Calculate the energy required to separate `._50Sn^120` into its constituents if `m_p`=1.007825amu, `m_(sn)`=119.902199amu, `m_n`=1.008665amu

Options

A. 1.02 MeV
B. 1021 MeV
C. 102.1 MeV
D. 0.1021 MeV

Answer

To calculate the energy required to separate \( _{50}^{120}Sn \) into its constituents, we will follow these steps:

### Step 1: Identify the constituents
The nucleus of \( _{50}^{120}Sn \) consists of:
- Protons: 50
- Neutrons: \( A - Z = 120 - 50 = 70 \)

### Step 2: Calculate the total mass of the constituents
Using the given masses:
- Mass of a proton, \( m_p = 1.007825 \, \text{amu} \)
- Mass of a neutron, \( m_n = 1.008665 \, \text{amu} \)

The total mass of the constituents (50 protons and 70 neutrons) is calculated as follows:

\[ \text{Total mass} = (50 \times m_p) + (70 \times m_n) \]

Substituting the values:

\[ \text{Total mass} = (50 \times 1.007825) + (70 \times 1.008665) \]

Calculating this gives:

\[ \text{Total mass} = 50.39125 + 70.60655 = 120.9978 \, \text{amu} \]

### Step 3: Calculate the mass defect
The mass defect \( \Delta m \) is the difference between the total mass of the constituents and the actual mass of the nucleus.

Given the mass of the nucleus \( m_{Sn} = 119.902199 \, \text{amu} \):

\[ \Delta m = \text{Total mass} - m_{Sn} \]

Substituting the values:

\[ \Delta m = 120.9978 - 119.902199 = 1.095601 \, \text{amu} \]

### Step 4: Convert mass defect to energy
Using Einstein's equation \( E = \Delta m c^2 \), we need to convert the mass defect from amu to energy in MeV.

1 amu is equivalent to \( 931.5 \, \text{MeV/c}^2 \). Therefore, the energy equivalent of the mass defect is:

\[ E = \Delta m \times 931.5 \, \text{MeV} \]

Substituting the value of \( \Delta m \):

\[ E = 1.095601 \times 931.5 \approx 1028.4 \, \text{MeV} \]

### Step 5: Convert to Mega Electron Volts
Since \( 1 \, \text{MeV} = 10^6 \, \text{eV} \), we can express the energy in Mega Electron Volts:

\[ E \approx 1.0284 \, \text{MeV} \]

### Final Answer
The energy required to separate \( _{50}^{120}Sn \) into its constituents is approximately **1.0284 MeV**.

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