To solve the problem step by step, we will use the photoelectric effect equation and the given values.
### Step 1: Understand the given data
- Work function \( W = 1 \, \text{eV} \)
- Wavelength \( \lambda = 3000 \, \text{Å} \)
### Step 2: Convert the work function to joules
The work function in joules can be calculated using the conversion \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[
W = 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}
\]
### Step 3: Convert the wavelength from angstroms to meters
1 Ångström (Å) = \( 10^{-10} \, \text{m} \), therefore:
\[
\lambda = 3000 \, \text{Å} = 3000 \times 10^{-10} \, \text{m} = 3 \times 10^{-7} \, \text{m}
\]
### Step 4: Use the photoelectric equation
The photoelectric equation is given by:
\[
E = W + KE
\]
where \( E \) is the energy of the incident photons, \( W \) is the work function, and \( KE \) is the kinetic energy of the emitted photoelectrons.
The energy of the photons can be expressed as:
\[
E = \frac{hc}{\lambda}
\]
where \( h \) is Planck's constant (\( 6.63 \times 10^{-34} \, \text{J s} \)) and \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).
### Step 5: Calculate the energy of the photons
Substituting the values into the equation:
\[
E = \frac{(6.63 \times 10^{-34} \, \text{J s})(3 \times 10^8 \, \text{m/s})}{3 \times 10^{-7} \, \text{m}}
\]
Calculating this gives:
\[
E = \frac{(6.63 \times 3) \times 10^{-26}}{3 \times 10^{-7}} = \frac{19.89 \times 10^{-26}}{3 \times 10^{-7}} = 6.63 \times 10^{-19} \, \text{J}
\]
### Step 6: Calculate the kinetic energy
Using the photoelectric equation:
\[
KE = E - W
\]
Substituting the values:
\[
KE = (6.63 \times 10^{-19} \, \text{J}) - (1.6 \times 10^{-19} \, \text{J}) = 5.03 \times 10^{-19} \, \text{J}
\]
### Step 7: Relate kinetic energy to maximum velocity
The kinetic energy of the emitted electrons can also be expressed as:
\[
KE = \frac{1}{2} mv^2
\]
where \( m \) is the mass of the electron (\( 9.1 \times 10^{-31} \, \text{kg} \)) and \( v \) is the maximum velocity of the photoelectrons.
Rearranging gives:
\[
v = \sqrt{\frac{2 \times KE}{m}}
\]
### Step 8: Substitute the values to find \( v \)
Substituting the values:
\[
v = \sqrt{\frac{2 \times (5.03 \times 10^{-19} \, \text{J})}{9.1 \times 10^{-31} \, \text{kg}}}
\]
Calculating this gives:
\[
v = \sqrt{\frac{1.006 \times 10^{-18}}{9.1 \times 10^{-31}}} \approx \sqrt{1.105 \times 10^{12}} \approx 10^{6} \, \text{m/s}
\]
### Final Answer
The maximum velocity of the photoelectrons is approximately:
\[
v \approx 10^6 \, \text{m/s}
\]
