Find the binding energy per nucleon of `` 79^197Au` if its atomic mass is 196.96 u.

Explanation

To find the binding energy per nucleon of \( ^{197}_{79}Au \) (Gold), we will follow these steps:

### Step 1: Identify the number of protons and neutrons
The atomic number \( Z \) of Gold (Au) is 79, which means it has 79 protons. The mass number \( A \) is 197, which is the total number of protons and neutrons.

To find the number of neutrons \( N \):
\[ N = A - Z = 197 - 79 = 118 \]

### Step 2: Calculate the mass of the nucleus
The mass of the nucleus can be calculated using the formula:
\[ m = Z \cdot m_p + N \cdot m_n \]
where:
- \( m_p \) is the mass of a proton (approximately \( 1.007276 \, u \)),
- \( m_n \) is the mass of a neutron (approximately \( 1.008665 \, u \)).

Substituting the values:
\[ m = 79 \cdot 1.007276 + 118 \cdot 1.008665 \]
Calculating each term:
\[ m = 79 \cdot 1.007276 = 79.574784 \, u \]
\[ m = 118 \cdot 1.008665 = 118.02087 \, u \]
Now adding these together:
\[ m = 79.574784 + 118.02087 = 197.595654 \, u \]

### Step 3: Calculate the binding energy
The binding energy \( BE \) can be calculated using the formula:
\[ BE = \left( Z \cdot m_p + N \cdot m_n - m \right) \cdot c^2 \]
We will convert the binding energy to MeV using the conversion factor \( 1 \, u \approx 931.5 \, MeV/c^2 \).

Substituting the values:
\[ BE = \left( 197.595654 - 196.96 \right) \cdot 931.5 \, MeV \]
Calculating the difference:
\[ BE = (197.595654 - 196.96) = 0.635654 \, u \]
Now converting to MeV:
\[ BE = 0.635654 \cdot 931.5 \approx 592.3 \, MeV \]

### Step 4: Calculate the binding energy per nucleon
The binding energy per nucleon \( BE/A \) is calculated by dividing the total binding energy by the mass number \( A \):
\[ BE/A = \frac{BE}{A} = \frac{592.3 \, MeV}{197} \]
Calculating this gives:
\[ BE/A \approx 3.0 \, MeV/nucleon \]

### Final Answer
The binding energy per nucleon of \( ^{197}_{79}Au \) is approximately \( 3.0 \, MeV/nucleon \).

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