Potassium-40 can decay in three modes: ⁻ -emission, ⁺ -emission or electron capture. The atomic masses of ⁴⁰₁₈ Ar , ⁴⁰₁₉ K and ⁴⁰₂₀ Ca are 39.9624 u , 39.9640 u and 39.9626 u respectively. Compute the Q -values for each of the three decay modes respectively.

Step-by-step solution

The Q -value for ⁻ -emission is given by the difference in atomic masses of the parent and daughter nuclei: Q_ ⁻ = [M(⁴⁰₁₉ K ) - M(⁴⁰₂₀ Ca )] c^2 Using 1 u = 931 MeV /c^2 , we get: Q_ ⁻ = (39.9640 - 39.9626) 931 = 0.0014 931 = 1.3034 MeV The Q -value for ⁺ -emission accounts for the mass of two electrons (one from the positron emitted and one from the difference in atomic electron count): Q_ ⁺ = [M(⁴⁰₁₉ K ) - M(⁴⁰₁₈ Ar ) - 2m_e] c^2 Using m_e c^2 = 0.511 MeV , we get: Q_ ⁺ = (39.9640 - 39.9624) 931 - 2(0.511) = 0.0