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.	$\small[Q_1=\left(M_x-M_y\right) c^2 \text { and } Q_2=\left[M_x-M_y-2 m_e\right] c^2$
2.	$\small[Q_1=\left(M_x-M_y\right) c^2 \text { and } Q_2=\left(M_x-M_y\right) c^2$
3.	$\small[Q_1=\left(M_x-M_y-2 m_e\right)c^2 \text { and } Q_2=\left(M_x-M_y+2 m_e\right) c^2$
4.	$\small[Q_1=\left(M_x-M_y+2 m_e\right) c^2 \text { and } Q_2=\left(M_x-M_y+2 m_e\right) c^4$

Hint:

Q -value = (mass of reactants - mass of products) c^{2} J

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

Step 1: Find the

Q -

$Q\text -$ value for a

β^{-}

$\beta^{-}$ decay

β^{-}

$\beta^{-}$ decay is represented as;

_{z} X^{A} \to_{z + 1} Y^{A} +_{- 1} e^{0} + \bar{v} + Q_{1}

${ }_z X^A \rightarrow_{z+1} Y^A+{ }_{-1} e^0+\bar{v}+Q_1$

Q_{1} = [m_{N} (_{Z} X^{A}) - m_{N} (_{Z + 1} Y^{A}) - m_{e}] c^{2}

$Q_1=\left[m_N\left({ }_Z X^A\right)-m_N\left({ }_{Z+1} Y^A\right)-m_e\right] c^2$

Q_{1} = [m_{N} (_{Z} X^{A}) + Z m_{e} - m_{N} (_{Z + 1} Y^{A}) - (Z + 1) m_{e}] 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} = [m (_{Z} X^{A}) - m (_{Z + 1} Y^{A})] c^{2}

$Q_1=\left[m\left({ }_Z X^A\right)-m\left({ }_{Z+1} Y^A\right)\right] c^2$

\Rightarrow Q_{1} = (M_{x} - M_{y}) c^{2}

$\Rightarrow Q_1=\left(M_x-M_y\right) c^2$

Step 2: Find the

Q -

$Q\text -$ value for a

β^{+}

$\beta^{+}$ decay

β^{+}

$\beta^{+}$ decay is represented as;

_{Z} X^{A} =_{Z - 1} Y^{A} +_{1} e^{0} + v + Q_{2}

${ }_Z X^A={ }_{Z-1} Y^A+{ }_1 e^0+v+Q_2$

Q_{2} = [m_{N} (_{Z} X^{A}) - m_{N} (_{Z - 1} Y^{A}) - m_{e}] c^{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} = [m_{N} (_{Z} X^{A}) + Z m_{e} - m_{N} (_{Z - 1} Y^{A}) - (Z - 1) m_{e} - 2 m_{e}] 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} = [m (_{Z} X^{A}) - m (_{Z - 1} Y^{A}) - 2 m_{e}] 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} = (M_{x} - M_{y} - 2 m_{e}) 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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