kinetic energy of emitted beta particle formula - For the \beta ^{+} (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K–shell, is captured by the nucleus and a neutrino is emitted). e^{+} + ^{A}_{Z} X \rightarrow ^{A}_{Z - 1} Y + v Show that if \beta^{+} emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.

For the \beta ^{+} (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K–shell, is captured by the nucleus and a neutrino is emitted). e^{+} + ^{A}_{Z} X \rightarrow ^{A}_{Z - 1} Y + v Show that if \beta^{+} emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.

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Anonym0us (Student, UG1 Grade)

Qalaxia Master Bot (last edited 1 year ago)

I found an answer from www.britannica.com

radioactivity | Definition, Types, Applications, & Facts | Britannica

Radioactivity, property exhibited by certain types of matter of emitting energy ... Radioactive decay is a property of several naturally occurring elements as ... The alpha particle is actually the nucleus of a helium-4 atom, with two positive charges ⁴/2He. ... reactions: beta-plus decay, or positron emission, and electron capture.

For more information, see radioactivity | Definition, Types, Applications, & Facts | Britannica

Swetha (last edited 1 year ago) (Student, UG1 Grade)

Nuclear reactions

When a neutron is converted into a proton, the process is known as beta decay.

converting the original nuclide into a nuclide isobar

The Beta decay general equation: _Z^AX\ \ \rightarrow_{\ \ \ Z-1}^{\ \ A}Y\ +e^++\ \overline{v}\ +Q

Electron capture equation: e^{+} + ^{A}_{Z} X \rightarrow ^{A}_{Z-1} Y + v

Step 1: Finding the energy released in beta decay( \beta^{+} emission) reaction

Energy released (or Q -value ) Q_1 = \text{ nuclear mass of } ^{A}_{Z} X - \text{nuclear mass of } ^{A}_{Z-1} Y - m_e

When calculating Q-values in a beta decay, convert nuclear mass to atomic mass.

To get the Q value in terms of atomic masses, subtract Z m_{e} from the atomic mass of ^{A}_{Z} X and (Z - 1)m_e from the atomic mass of ^{A}_{Z-1} Y .

[math] Q_1 = [\text{ Atomic mass of } ^{A}_{Z} X - Zm_e - [\text{ Atomic mass of } ^{A}_{Z-1} Y - (Z-1)m_e] - m_e]c^2 [/math]

[math] Q_1 = [\text{ Atomic mass of } ^{A}_{Z} X - \text{Atomic mass of } ^{A}_{Z-1} Y -Zm_e +Zm_e-m_e - m_e] c^2 [/math]

[math]Q_1=\left[\text{ m}\ \left(_Z^AX\right)-m\left(_{Z-1}^A\ Y\right)-2m_e\right]c^2[/math]

Step 2: Finding the energy released in electron capture reaction

Energy released [math] Q_2 = [M (^{A}_{Z} X) + m_e - M (^{A}_{Z-1} Y)]c^2 [/math]

converting nuclear mass to atomic mass.

[math] Q_2 = [m (^{A}_{Z} X) + m_e - Zm_e - m (^{A}_{Z-1} Y) + (Z - 1)m_e]c^2 [/math]

[math] Q_2 = [m (^{A}_{Z} X) + m_e - m_e - m (^{A}_{Z-1} Y)]c^2 [/math]

[math] Q_2 = [m (^{A}_{Z} X) - m (^{A}_{Z-1} Y)]c^2 [/math]

Thus, it can be concluded that Q_1 > 0 and Q_2 > 0 but Q_2 > 0 this does not suggest Q_1 > 0

This means that if \beta^{+} emission is energetically allowable, the electron capture process is also allowable, but not vice versa. This is due to the fact that for an energetically allowable nuclear reaction, the Q-value must be positive.