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Step-by-Step Solution
Step 1: Identify the mass defect
The problem states that the mass of the $_3^7Li$ nucleus is 0.042 u less than the sum of the masses of its constituent nucleons. Hence, the mass defect $\Delta M$ is:
$$\Delta M = 0.042 \, u.$$
Step 2: Convert the mass defect from atomic mass units to energy (MeV)
We use the conversion factor:
$$ 1 \, u = 931.5 \, \text{MeV}/c^2. $$
Thus,
$$ \Delta M \times 931.5 \, \text{MeV}/c^2 = 0.042 \times 931.5 \, \text{MeV}/c^2 = 39.1 \, \text{MeV}/c^2. $$
Since we are interested in binding energy, the factor of $c^2$ is typically implicit, so we can write this directly as 39.1 MeV.
Step 3: Determine the number of nucleons
The $_3^7Li$ nucleus has 3 protons and 4 neutrons, giving a total of 7 nucleons.
Step 4: Calculate the binding energy per nucleon
To find the binding energy per nucleon, we divide the total binding energy by the number of nucleons:
$$ E_{\text{bn}} = \frac{39.1 \,\text{MeV}}{7} \approx 5.6 \,\text{MeV}. $$
