Determine the maximum kinetic energy of the beta particle emitted during the following decay scheme:
^{12}\text{N} \to ^{12}\textC^{*} + e^{+} + \nu $<br>^{12}\textC^{*} \to ^{12}\text{C} + \gamma\;(4.43 \text{ MeV})$ .
The atomic mass of ^{12}\text{N} $is given as$ 12.018613 \text{ u}.

Question

Determine the maximum kinetic energy of the beta particle emitted during the following decay scheme:
^{12}\text{N} \to ^{12}\textC^{*} + e^{+} + \nu

<br>^{12}\textC^{*} \to ^{12}\text{C} + \gamma\;(4.43 \text{ MeV})

.
The atomic mass of ^{12}\text{N}

is given as

12.018613 \text{ u}.

Quantrex Academy · Accepted Answer

Correct answer: C. 11.88 MeV

The maximum kinetic energy of the emitted positron (

e^{+}

) is equal to the

Q

-value of the decay process.

For a

\beta^{+}

decay, the

Q

-value in terms of atomic masses is given by:

Q = [m(^{12}\text{N}) - m(^{12}\textC^{*}) - 2m_e]c^2

Since the ^{12}\textC^{*}

nucleus is in an excited state and subsequently emits a gamma ray of energy

E_{\gamma} = 4.43 \text{ MeV}

to reach the ground state, its mass can be written as:<br>

m(^{12}\textC^{*})c^2 = m(^{12}\text{C})c^2 + E_{\gamma}

<br><br>Substituting this into the

Q

-value equation, we get:<br>

K_\max = Q = [m(^{12}\text{N}) - m(^{12}\text{C})]c^2 - 2m_e c^2 - E_{\gamma}

<br><br>Given values:<br>Atomic mass of ^{12}\text{N} = 12.018613 \text{ u}

Atomic mass of ^{12}\text{C} = 12.000000 \text{ u}

(by definition)<br>Mass difference

\Delta m = 12.018613 - 12.000000 = 0.018613 \text{ u}

<br><br>Converting the mass difference into energy (using

1 \text{ u} \approx 931.5 \text{ MeV}/c^2

):<br>

\Delta m c^2 = 0.018613 \times 931.5 \text{ MeV} \approx 17.338 \text{ MeV}

<br><br>The rest mass energy of two electrons is:<br>

2m_e c^2 = 2 \times 0.511 \text{ MeV} = 1.022 \text{ MeV}

<br><br>Now, substituting the energies into the

K_\max

equation:<br>

K_\max = 17.338 \text{ MeV} - 1.022 \text{ MeV} - 4.43 \text{ MeV}

<br><br>

K_\max = 11.886 \text{ MeV} \approx 11.88 \text{ MeV}

Determine the maximum kinetic energy of the beta particle emitted...