Calculate the threshold frequency of metal if the binding energy is `180.69 kJ mol^(-1)` of electron.

Explanation

To calculate the threshold frequency of a metal when given the binding energy, we can follow these steps:

### Step 1: Convert Binding Energy from kJ/mol to J/atom
The binding energy is given as \( BE = 180.69 \, \text{kJ/mol} \). We need to convert this to joules per atom.

1. **Convert kJ to J**:
\[ 180.69 \, \text{kJ/mol} = 180.69 \times 10^3 \, \text{J/mol} \]

2. **Use Avogadro's number** to find the energy per atom:
\[ \text{Energy per atom} = \frac{180.69 \times 10^3 \, \text{J/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} \]
\[ \text{Energy per atom} \approx 3.00 \times 10^{-19} \, \text{J} \]

### Step 2: Use the Energy-Frequency Relationship
The relationship between energy and frequency is given by the equation:
\[ E = h \nu \]
where:
- \( E \) is the energy,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( \nu \) is the frequency.

We can rearrange this equation to solve for frequency:
\[ \nu = \frac{E}{h} \]

### Step 3: Calculate the Threshold Frequency
Substituting the values we have:
\[ \nu = \frac{3.00 \times 10^{-19} \, \text{J}}{6.626 \times 10^{-34} \, \text{J s}} \]
\[ \nu \approx 4.52 \times 10^{14} \, \text{Hz} \]

### Final Answer
The threshold frequency of the metal is approximately \( 4.52 \times 10^{14} \, \text{Hz} \).

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