The binding energy per nucleon of `._(3)^(7) Li` and `._(2)^(4)He` nuclei are `5.60` MeV and `7.06` MeV, respectively. In the nuclear reaction `._(3)^(7)Li+._(1)^(1)H rarr ._(2)^(4)He+._(2)^(4)He+Q`, the value of energy `Q` released is

Options

A. `19.6` MeV
B. `-2.4` MeV
C. `8.4` MeV
D. `17.3` MeV

Answer

To find the energy \( Q \) released in the nuclear reaction \( \text{ }_3^7\text{Li} + \text{ }_1^1\text{H} \rightarrow \text{ }_2^4\text{He} + \text{ }_2^4\text{He} + Q \), we can follow these steps:

### Step 1: Identify the binding energies and nucleons
- The binding energy per nucleon of \( \text{ }_3^7\text{Li} \) is \( 5.60 \) MeV.
- The binding energy per nucleon of \( \text{ }_2^4\text{He} \) is \( 7.06 \) MeV.
- The number of nucleons in \( \text{ }_3^7\text{Li} \) is \( 7 \).
- The number of nucleons in \( \text{ }_2^4\text{He} \) is \( 4 \).

### Step 2: Calculate the total binding energy of the reactants
The total binding energy of the reactants (lithium and hydrogen) is calculated as follows:
- For \( \text{ }_3^7\text{Li} \):
\[ \text{Total Binding Energy of Li} = \text{Binding Energy per Nucleon} \times \text{Number of Nucleons} = 5.60 \, \text{MeV} \times 7 = 39.2 \, \text{MeV} \]
- For \( \text{ }_1^1\text{H} \):
\[ \text{Total Binding Energy of H} = 0 \, \text{MeV} \quad (\text{since } \text{H} \text{ has no binding energy}) \]

Thus, the total binding energy of the reactants is:
\[ \text{Total Binding Energy of Reactants} = 39.2 \, \text{MeV} + 0 \, \text{MeV} = 39.2 \, \text{MeV} \]

### Step 3: Calculate the total binding energy of the products
The total binding energy of the products (two \( \text{ }_2^4\text{He} \) nuclei) is calculated as follows:
\[ \text{Total Binding Energy of Products} = 2 \times \text{Binding Energy per Nucleon of He} \times \text{Number of Nucleons in He} = 2 \times 7.06 \, \text{MeV} \times 4 = 56.48 \, \text{MeV} \]

### Step 4: Calculate the energy \( Q \) released
The energy \( Q \) released in the reaction can be calculated using the difference in binding energies:
\[ Q = \text{Total Binding Energy of Products} - \text{Total Binding Energy of Reactants} \]
Substituting the values:
\[ Q = 56.48 \, \text{MeV} - 39.2 \, \text{MeV} = 17.28 \, \text{MeV} \]

### Final Answer
Thus, the value of energy \( Q \) released is approximately \( 17.3 \, \text{MeV} \).

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