To solve the problem of finding the work function of the metal in the photoelectric effect, we will follow these steps:
### Step 1: Understand the relationship between kinetic energy, work function, and incident energy.
In the photoelectric effect, the maximum kinetic energy (K.E.) of the emitted electrons can be expressed as:
\[ K.E. = E_{incident} - \phi \]
where \( E_{incident} \) is the energy of the incident photons and \( \phi \) is the work function of the metal.
### Step 2: Calculate the energy of the incident photons.
The energy of the incident photons can be calculated using the formula:
\[ E_{incident} = \frac{12400 \, \text{eV} \cdot \text{Å}}{\lambda} \]
Given that the wavelength \( \lambda = 4000 \, \text{Å} \), we can substitute this value into the formula:
\[ E_{incident} = \frac{12400 \, \text{eV} \cdot \text{Å}}{4000 \, \text{Å}} \]
### Step 3: Perform the calculation for \( E_{incident} \).
Calculating \( E_{incident} \):
\[ E_{incident} = \frac{12400}{4000} = 3.1 \, \text{eV} \]
### Step 4: Calculate the maximum kinetic energy of the electrons.
The maximum kinetic energy can be calculated using the formula:
\[ K.E. = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is the maximum speed of the electrons. Given \( v = 6 \times 10^5 \, \text{m/s} \) and the mass of the electron \( m = 9.1 \times 10^{-31} \, \text{kg} \):
\[ K.E. = \frac{1}{2} \times 9.1 \times 10^{-31} \times (6 \times 10^5)^2 \]
### Step 5: Perform the calculation for \( K.E. \).
Calculating \( K.E. \):
1. Calculate \( (6 \times 10^5)^2 = 36 \times 10^{10} \).
2. Now, calculate \( K.E. = \frac{1}{2} \times 9.1 \times 10^{-31} \times 36 \times 10^{10} \).
3. This gives:
\[ K.E. = 0.5 \times 9.1 \times 36 \times 10^{-21} \]
\[ K.E. = 163.8 \times 10^{-21} \, \text{J} \]
### Step 6: Convert \( K.E. \) from Joules to electron volts.
To convert Joules to electron volts, use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ K.E. = \frac{163.8 \times 10^{-21}}{1.6 \times 10^{-19}} \]
\[ K.E. \approx 1.024 \, \text{eV} \]
### Step 7: Substitute values into the equation for work function.
Now, substitute the values of \( E_{incident} \) and \( K.E. \) into the equation:
\[ K.E. = E_{incident} - \phi \]
\[ 1.024 = 3.1 - \phi \]
### Step 8: Solve for the work function \( \phi \).
Rearranging the equation gives:
\[ \phi = 3.1 - 1.024 \]
\[ \phi \approx 2.076 \, \text{eV} \]
### Final Answer:
The work function of the metal is approximately \( 2.076 \, \text{eV} \).
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