Calculate the mass of an `alpha-particle.Its binding energy is 28.2 meV.

Explanation

To calculate the mass of an alpha particle given its binding energy, we can follow these steps:

### Step 1: Understand the Composition of an Alpha Particle
An alpha particle consists of 2 protons and 2 neutrons. Therefore, we can express the mass of the alpha particle (m) in terms of the masses of protons (mp) and neutrons (mn).

### Step 2: Write the Binding Energy Equation
The binding energy (BE) of the alpha particle is given by the equation:
\[ BE = \Delta m \cdot c^2 \]
where \(\Delta m\) is the mass defect, and \(c\) is the speed of light.

### Step 3: Define the Mass Defect
The mass defect \(\Delta m\) can be calculated as:
\[ \Delta m = (2 \cdot m_p + 2 \cdot m_n) - m \]
where:
- \(m_p\) is the mass of a proton (approximately 1.007276 u),
- \(m_n\) is the mass of a neutron (approximately 1.008665 u),
- \(m\) is the mass of the alpha particle.

### Step 4: Substitute the Values
We know the binding energy is given as 28.2 MeV. We also know that 1 atomic mass unit (u) corresponds to 931 MeV/c². Therefore, we can convert the binding energy into mass units:
\[ BE = 28.2 \text{ MeV} = \frac{28.2}{931} \text{ u} \cdot c^2 \]
Calculating this gives:
\[ BE \approx 0.0303 \text{ u} \cdot c^2 \]

### Step 5: Set Up the Equation
Now we can set up the equation:
\[ 0.0303 = (2 \cdot 1.007276 + 2 \cdot 1.008665) - m \]

### Step 6: Calculate the Total Mass of Protons and Neutrons
Calculating the total mass of the protons and neutrons:
\[ 2 \cdot 1.007276 + 2 \cdot 1.008665 = 2.014552 + 2.01733 = 4.031882 \text{ u} \]

### Step 7: Solve for the Mass of the Alpha Particle
Now substituting this back into our equation:
\[ 0.0303 = 4.031882 - m \]
Rearranging gives:
\[ m = 4.031882 - 0.0303 \]
Calculating this gives:
\[ m \approx 4.001582 \text{ u} \]

### Step 8: Round the Result
The mass of the alpha particle can be rounded to:
\[ m \approx 4.00 \text{ u} \]

### Final Result
Thus, the mass of the alpha particle is approximately **4.00 u**.

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