The work function of a certain metal is 2.3 eV. If light of wave number `2 xx 10^(6)m^(-1)` falls on it, the kinetic energies of fastest and slowest ejected electron will be respectively.

Options

A. 2.48 eV , 0.18 eV
B. 0.18 eV, zero
C. 2.30 eV, 0.18 eV
D. 0.18 eV , 0.18 eV

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Data
- Work function (φ₀) of the metal = 2.3 eV
- Wave number (ν) = 2 x 10^6 m⁻¹

### Step 2: Calculate the Wavelength (λ)
The wave number (ν) is related to the wavelength (λ) by the formula:
\[ ν = \frac{1}{λ} \]
Thus, we can find the wavelength:
\[ λ = \frac{1}{ν} = \frac{1}{2 \times 10^6} = 5 \times 10^{-7} \text{ m} \]

### Step 3: Calculate the Energy of One Photon (E)
The energy of a photon can be calculated using the formula:
\[ E = \frac{hc}{λ} \]
Where:
- \(h\) (Planck's constant) = \(6.63 \times 10^{-34} \text{ J s}\)
- \(c\) (speed of light) = \(3 \times 10^8 \text{ m/s}\)

Substituting the values:
\[ E = \frac{(6.63 \times 10^{-34}) \times (3 \times 10^8)}{5 \times 10^{-7}} \]
Calculating this gives:
\[ E = \frac{1.989 \times 10^{-25}}{5 \times 10^{-7}} = 3.978 \times 10^{-19} \text{ J} \]

### Step 4: Convert Energy from Joules to Electron Volts
To convert the energy from joules to electron volts, we use the conversion factor:
\[ 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \]
Thus:
\[ E = \frac{3.978 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 2.48625 \text{ eV} \]

### Step 5: Calculate the Kinetic Energy of the Fastest Ejected Electron
The kinetic energy (KE) of the fastest ejected electron can be calculated using the formula:
\[ KE = E - φ₀ \]
Substituting the values:
\[ KE = 2.48625 \text{ eV} - 2.3 \text{ eV} = 0.18625 \text{ eV} \]

### Step 6: Determine the Kinetic Energy of the Slowest Ejected Electron
The slowest ejected electron has a minimum kinetic energy of:
\[ KE_{\text{min}} = 0 \text{ eV} \]

### Final Answer
- Kinetic energy of the fastest ejected electron = 0.18625 eV (approximately 0.18 eV)
- Kinetic energy of the slowest ejected electron = 0 eV