Question

The nucleus ${}_6{C^{12}}$ absorbs an energetic neutron and emits a beta particle ($\beta$ ).The resulting nucleus is
A. ${}_7{N^{14}}$
B. ${}_5{B^{13}}$
C. ${}_7{N^{13}}$
D. ${}_6{C^{13}}$

Answer 1

To solve this problem write the equation for the reaction when a beta particle leaves a nucleus. An electron with high speed is called a beta particle when it leaves from the nucleus. The general equation for beta decay is given by, $_Z{X^A}\; \to {\;_{Z + 1}}X{\prime ^A}\; + {\;_{ - 1}}{e^0}\; + \;{\bar \nu _e}$ where $A$ and $Z$ are the mass number and atomic number of the decaying nucleus, and $X$ and $X\prime$ are the initial and final elements, respectively.

Complete step by step answer:
We know, in ${\beta ^ - }$ decay, the weak interaction converts an atomic nucleus into a nucleus with atomic number increased by one, while emitting an electron (${e^ - }$) and an electronic antineutrino ($\;{\bar \nu _e}$). ${\beta ^ - }$ decay generally occurs in neutron-rich nuclei.

Here, we have nucleus ${}_6{C^{12}}$ which absorbs an energetic neutron and emits a beta particle ($\beta$ ).So, the reaction equation can be written as, ${}_6{C^{12}}{ + _0}{n^1} \to {\;_Z}{X^A}\; + {\;_{ - 1}}{e^0}\;$ .Here, let’s assume the daughter nucleus will have the mass number $A$ and atomic number $Z$. Now, we can balance the mass number and atomic number for both sides of the equation. So, balancing mass number we get,
$12 + 1 = A + 0$
$\Rightarrow A = 13$
Now, balancing the atomic number of the equation from both sides we get,
$6 + 0 = Z - 1$
$\therefore Z = 7$
So, the resulting nucleus has an atomic number $7$. So, it will be a nucleus of nitrogen.Thus, the resulting nucleus will be, ${}_7{N^{13}}$

Hence, option C is correct.

Note: ${\beta ^ - }$ decay occurs when the nucleus has a number of neutrons, greater than protons. Here, the carbon nucleus has an equal number of protons and neutrons, but ${\beta ^ - }$ occurs due to absorption of extra neutrons. Also, ${\beta ^ + }$decay occurs when the nucleus has a number of protons, greater than neutrons. ${\beta ^ + }$ decay can only happen inside nuclei when the daughter nucleus has a greater binding energy (and therefore a lower total energy) than the mother nucleus.