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Step-by-Step Solution
Step 1: Identify the masses involved
• Mass of proton, $M_p = 1.0073 \,\text{u}$
• Mass of neutron, $M_n = 1.0087 \,\text{u}$
• Mass of helium nucleus, $M_{\text{He}} = 4.0015 \,\text{u}$
Step 2: Calculate the total mass of constituent nucleons
The helium nucleus contains 2 protons and 2 neutrons, so the total mass of its nucleons would be
$ \text{Mass of constituent nucleons} = 2 \times M_p + 2 \times M_n.$
Substitute the values:
$ = 2 \times 1.0073 + 2 \times 1.0087$
Step 3: Calculate the mass defect
The mass defect $\Delta m$ is given by
$ \Delta m = \bigl(2 M_p + 2 M_n \bigr) - M_{\text{He}}.$
Substituting the numbers:
$\Delta m = \bigl(2 \times 1.0073 + 2 \times 1.0087 \bigr) - 4.0015.$
Compute carefully:
$= (2.0146 + 2.0174) - 4.0015 = 4.0320 - 4.0015 = 0.0305 \,\text{u (approximately)}. $
(Note: Slight variations in rounding may occur; the final answer remains consistent around 0.0307 u as in the provided calculation.)
Step 4: Convert the mass defect into energy
To convert mass (in atomic mass units, u) into energy (in MeV), we use the relation:
$ E = \Delta m \times 931 \,\text{MeV/u}. $
Substituting $\Delta m = 0.0305 \,\text{u}$:
$ E \approx 0.0305 \times 931 \,\text{MeV} = 28.4 \,\text{MeV (approximately)}. $
Final Answer
Thus, the binding energy of the $\,{}_2^4{\text{He}}$ nucleus is about 28.4 MeV.
