Ncert Exercises & Solutions

\(M_x\) and \(M_y\) denote the atomic masses of the parent and the daughter nuclei respectively in radioactive decay. The \(Q\text -\)value for a \(\beta^{-}\) decay is \(Q_1\) and that for a \(\beta^{+}\) decay is \(Q_2.\) If \(m_e\) denotes the mass of an electron, then which of the following statements is correct?
1. \(Q_1=\left(M_x-M_y\right) c^2 \text { and } Q_2=\left[M_x-M_y-2 m_e\right] c^2 \)
2. \( Q_1=\left(M_x-M_y\right) c^2 \text { and } Q_2=\left(M_x-M_y\right) c^2 \)
3. \(Q_1=\left(M_x-M_y-2 m_e\right)c^2 \text { and } Q_2=\left(M_x-M_y+2 m_e\right) c^2 \)
4. \(Q_1=\left(M_x-M_y+2 m_e\right) c^2 \text { and } Q_2=\left(M_x-M_y+2 m_e\right) c^4 \)

Hint: \(\text { Q-value }=\left(\text { Mass of reactants }- \text { mass of products}\right) {c}^2~ \text {J} \)

Step 1: Find the \(Q\text -\) value for a \(\beta^{-}\) decay
\(\beta^{-}\) decay is represented as;
\( { }_z X^A \rightarrow_{z+1} Y^A+{ }_{-1} e^0+\bar{v}+Q_1 \)
\( Q_1=\left[m_N\left({ }_Z X^A\right)-m_N\left({ }_{Z+1} Y^A\right)-m_e\right] c^2 \)
\(Q_1=\left[m_N\left({ }_Z X^A\right)+Z m_e-m_N\left({ }_{Z+1} Y^A\right)-(Z+1) m_e\right] c^2 \)
\(Q_1=\left[m\left({ }_Z X^A\right)-m\left({ }_{Z+1} Y^A\right)\right] c^2 \)
\(\Rightarrow Q_1=\left(M_x-M_y\right) c^2\)

Step 2: Find the \(Q\text -\) value for a \(\beta^{+}\) decay
\(\beta^{+}\)decay is represented as;
\({ }_Z X^A={ }_{Z-1} Y^A+{ }_1 e^0+v+Q_2 \)
\(Q_2=\left[m_N\left({ }_Z X^A\right)-m_N\left({ }_{Z-1} Y^A\right)-m_e\right] c^2 \)
\(Q_2=\left[m_N\left({ }_Z X^A\right)+Z m_e-m_N\left({ }_{Z-1} Y^A\right)-(Z-1) m_e-2 m_e\right] c^2 \)
\(Q_2=\left[m\left({ }_Z X^A\right)-m\left({ }_{Z-1} Y^A\right)-2 m_e\right] c^2 \)
\(\Rightarrow Q_2=\left(M_x-M_y-2 m_e\right) c^2 \)
Hence, option (1) is the correct answer.

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