Sure! Let's find the energy of photons given the wavelengths for red and blue light.
1. Constants and Given Values:
- Planck's constant [tex]\( h = 6.626 \times 10^{-34} \)[/tex] Joule seconds (J·s)
- Speed of light [tex]\( c = 3.0 \times 10^8 \)[/tex] meters per second (m/s)
- Conversion factor [tex]\( 1 \)[/tex] electron-volt [tex]\( (eV) = 1.60 \times 10^{-19} \)[/tex] Joules (J)
- Wavelength of red light [tex]\( \lambda_{\text{red}} = 600 \)[/tex] nanometers (nm) [tex]\( = 600 \times 10^{-9} \)[/tex] meters (m)
- Wavelength of blue light [tex]\( \lambda_{\text{blue}} = 400 \)[/tex] nanometers (nm) [tex]\( = 400 \times 10^{-9} \)[/tex] meters (m)
2. Energy of a Photon:
The energy [tex]\( E \)[/tex] of a photon can be calculated using the formula:
[tex]\[
E = \frac{h \cdot c}{\lambda}
\][/tex]
3. Calculating the Energy of a Red Photon:
Substituting the values:
[tex]\[
E_{\text{red}} = \frac{6.626 \times 10^{-34} \, \text{J·s} \times 3.0 \times 10^8 \, \text{m/s}}{600 \times 10^{-9} \, \text{m}}
\][/tex]
This calculation gives:
[tex]\[
E_{\text{red}} = 3.313 \times 10^{-19} \, \text{J}
\][/tex]
4. Converting Energy of a Red Photon to Electron-Volts:
[tex]\[
E_{\text{red}} (\text{eV}) = \frac{E_{\text{red}} (\text{J})}{1.60 \times 10^{-19} \, \text{J/eV}}
\][/tex]
So,
[tex]\[
E_{\text{red}} (\text{eV}) = \frac{3.313 \times 10^{-19} \, \text{J}}{1.60 \times 10^{-19} \, \text{J/eV}} = 2.070625 \, \text{eV}
\][/tex]
5. Calculating the Energy of a Blue Photon:
Substituting the values:
[tex]\[
E_{\text{blue}} = \frac{6.626 \times 10^{-34} \, \text{J·s} \times 3.0 \times 10^8 \, \text{m/s}}{400 \times 10^{-9} \, \text{m}}
\][/tex]
This calculation gives:
[tex]\[
E_{\text{blue}} = 4.9695 \times 10^{-19} \, \text{J}
\][/tex]
6. Converting Energy of a Blue Photon to Electron-Volts:
[tex]\[
E_{\text{blue}} (\text{eV}) = \frac{E_{\text{blue}} (\text{J})}{1.60 \times 10^{-19} \, \text{J/eV}}
\][/tex]
So,
[tex]\[
E_{\text{blue}} (\text{eV}) = \frac{4.9695 \times 10^{-19} \, \text{J}}{1.60 \times 10^{-19} \, \text{J/eV}} = 3.1059375 \, \text{eV}
\][/tex]
To summarize:
- The energy of a red photon (600 nm) is:
- [tex]\( 3.313 \times 10^{-19} \, \text{J} \)[/tex]
- [tex]\( 2.070625 \, \text{eV} \)[/tex]
- The energy of a blue photon (400 nm) is:
- [tex]\( 4.9695 \times 10^{-19} \, \text{J} \)[/tex]
- [tex]\( 3.1059375 \, \text{eV} \)[/tex]
