To determine the work function of the metal given the wavelength of light that just causes the emission of photoelectrons, we need to calculate the energy of the photons corresponding to this wavelength. The work function is the minimum energy needed to emit an electron from the metal surface.
### Step-by-Step Solution:
1. Given Data:
- Wavelength of light, [tex]\( \lambda = 6.5 \times 10^{-7} \)[/tex] meters.
- Planck's constant, [tex]\( h = 6.626 \times 10^{-34} \)[/tex] joule-seconds (J·s).
- Speed of light, [tex]\( c = 3 \times 10^{8} \)[/tex] meters per second (m/s).
- Elementary charge, [tex]\( e = 1.602 \times 10^{-19} \)[/tex] coulombs (C), or joules per electron volt (J/eV).
2. Calculate the Energy of a Photon:
The energy [tex]\( E \)[/tex] of a photon is given by the equation:
[tex]\[
E = \frac{h \cdot c}{\lambda}
\][/tex]
Plugging in the given values, we have:
[tex]\[
E = \frac{6.626 \times 10^{-34} \, \text{J·s} \times 3 \times 10^8 \, \text{m/s}}{6.5 \times 10^{-7} \, \text{m}}
\][/tex]
3. Energy in Joules:
Solving the equation above, we get the energy of the photon in joules:
[tex]\[
E \approx 3.058 \times 10^{-19} \, \text{J}
\][/tex]
4. Convert Energy to Electron Volts:
To convert the energy from joules to electron volts (eV), use the relationship:
[tex]\[
1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}
\][/tex]
Thus, the energy in electron volts is:
[tex]\[
E \approx \frac{3.058 \times 10^{-19} \, \text{J}}{1.602 \times 10^{-19} \, \text{J/eV}}
\][/tex]
5. Energy in Electron Volts:
Solving the equation above, we get the energy of the photon in electron volts:
[tex]\[
E \approx 1.909 \, \text{eV}
\][/tex]
### Summary:
- The work function of the metal in joules is [tex]\( 3.058 \times 10^{-19} \, \text{J} \)[/tex].
- The work function of the metal in electron volts is [tex]\( 1.909 \, \text{eV} \)[/tex].
This detailed solution outlines how the energy of a photon (and hence the work function) can be calculated using the given wavelength and fundamental constants.
