Consider two arbitaray decay equation and mark the correct alternative` s` given below. (i) `._(92)^(230)U rarr n+._(92)^(229)U` (ii)`._(92_^(230)U rarr P +._(91)^(229)Pa`
Given: `M(._(92)^(230)U) =230.033927 u,M(._(92)^(229)U)=229.03349 u, m_(n)=1.008665u`, `M(._(91)^(229)Pa) =229.032089, m_p`=1.007825, 1 am u =931.5 MeV`.

Options

A. Only decay (i) is possible.
B. Only decay (ii) is possible.
C. Both decay are possible.
D. Neither of the two decay is possible.

Explanation

To solve the problem, we need to calculate the Q value for both decay reactions and determine whether they are possible based on the Q values obtained. The Q value is calculated using the formula:

\[ Q = (M_{\text{reactants}} - M_{\text{products}}) \times c^2 \]

where \( c^2 \) is given as \( 931.5 \, \text{MeV/u} \).

### Step 1: Calculate Q value for the first reaction

**Reaction (i):**
\[ \,_{92}^{230}\text{U} \rightarrow n + \,_{92}^{229}\text{U} \]

**Mass of reactants:**
- Mass of \( \,_{92}^{230}\text{U} = 230.033927 \, u \)

**Mass of products:**
- Mass of neutron \( m_n = 1.008665 \, u \)
- Mass of \( \,_{92}^{229}\text{U} = 229.03349 \, u \)

**Total mass of products:**
\[ M_{\text{products}} = m_n + M( \,_{92}^{229}\text{U}) = 1.008665 \, u + 229.03349 \, u = 230.042155 \, u \]

**Calculate Q value:**
\[ Q_1 = M_{\text{reactants}} - M_{\text{products}} = 230.033927 \, u - 230.042155 \, u \]
\[ Q_1 = -0.008228 \, u \]
\[ Q_1 \text{ (in MeV)} = -0.008228 \, u \times 931.5 \, \text{MeV/u} \approx -7.67 \, \text{MeV} \]

### Step 2: Calculate Q value for the second reaction

**Reaction (ii):**
\[ \,_{92}^{230}\text{U} \rightarrow p + \,_{91}^{229}\text{Pa} \]

**Mass of reactants:**
- Mass of \( \,_{92}^{230}\text{U} = 230.033927 \, u \)

**Mass of products:**
- Mass of proton \( m_p = 1.007825 \, u \)
- Mass of \( \,_{91}^{229}\text{Pa} = 229.032089 \, u \)

**Total mass of products:**
\[ M_{\text{products}} = m_p + M( \,_{91}^{229}\text{Pa}) = 1.007825 \, u + 229.032089 \, u = 230.039914 \, u \]

**Calculate Q value:**
\[ Q_2 = M_{\text{reactants}} - M_{\text{products}} = 230.033927 \, u - 230.039914 \, u \]
\[ Q_2 = -0.005987 \, u \]
\[ Q_2 \text{ (in MeV)} = -0.005987 \, u \times 931.5 \, \text{MeV/u} \approx -5.57 \, \text{MeV} \]

### Step 3: Conclusion

Both Q values for the reactions are negative:
- \( Q_1 \approx -7.67 \, \text{MeV} \)
- \( Q_2 \approx -5.57 \, \text{MeV} \)

Since both reactions have negative Q values, they are not possible.

### Final Answer:
Neither of the two decays is possible.

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