Step 1: Understanding the Concept:
In any balanced nuclear reaction, both the total mass number (represented by \(A\), the superscript) and the total atomic number (represented by \(Z\), the subscript) must be conserved.
By applying these conservation laws, we can deduce the properties of unknown particles emitted during radioactive decay.
Step 2: Key Formula or Approach:
The nuclear reaction is:
\[ {}_{Z_1}^{A_1}\text{Parent} \longrightarrow {}_{Z_2}^{A_2}\text{Daughter} + {}_{Z_\beta}^{A_\beta}\beta + {}_{Z_X}^{A_X}\text{X} \]
We enforce two rules:
1. Conservation of mass number: \(A_1 = A_2 + A_\beta + A_X\)
2. Conservation of atomic number: \(Z_1 = Z_2 + Z_\beta + Z_X\)
Step 3: Detailed Explanation:
Let's analyze the given equation: \({}_6\text{C}^{11} \longrightarrow {}_5\text{B}^{11} + \beta + \text{X}\).
First, apply the conservation of mass number (\(A\)):
\[ 11 = 11 + A_\beta + A_X \]
A beta particle (whether an electron or a positron) has a mass number of zero (\(A_\beta = 0\)).
\[ 11 = 11 + 0 + A_X \implies A_X = 0 \]
The unknown particle X has no mass number.
Next, apply the conservation of atomic number (\(Z\)):
\[ 6 = 5 + Z_\beta + Z_X \]
Here, we must identify the type of beta decay. Since the atomic number decreases from 6 to 5 (a proton turns into a neutron), the emitted \(\beta\) particle must carry a positive charge to balance the equation.
Thus, the \(\beta\) particle is a positron (\(\beta^+\) or \({}_{+1}e^0\)), which means \(Z_\beta = +1\).
\[ 6 = 5 + 1 + Z_X \]
\[ 6 = 6 + Z_X \implies Z_X = 0 \]
The unknown particle X has a mass number of 0 and an atomic charge of 0.
In nuclear physics, \(\beta^+\) (positron) emission is always accompanied by the emission of an electron neutrino (\(\nu_e\)) to conserve lepton number.
Conversely, \(\beta^-\) (electron) emission is accompanied by an antineutrino (\(\bar{\nu}_e\)).
Since this is a \(\beta^+\) decay, particle X must be a neutrino.
Step 4: Final Answer:
The particle 'x' is a neutrino.