Question

The mass of ${}_{27}^{57}Co: 56.936296u; {}_{26}^{57}Fe: 56.935399u$ are given. Two possible nuclear reactions are proposed

(i) ${}_{27}^{57}Co \rightarrow {}_{26}^{57}Fe + {}_{+1}^{0}e + v$ (positron emission) (ii) ${}_{27}^{57}Co + {}_{-1}^{0}e \rightarrow {}_{26}^{57}Fe + v$ (K-shell capture)

Of these two reactions,

• A. (i) is possible but (ii) is not possible
• B. (i) is not possible but (ii) is possible
• C. (i) is possible as well as (ii) is possible
• D. (i) is not possible as well as (ii) is not possible

Answer 1

Answer: (B)

B

Answer 2

The possibility of a nuclear reaction is determined by the conservation of energy, which translates to a comparison of masses. When atomic masses ($M$) are given, we must correctly use the formulas for positron emission and electron capture, accounting for electron masses ($m_e$) and electron binding energies ($B_e$).

1. Positron Emission: The reaction is ${}_{27}^{57}Co \rightarrow {}_{26}^{57}Fe + e^+ + v$. The condition for this reaction to be energetically possible, using atomic masses, is: $M(^{57}_{27}Co) > M(^{57}_{26}Fe) + 2m_e$ Given masses: $M(^{57}_{27}Co) = 56.936296u$ and $M(^{57}_{26}Fe) = 56.935399u$. Mass of positron $m_e = 0.000548579909u$. Checking the condition: $56.936296u > 56.935399u + 2 \times 0.000548579909u$ $56.936296u > 56.935399u + 0.001097159818u$ $56.936296u > 56.936496159818u$ This inequality is false. Therefore, positron emission is not possible.

2. K-shell Electron Capture: The reaction is ${}_{27}^{57}Co + e^- \rightarrow {}_{26}^{57}Fe + v$. The condition for electron capture using atomic masses is: $M(^{57}_{27}Co) + B_{e,K}(Co) > M(^{57}_{26}Fe) + B_{e,K}(Fe)$ Rearranging this, we get: $M(^{57}_{27}Co) - M(^{57}_{26}Fe) > B_{e,K}(Fe) - B_{e,K}(Co)$. The mass difference is: $56.936296u - 56.935399u = 0.000897u$. The K-shell electron binding energies are approximately $B_{e,K}(Co) \approx 7.709 \text{ keV}$ and $B_{e,K}(Fe) \approx 7.111 \text{ keV}$. The difference is $B_{e,K}(Fe) - B_{e,K}(Co) \approx 7.111 \text{ keV} - 7.709 \text{ keV} = -0.598 \text{ keV}$. Converting this energy difference to atomic mass units: $-0.598 \text{ keV} \times \frac{1 \text{ u}}{931494.102 \text{ keV}} \approx -0.000000642 \text{ u}$. Checking the condition: $0.000897u > -0.000000642u$ This inequality is true. Therefore, K-shell electron capture is possible.

Since reaction (i) is not possible and reaction (ii) is possible, the correct option is (B).