When light of wavelength lesser than 6000 Å is incident on a metal, electrons are emitted. The approximate work - function of the metal is :

Options

A. 1 eV
B. 2 eV
C. 4 eV
D. 6 eV

Answer

To solve the problem regarding the work function of a metal when light of wavelength less than 6000 Å is incident on it, we can follow these steps:

### Step 1: Understand the relationship between wavelength and energy
The energy 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 (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength in meters.

### Step 2: Convert the wavelength from Ångströms to meters
The given wavelength is 6000 Å. To convert this to meters:
\[ 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \]

### Step 3: Calculate the energy of the photon
Now, substitute the values of \( h \), \( c \), and \( \lambda \) into the energy formula:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^8 \, \text{m/s})}{6 \times 10^{-7} \, \text{m}} \]

Calculating this gives:
\[ E = \frac{1.9878 \times 10^{-25}}{6 \times 10^{-7}} \]
\[ E \approx 3.31 \times 10^{-19} \, \text{J} \]

### Step 4: Convert energy from Joules to electron volts
To convert Joules to electron volts, use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E \approx \frac{3.31 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \]
\[ E \approx 2.07 \, \text{eV} \]

### Step 5: Determine the work function
The work function \( \phi \) of the metal is the minimum energy required to eject an electron. Since electrons are emitted when the incident light has a wavelength less than 6000 Å, this implies that the energy of the photons is greater than the work function of the metal:
\[ \phi < E \]
Thus, the work function must be less than approximately 2.07 eV.

### Step 6: Approximate the work function
Given that the question asks for the approximate work function, we can conclude that the work function is likely around 2 eV, as this is the closest reasonable value below 2.07 eV.

### Final Answer
The approximate work function of the metal is **2 eV**.

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