To solve the problem, we will follow these steps:
### Step 1: Understand the relationship between energy, work function, and kinetic energy
The maximum kinetic energy (K.E. max) of the emitted photoelectron can be calculated using the equation derived from the photoelectric effect:
\[
K.E._{\text{max}} = E - \phi
\]
where:
- \(E\) is the energy of the incoming photons,
- \(\phi\) is the work function of the metal.
### Step 2: Calculate the energy of the incoming photons
The energy \(E\) of the photons can be calculated using the formula:
\[
E = \frac{hc}{\lambda}
\]
where:
- \(h\) is Planck's constant,
- \(c\) is the speed of light,
- \(\lambda\) is the wavelength of the light.
Given:
- \(h = 6.6 \times 10^{-34} \, \text{Js}\),
- \(c = 3 \times 10^8 \, \text{m/s}\),
- \(\lambda = 1320 \, \text{Å} = 1320 \times 10^{-10} \, \text{m}\).
### Step 3: Substitute the values to find \(E\)
Substituting the values into the energy formula:
\[
E = \frac{(6.6 \times 10^{-34} \, \text{Js}) \times (3 \times 10^8 \, \text{m/s})}{1320 \times 10^{-10} \, \text{m}}
\]
Calculating \(E\):
\[
E = \frac{(6.6 \times 3) \times 10^{-34 + 8 + 10}}{1320}
\]
\[
E = \frac{19.8 \times 10^{-16}}{1320} \approx 1.5 \times 10^{-18} \, \text{J}
\]
### Step 4: Calculate the maximum kinetic energy
Now, we substitute \(E\) and \(\phi\) into the kinetic energy formula:
\[
K.E._{\text{max}} = E - \phi
\]
Given \(\phi = 2.2 \times 10^{-19} \, \text{J}\):
\[
K.E._{\text{max}} = (1.5 \times 10^{-18} \, \text{J}) - (2.2 \times 10^{-19} \, \text{J})
\]
\[
K.E._{\text{max}} = 1.5 \times 10^{-18} - 0.22 \times 10^{-18} = 1.28 \times 10^{-18} \, \text{J}
\]
### Step 5: Convert kinetic energy from Joules to electron volts
To convert Joules to electron volts, we use the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\):
\[
K.E._{\text{max}} \, (\text{in eV}) = \frac{1.28 \times 10^{-18} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \approx 8 \, \text{eV}
\]
### Final Answer
The maximum kinetic energy of the emitted photoelectron is approximately \(8 \, \text{eV}\).
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