What is the binding energy (in J/mol or KJ/mol) of an electron in a metal. Whose threshold frequency for photon electron is `2.5xx10^(14)s^(-1)` ?

Options

A. 99.38 KJ/mol
B. `1.66 xx10^(-19)` J/ mol
C. `2.75 xx10^(-43)` J/mol
D. `7.22xx10^(17)` kJ/mol

Answer

To find the binding energy of an electron in a metal given the threshold frequency, we can follow these steps:

### Step 1: Understand the relationship between binding energy and threshold frequency
The binding energy (or work function, \( \phi \)) of an electron in a metal can be calculated using the formula:
\[ \phi = h \nu \]
where:
- \( \phi \) is the binding energy in joules,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( \nu \) is the threshold frequency in \( \text{s}^{-1} \).

### Step 2: Substitute the values into the formula
Given the threshold frequency \( \nu = 2.5 \times 10^{14} \, \text{s}^{-1} \), we can substitute this value into the equation:
\[ \phi = 6.626 \times 10^{-34} \, \text{J s} \times 2.5 \times 10^{14} \, \text{s}^{-1} \]

### Step 3: Perform the multiplication
Calculating the above:
\[ \phi = 6.626 \times 2.5 \times 10^{-34 + 14} = 6.626 \times 2.5 \times 10^{-20} \]
Calculating \( 6.626 \times 2.5 \):
\[ 6.626 \times 2.5 = 16.565 \]
Thus,
\[ \phi = 16.565 \times 10^{-20} \, \text{J} = 1.6565 \times 10^{-19} \, \text{J} \]

### Step 4: Convert the energy to per mole
To convert the energy from joules to joules per mole, we multiply by Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)):
\[ \text{Binding energy per mole} = \phi \times N_A = 1.6565 \times 10^{-19} \, \text{J} \times 6.022 \times 10^{23} \, \text{mol}^{-1} \]

### Step 5: Calculate the total binding energy
Calculating this gives:
\[ \text{Binding energy per mole} = 1.6565 \times 6.022 \times 10^{4} \, \text{J/mol} \]
Calculating \( 1.6565 \times 6.022 \):
\[ 1.6565 \times 6.022 \approx 9.95 \times 10^{4} \, \text{J/mol} = 99.5 \, \text{kJ/mol} \]

### Step 6: Round to the appropriate significant figures
Thus, the binding energy of the electron in the metal is approximately:
\[ \text{Binding energy} \approx 99.38 \, \text{kJ/mol} \]

### Final Answer
The binding energy of an electron in the metal is approximately **99.38 kJ/mol**.

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