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Atoms and Nuclei question

2016 · Shift 1 · Q52
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Atoms and Nuclei question

2016 · Shift 1 · Q52

JEE AdvancedPhysicsAtoms and NucleiNumerical+3 / −1
The isotope 512B_5^{12}B512​B having a mass 12.014 u undergoes β\betaβ-decay to 612C_6^{12}C612​C. 612C_6^{12}C612​C has an excited state of the nucleus (612C_6^{12}C612​C*) at 4.041 MeV above its ground state. If 512B_5^{12}B512​B decays to 612C_6^{12}C612​C*, the maximum kinetic energy of the β\betaβ-particle in units of MeV is (1u = 931.5 MeV/c2, where c is the speed of light in vacuum).
Numerical answer
View written solutionFree

Correct answer: 9

  1. Write the decay equation

For β−\beta^-β− decay,

512B→612C∗+e−+νˉ{}_5^{12}B \to {}_6^{12}C^* + e^- + \bar\nu512​B→612​C∗+e−+νˉ

where 612C∗{}_6^{12}C^*612​C∗ is the excited nucleus at 4.041 MeV4.041\,\text{MeV}4.041MeV above the ground state.

  1. Use atomic masses

Given atomic mass of 12B{}^{12}B12B:

M(12B)=12.014 uM({}^{12}B)=12.014\,uM(12B)=12.014u

For carbon-12, the atomic mass is exactly

M(12C)=12.000 uM({}^{12}C)=12.000\,uM(12C)=12.000u

In β−\beta^-β− decay, when atomic masses are used, the QQQ-value is

Q=[M(12B)−M(12C)]c2Q = \big[M({}^{12}B)-M({}^{12}C)\big]c^2Q=[M(12B)−M(12C)]c2

So,

Q=(12.014−12.000)u c2=0.014×931.5 MeVQ = (12.014-12.000)u\,c^2 = 0.014\times 931.5\,\text{MeV}Q=(12.014−12.000)uc2=0.014×931.5MeV Q=13.041 MeVQ = 13.041\,\text{MeV}Q=13.041MeV
  1. Account for excitation energy

Since the daughter nucleus is produced in an excited state of energy 4.041 MeV4.041\,\text{MeV}4.041MeV, the available kinetic energy becomes

Q′=13.041−4.041=9.000 MeVQ' = 13.041 - 4.041 = 9.000\,\text{MeV}Q′=13.041−4.041=9.000MeV
  1. Maximum kinetic energy of the β\betaβ-particle

The neutrino can carry away variable energy. The maximum kinetic energy of the β\betaβ-particle occurs when the neutrino carries negligible energy and nuclear recoil is ignored.

Hence,

Kβ,max⁡=9.000 MeVK_{\beta,\max} = 9.000\,\text{MeV}Kβ,max​=9.000MeV
  1. Final integer answer
9\boxed{9}9​
  1. Comparison with stored answer

Stored correct answer = 999

Our derived answer matches it.

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