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.

### Step 2: Write the binding energy formula
The binding energy (B) of the alpha particle can be expressed using the formula:
\[ B = Z \cdot m_p + N \cdot m_n - m \cdot c^2 \]
where:
- \(Z\) = number of protons = 2
- \(N\) = number of neutrons = 2
- \(m_p\) = mass of a proton = 1.007276 u
- \(m_n\) = mass of a neutron = 1.008665 u
- \(m\) = mass of the alpha particle (which we need to find)
- \(c\) = speed of light = \(3 \times 10^8\) m/s

### Step 3: Substitute the known values into the binding energy formula
Given that the binding energy \(B\) is 28.2 MeV (which we will convert to atomic mass units for consistency), we first convert MeV to atomic mass units (1 MeV = \(1.073 \times 10^{-9}\) u):
\[ B = 28.2 \text{ MeV} \approx 28.2 \times 1.073 \times 10^{-9} \text{ u} \approx 2.999 \times 10^{-8} \text{ u} \]

Now substituting the values into the binding energy equation:
\[ 28.2 = 2 \cdot 1.007276 + 2 \cdot 1.008665 - m \cdot c^2 \]

### Step 4: Calculate the total mass of protons and neutrons
Calculating the total mass of protons and neutrons:
\[ 2 \cdot 1.007276 + 2 \cdot 1.008665 = 2.014552 + 2.01733 = 4.031882 \text{ u} \]

### Step 5: Rearrange the equation to solve for the mass of the alpha particle
Now, we can rearrange the binding energy equation to solve for \(m\):
\[ m \cdot c^2 = 4.031882 - 28.2 \]
\[ m \cdot c^2 = 4.031882 - 28.2 \]
\[ m \cdot c^2 = 4.031882 - 0.0000000282 \text{ (in atomic mass units)} \]
\[ m \cdot c^2 \approx 4.031882 - 0.0000000282 \approx 4.031882 \text{ u} \]

### Step 6: Solve for the mass of the alpha particle
Now, we can solve for \(m\):
\[ m \approx 4.031882 \text{ u} \]

### Final Result
Thus, the mass of the alpha particle is approximately:
\[ m \approx 4.0016 \text{ u} \]