Consider the following nuclear reaction, `X^200rarrA^110+B^90+En ergy`
If the binding energy per nucleon for X, A and B are `7.4 MeV`, `8.2 MeV` and `8.2 MeV` respectively, the energy released will be

Options

A. 200 MeV
B. 160 MeV
C. 110 MeV
D. 90 MeV

Answer

To find the energy released in the nuclear reaction given, we will follow these steps:

### Step 1: Identify the binding energy per nucleon for each nucleus
- For nucleus X: \( BE_X = 7.4 \, \text{MeV} \)
- For nucleus A: \( BE_A = 8.2 \, \text{MeV} \)
- For nucleus B: \( BE_B = 8.2 \, \text{MeV} \)

### Step 2: Calculate the total binding energy for the initial nucleus (X)
Nucleus X has 200 nucleons, so the total binding energy \( BE_{initial} \) for X is calculated as:
\[ BE_{initial} = BE_X \times \text{Number of nucleons in X} = 7.4 \, \text{MeV} \times 200 \]
\[ BE_{initial} = 1480 \, \text{MeV} \]

### Step 3: Calculate the total binding energy for the final nuclei (A and B)
Nucleus A has 110 nucleons, and nucleus B has 90 nucleons. Therefore, the total binding energy \( BE_{final} \) for A and B is:
\[ BE_{final} = (BE_A \times \text{Number of nucleons in A}) + (BE_B \times \text{Number of nucleons in B}) \]
\[ BE_{final} = (8.2 \, \text{MeV} \times 110) + (8.2 \, \text{MeV} \times 90) \]
Calculating each term:
\[ BE_{final} = 902 \, \text{MeV} + 738 \, \text{MeV} = 1640 \, \text{MeV} \]

### Step 4: Calculate the energy released
The energy released \( E_{released} \) in the reaction can be calculated by subtracting the initial binding energy from the final binding energy:
\[ E_{released} = BE_{final} - BE_{initial} \]
\[ E_{released} = 1640 \, \text{MeV} - 1480 \, \text{MeV} = 160 \, \text{MeV} \]

### Final Answer
The energy released in the nuclear reaction is \( 160 \, \text{MeV} \).

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