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-value for a $β^{-}$ decay is Q₁ and that for a $β^{+}$ decay is Q₂. If $m_{e}$ denotes the mass of an electron, then which of the following statements is correct?

1. $Q_{1} = (M_{x} - M_{y}) c^{2} and Q_{2} = [M_{x} - M_{y} - 2 m_{e}] c^{2}$

2. $Q_{1} = (M_{x} - M_{y}) c^{2} and Q_{2} = (M_{x} - M_{y}) c^{2}$

3. $Q_{1} = (M_{x} - M_{y} - 2 m_{e}) c^{2} and Q_{2} = (M_{x} - M_{y} + 2 m_{e}) c^{2}$

4. $Q_{1} = (M_{x} - M_{y} + 2 m_{e}) c^{2} and Q_{2} = (M_{x} - M_{y} + 2 m_{e}) c^{2}$

(1) Hint : The Q - value can be found using the mass - energy equivalent formula .

Step 1 : Find the Q - value for β^{+} decay .

Let the nucleus is_{z} X^{A} . β^{+} decay is represented as;

z X^{A} \to_{z - 1} Y^{A} +_{+ 1} e^{0} + ν + Q_{2}

∴ Q_{2} = [m_{n} (_{z} X^{A}) - m_{n} (_{z - 1} Y^{A}) - m_{e}] c^{2}

= [m_{n} (_{z} X^{A}) + {zm}_{e} - m_{n} (_{z - 1} Y^{A}) - (z - 1) m_{e} - 2 m_{e}] c^{2}

= [m (_{z} X^{A}) - m (_{z - 1} Y^{A}) - 2 m_{e}] c^{2}

= (M_{x} - M_{y} - 2 m_{e}) c^{2}

Step 1 : Find the Q - value for β^{-} decay .

β^{-} decay is represented as =_{z} X^{A} \to_{z + 1} A^{Y} +_{- 1} e^{0} + \bar{v} + α_{1}

α_{1} = [m_{n} (_{z} X^{A}) - m_{n} (_{z + 1} Y^{A}) - m_{e}] c^{2}

= [m_{n} (_{z} X^{A}) + {zm}_{e} - m_{n} (_{z + 1} Y^{A}) - (z + 1) m_{e}] c^{2}

= [m (_{z} X^{A}) - m (_{z - 1} Y^{A})] c^{2}

= (M_{x} - M_{y}) c^{2}

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