To find the work function of the metal, we can use the formula that relates the total energy of the incident photons to the work function and the kinetic energy of the emitted photoelectrons. The equation is:
\[ \text{Total Energy} = \text{Work Function} + \text{Kinetic Energy} \]
From this, we can rearrange it to find the work function (\( \phi \)):
\[ \phi = \text{Total Energy} - \text{Kinetic Energy} \]
### Step 1: Calculate the Total Energy of the Photons
The total energy of the photons can be calculated using the equation:
\[ \text{Total Energy} = \frac{hc}{\lambda} \]
Where:
- \( h = 6.626 \times 10^{-34} \, \text{Js} \) (Planck's constant)
- \( c = 3 \times 10^{8} \, \text{ms}^{-1} \) (speed of light)
- \( \lambda = 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} \)
Substituting the values:
\[
\text{Total Energy} = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^{8} \, \text{ms}^{-1})}{4000 \times 10^{-10} \, \text{m}}
\]
Calculating this gives:
\[
\text{Total Energy} = \frac{(6.626 \times 3) \times 10^{-26}}{4000 \times 10^{-10}} = \frac{19.878 \times 10^{-26}}{4 \times 10^{-7}} = 4.9695 \times 10^{-19} \, \text{J}
\]
### Step 2: Calculate the Kinetic Energy of the Photoelectrons
The kinetic energy (\( KE \)) of the emitted photoelectrons can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
Where:
- \( m = 9 \times 10^{-31} \, \text{kg} \) (mass of the electron)
- \( v = 6 \times 10^{5} \, \text{ms}^{-1} \) (velocity of the photoelectrons)
Substituting the values:
\[
KE = \frac{1}{2} (9 \times 10^{-31} \, \text{kg}) (6 \times 10^{5} \, \text{ms}^{-1})^2
\]
Calculating this gives:
\[
KE = \frac{1}{2} (9 \times 10^{-31}) (36 \times 10^{10}) = 162 \times 10^{-21} \, \text{J} = 1.62 \times 10^{-19} \, \text{J}
\]
### Step 3: Calculate the Work Function
Now we can substitute the total energy and kinetic energy into the equation for work function:
\[
\phi = \text{Total Energy} - KE
\]
Substituting the values we calculated:
\[
\phi = (4.9695 \times 10^{-19} \, \text{J}) - (1.62 \times 10^{-19} \, \text{J})
\]
Calculating this gives:
\[
\phi = 3.3495 \times 10^{-19} \, \text{J}
\]
### Step 4: Convert Work Function to Electron Volts
To convert the work function from joules to electron volts, we use the conversion factor:
\[
1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}
\]
So,
\[
\phi \, (\text{in eV}) = \frac{3.3495 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \approx 2.093 \, \text{eV}
\]
### Final Answer
The work function of the metal is approximately \( \phi \approx 2.1 \, \text{eV} \).
