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Chapter 40: Problem 64

What is the total energy released in the decay \(n \rightarrow p+e^{-}+\bar{\nu}_{e} ?\)

Short Answer

Expert verified
Based on the given step-by-step solution, the short answer is: The total energy released in the decay of a neutron into a proton, an electron, and an electron anti-neutrino is approximately 2.211 × 10⁻¹³ Joules.

Step by step solution

01

Masses of the particles involved

Firstly, we need to know the masses of the particles involved in the decay: neutron, proton, and electron. The electron anti-neutrino's mass can be considered negligible. The masses (in atomic mass units) are as follows: - Mass of neutron (m_n): 1.008665 u - Mass of proton (m_p): 1.007276 u - Mass of electron (m_e): 0.000549 u
02

Conversion of atomic mass units to kilograms

To use the mass-energy equivalency formula, we need to convert the masses of the particles from atomic mass units (u) to kilograms (kg). We can do that using the following conversion factor: 1 u = 1.66053906660 × 10⁻²⁷ kg. Thus, - m_n = 1.008665 u × 1.66053906660 × 10⁻²⁷ kg/u = 1.6749 × 10⁻²⁷ kg - m_p = 1.007276 u × 1.66053906660 × 10⁻²⁷ kg/u = 1.6726 × 10⁻²⁷ kg - m_e = 0.000549 u × 1.66053906660 × 10⁻²⁷ kg/u = 9.1094 × 10⁻³¹ kg
03

Calculate the energy before and after decay

Now, we can calculate the energy before and after the decay using the mass-energy equivalency formula (E = mc²). The energy before the decay is the energy of the neutron, and the energy after the decay is the sum of the energies for the proton, electron, and electron anti-neutrino (which has negligible energy). - Energy before decay (E_n): E_n = m_n × c² = (1.6749 × 10⁻²⁷ kg) × (3 × 10⁸ m/s)² = 1.5054 × 10⁻¹⁰ J - Energy after decay (E_p + E_e): E_p = m_p × c² = (1.6726 × 10⁻²⁷ kg) × (3 × 10⁸ m/s)² = 1.5032 × 10⁻¹⁰ J E_e = m_e × c² = (9.1094 × 10⁻³¹ kg) × (3 × 10⁸ m/s)² = 8.1871 × 10⁻¹⁴ J
04

Calculate the total energy released

Now that we have the energy before and after the decay, we can find the total energy released in the process by subtracting the final energy from the initial energy. Total energy released (ΔE) = E_n - (E_p + E_e) = 1.5054 × 10⁻¹⁰ J - (1.5032 × 10⁻¹⁰ J + 8.1871 × 10⁻¹⁴ J) = 2.211 × 10⁻¹³ J Thus, the total energy released in the decay of a neutron into a proton, electron, and an electron anti-neutrino is approximately 2.211 × 10⁻¹³ Joules.

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