To find the work function of the metal, we can follow these steps:
### Step 1: Understand the relationship between energy, work function, and kinetic energy
The energy of the incident photon (electromagnetic radiation) can be expressed as:
\[ E = h \nu = \frac{hc}{\lambda} \]
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 (\( 3.0 \times 10^{8} \, \text{m/s} \)),
- \( \lambda \) is the wavelength of the radiation (in meters).
The kinetic energy (KE) of the emitted electrons is given as:
\[ KE = E - \Phi \]
Where:
- \( \Phi \) is the work function of the metal.
### Step 2: Convert the wavelength from nanometers to meters
Given the wavelength \( \lambda = 330 \, \text{nm} \):
\[ \lambda = 330 \times 10^{-9} \, \text{m} \]
### Step 3: Calculate the energy of the photon
Using the formula:
\[ E = \frac{hc}{\lambda} \]
Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{Js})(3.0 \times 10^{8} \, \text{m/s})}{330 \times 10^{-9} \, \text{m}} \]
Calculating this gives:
\[ E = 6.023 \times 10^{-19} \, \text{J} \]
### Step 4: Convert the kinetic energy from eV to joules
The kinetic energy is given as \( 0.2 \, \text{eV} \). To convert this to joules:
\[ KE = 0.2 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 3.2 \times 10^{-20} \, \text{J} \]
### Step 5: Rearrange the equation to find the work function
Using the equation:
\[ \Phi = E - KE \]
Substituting the values we calculated:
\[ \Phi = 6.023 \times 10^{-19} \, \text{J} - 3.2 \times 10^{-20} \, \text{J} \]
Calculating this gives:
\[ \Phi = 5.703 \times 10^{-19} \, \text{J} \]
### Step 6: Convert the work function from joules to eV
To convert the work function back to eV:
\[ \Phi = \frac{5.703 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \approx 3.56 \, \text{eV} \]
### Final Answer
The work function of the metal is approximately \( 3.56 \, \text{eV} \).
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