Question

M_x and M_y denote the atomic masses of the parent and the daughter nuclei respectively in a 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?

Options

• Q₁ = (M_x – M_y) c² and Q₂ = [M_x – M_y – 2m_e]c²

• Q₁ = (M_x – M_y) c² and Q₂ = (M_x – M_y)c²

• Q₁ = (M_x – M_y – 2m_e) c² and Q₂ = (M_x – M_y + 2 m_e)c²

• Q₁ = (M_x – M_y + 2m_e) c² and Q₂ = (M_x – M_y + 2 m_e)c²

Answer 1

Q₁ = (M_x – M_y) c² and Q₂ = [M_x – M_y – 2m_e]c² 

Explanation:

Q value or energy of nuclear reaction: The energy absorbed or released during a nuclear reaction is known as the Q-value of nuclear reaction.

Q-value = (Mass of reactants – mass of products)c² Joules

= (Mass of reactants – mass of products) amu

If Q < 0, the nuclear reaction is known as endothermic. (The energy is absorbed in the reaction)

If Q > 0, the nuclear reaction is known as exothermic. (The energy is released in the reaction)

Let the nucleus be _ZX^A

β^– decay is represented as: _zX^A → _z+1A^Y + _–1e⁰ + `barv` + Q₁

Q₁ = [m_n(_zX^A) – m_n(_z+1Y^A) – m_e]c²

= [m_n(_zX^A) + Zm_e – m_n(_z+1Y^A) – (Z + 1)m_e]c²

= [m(_zX^A) – m(_z–1Y^A)]c²

⇒ Q₁ = (M_x – M_y)c²

β⁺ decay is represented as: _zX^A → _z–1Y^A + _–1e⁰ + v + Q₂

Q₂ = [m_n(_zX^A) – m_n(_z–1Y^A) – m_e]c²

= [m_n(_zX^A) + Zm_e – m_n(_z–1Y^A) – (Z – 1)m_e – m_e]c²

= [m(_zX^A) – m(_z–1Y^A) – 2m_e]c²

⇒ Q₂ = (M_x – M_y – 2m_e)c²