Calculate the binding energy per nucleon in the nuclei of `._(15)P^(31)`. Given `m(._(15)P^(31))=30.97376u, m(._(0)n^(1))=1.00865 u, m(._(1)H^(1))=1.00782u`.

Explanation

To calculate the binding energy per nucleon for the nucleus of Phosphorus-31 (symbolized as \( _{15}^{31}P \)), we will follow these steps:

### Step 1: Determine the number of protons and neutrons
The atomic number \( Z \) of phosphorus is 15, which means it has 15 protons. The mass number \( A \) is 31, which means the total number of nucleons (protons + neutrons) is 31.

To find the number of neutrons \( N \):
\[ N = A - Z = 31 - 15 = 16 \]

### Step 2: Calculate the total mass of the individual nucleons
The total mass of the nucleons can be calculated using the masses of protons and neutrons:
- Mass of one proton \( m_p = 1.00782 \, u \)
- Mass of one neutron \( m_n = 1.00865 \, u \)

The total mass of 15 protons and 16 neutrons is:
\[ \text{Total mass} = (15 \times m_p) + (16 \times m_n) \]
\[ = (15 \times 1.00782) + (16 \times 1.00865) \]
\[ = 15.1173 \, u + 16.1384 \, u = 31.2557 \, u \]

### Step 3: Calculate the mass defect \( \Delta m \)
The mass defect \( \Delta m \) is the difference between the total mass of the individual nucleons and the actual mass of the nucleus:
\[ \Delta m = \text{Total mass of nucleons} - \text{Mass of nucleus} \]
\[ = 31.2557 \, u - 30.97376 \, u = 0.28194 \, u \]

### Step 4: Convert the mass defect to binding energy
The binding energy \( E_b \) can be calculated using the formula:
\[ E_b = \Delta m \times 931 \, \text{MeV} \]
\[ = 0.28194 \, u \times 931 \, \text{MeV/u} = 262.48614 \, \text{MeV} \]

### Step 5: Calculate the binding energy per nucleon
To find the binding energy per nucleon, we divide the total binding energy by the number of nucleons:
\[ \text{Binding energy per nucleon} = \frac{E_b}{A} = \frac{262.48614 \, \text{MeV}}{31} \]
\[ = 8.47 \, \text{MeV/nucleon} \]

### Final Answer
The binding energy per nucleon in the nuclei of \( _{15}^{31}P \) is approximately \( 8.47 \, \text{MeV/nucleon} \).

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