A helium atom, a hydrogen atom and a neutron have mases of` 4.003 u, 1.008 u` and `1.009 u` (unified atomic mass units), respectively. Assuming that hydogen atoms and neutrons can be fuse to from helium, what is the binding energy of a helium nucleus?

Options

A. `2.01 u`
B. `3.031 u`
C. `1.017 u`
D. `0.031u`

Explanation

To find the binding energy of a helium nucleus formed by fusing hydrogen atoms and neutrons, we can follow these steps:

### Step 1: Write the Fusion Reaction
The fusion of hydrogen atoms and neutrons to form helium can be represented as:
\[ 2 \, ^1_1\text{H} + 2 \, ^1_0\text{n} \rightarrow \, ^4_2\text{He} \]
Where:
- \( ^1_1\text{H} \) is a hydrogen atom (mass = 1.008 u)
- \( ^1_0\text{n} \) is a neutron (mass = 1.009 u)
- \( ^4_2\text{He} \) is a helium atom (mass = 4.003 u)

### Step 2: Calculate the Total Mass of Reactants
The total mass of the reactants (2 hydrogen atoms and 2 neutrons) can be calculated as follows:
\[ \text{Total mass of reactants} = 2 \times (1.008 \, \text{u}) + 2 \times (1.009 \, \text{u}) \]
Calculating this gives:
\[ = 2.016 \, \text{u} + 2.018 \, \text{u} = 4.034 \, \text{u} \]

### Step 3: Calculate the Mass Defect
The mass defect (\( \Delta m \)) is the difference between the total mass of the reactants and the mass of the helium nucleus:
\[ \Delta m = \text{Total mass of reactants} - \text{Mass of helium nucleus} \]
Substituting the values:
\[ \Delta m = 4.034 \, \text{u} - 4.003 \, \text{u} = 0.031 \, \text{u} \]

### Step 4: Convert Mass Defect to Energy
The binding energy (\( E \)) can be calculated using Einstein's equation \( E = \Delta m c^2 \). To convert the mass defect from atomic mass units to energy, we use the conversion factor \( 1 \, \text{u} = 931.5 \, \text{MeV} \):
\[ E = \Delta m \times 931.5 \, \text{MeV/u} \]
Substituting the mass defect:
\[ E = 0.031 \, \text{u} \times 931.5 \, \text{MeV/u} \approx 28.9 \, \text{MeV} \]

### Final Answer
The binding energy of the helium nucleus is approximately **28.9 MeV**.

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