The photo electric threshold wavelength for a metal is `2750A^(@)`, The minimum energy of a photon that can produce photoelectric effect in the metal is

Options

A. 0.045eV
B. 0.45 eV
C. 4.5 eV
D. 0.0045 eV

Answer

To find the minimum energy of a photon that can produce the photoelectric effect in a metal with a threshold wavelength of 2750 Å, we can use the relationship between energy, wavelength, and Planck's constant. Here’s a step-by-step solution:

### Step 1: Understand the relationship between energy and wavelength
The energy \( E \) of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength of the photon.

### Step 2: Convert the wavelength from Angstroms to meters
The given threshold wavelength is \( 2750 \, \text{Å} \). We need to convert this to meters:
\[ \lambda = 2750 \, \text{Å} = 2750 \times 10^{-10} \, \text{m} = 2.75 \times 10^{-7} \, \text{m} \]

### Step 3: Substitute the values into the energy formula
Now we can substitute the values of \( h \), \( c \), and \( \lambda \) into the energy formula:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{Js})(3.00 \times 10^8 \, \text{m/s})}{2.75 \times 10^{-7} \, \text{m}} \]

### Step 4: Calculate the energy
Calculating the numerator:
\[ hc = 6.626 \times 10^{-34} \times 3.00 \times 10^8 = 1.9878 \times 10^{-25} \, \text{Jm} \]
Now, divide by the wavelength:
\[ E = \frac{1.9878 \times 10^{-25}}{2.75 \times 10^{-7}} \approx 7.23 \times 10^{-19} \, \text{J} \]

### Step 5: Convert energy from Joules to electron volts
To convert Joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \):
\[ E \approx \frac{7.23 \times 10^{-19}}{1.602 \times 10^{-19}} \approx 4.51 \, \text{eV} \]

### Final Answer
Thus, the minimum energy of a photon that can produce the photoelectric effect in the metal is approximately:
\[ E \approx 4.5 \, \text{eV} \]