² \]

Step 5: Identify the Correct Answer

Matching the calculations with the provided options, the correct option is: (a) \(Q_{1}=(M_{x}-M_{y})c^2\) and \(Q_{2}=(M_{x}-M_{y}-2m_{e})c^2\).

Key Concepts

Beta Decay

Beta decay is a fascinating process observed in radioactive substances. It involves the transformation of a neutron into a proton or vice versa. This process is driven by the weak nuclear force, one of the four fundamental forces of nature.

• Beta Minus (β⁻) Decay: In this type of decay, a neutron within the nucleus is converted into a proton.
• This transformation releases an electron (called a beta particle) and an antineutrino.
• Beta Plus (β⁺) Decay: Conversely, this process involves the decay of a proton into a neutron.
• It emits a positron (the antimatter counterpart of the electron) and a neutrino.

Beta decay helps an unstable nucleus move towards a more stable form by adjusting its neutron-to-proton ratio. These particles released in the processes ensure the conservation of charge and energy.

Q-value

The Q-value is a crucial concept in understanding nuclear reactions and decay processes. It represents the energy released or absorbed during a nuclear reaction or decay.

• Energy Release: The Q-value can be thought of as the net energy change, typically expressed in units of energy such as MeV (Mega-electronvolts).
• In a nuclear decay, a positive Q-value indicates that the process is exothermic, meaning it releases energy.

For beta decay, the Q-value calculation involves the difference in atomic masses between the parent and daughter nuclei. More precisely:

• Beta Minus Decay Q-value (Q₁): The energy released is calculated as \[ Q₁ = (M_{x} - M_{y})c^2 \]
• This reflects the mass-energy equivalence principle without directly considering the emitted electron's mass, as it's part of the energy balance.

The Q-value informs us about the reaction's energetics, indicating either energy release or input required for the decay.

Mass-Energy Equivalence

Mass-energy equivalence is a fundamental principle in physics, articulated by Einstein's famous equation, \( E=mc^2 \). This states that mass and energy are interchangeable; mass can be converted into energy and vice versa.

• Implications for Decay: In radioactive decay processes, such as beta decay, a tiny loss of mass results in a significant energy release.
• This released energy is what we refer to as the Q-value of the decay reaction. It reflects how the mass difference between the initial and final states translates into energy.

For instance, in beta decay, we compute the Q-value using the masses of the parent and daughter nuclei, showing the difference in mass times the square of the speed of light.This calculation helps in understanding how energy efficiencies occur in nuclear transformations.

Neutron to Proton Conversion

Neutron to proton conversion is a key aspect of beta decay. This process is fundamental in balancing nuclear reactions and maintaining stability within a nucleus.

• Beta Minus Decay: A neutron inside the nucleus changes into a proton.
• This results in the emission of an electron (beta particle) and an antineutrino, which helps conserve the reaction's energy and momentum.

Conversely, in beta plus decay, a proton converts into a neutron, thereby emitting a positron and a neutrino. This balance is crucial for the atomic nucleus, as it helps achieve a stable configuration by moderating the neutron-to-proton ratio. By understanding these conversions, students can grasp how nuclear decay processes function and their role in natural radioactive transformations.