To calculate the energy of a photon with a given wavelength, we can use the formula derived from Planck's relation:
[tex]\[ E = \frac{h \cdot c}{\lambda} \][/tex]
where:
- [tex]\(E\)[/tex] is the energy of the photon in joules (J),
- [tex]\(h\)[/tex] is Planck's constant ([tex]\(6.626 \times 10^{-34} \)[/tex] J·s),
- [tex]\(c\)[/tex] is the speed of light ([tex]\(3.00 \times 10^8\)[/tex] m/s),
- [tex]\(\lambda\)[/tex] is the wavelength of the photon in meters (m).
Given the wavelength [tex]\(\lambda\)[/tex] is 428 nanometers (nm), we first need to convert this into meters.
1 nanometer is [tex]\(1 \times 10^{-9}\)[/tex] meters. Therefore:
[tex]\[ \lambda = 428 \, \text{nm} = 428 \times 10^{-9} \, \text{m} \][/tex]
[tex]\[ \lambda = 4.28 \times 10^{-7} \, \text{m} \][/tex]
Next, we use the formula to find the energy [tex]\(E\)[/tex]:
1. Identify the constants:
- Planck's constant [tex]\( h = 6.626 \times 10^{-34} \, \text{J·s} \)[/tex]
- Speed of light [tex]\( c = 3.00 \times 10^8 \, \text{m/s} \)[/tex]
2. Substitute the values into the formula:
[tex]\[ E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{4.28 \times 10^{-7} } \][/tex]
After solving this expression, the energy [tex]\( E \)[/tex] of the photon is found to be approximately [tex]\(4.644392523364486 \times 10^{-19}\)[/tex] joules.
So, a photon with a wavelength of 428 nm has an energy of approximately [tex]\(4.644392523364486 \times 10^{-19}\)[/tex] J.
