To determine the wavelength of a photon given its energy, we can use the fundamental relationship between the energy of a photon [tex]\( E \)[/tex], Planck's constant [tex]\( h \)[/tex], and the speed of light [tex]\( c \)[/tex]. The formula that relates these quantities is:
[tex]\[ \lambda = \frac{h \cdot c}{E} \][/tex]
Where:
- [tex]\( \lambda \)[/tex] is the wavelength of the photon
- [tex]\( h \)[/tex] is Planck's constant ([tex]\( 6.626 \times 10^{-34} \)[/tex] Joules [tex]\(\cdot\)[/tex] seconds)
- [tex]\( c \)[/tex] is the speed of light ([tex]\( 3.0 \times 10^8 \)[/tex] meters per second)
- [tex]\( E \)[/tex] is the energy of the photon ([tex]\( 4.56 \times 10^{-19} \)[/tex] Joules)
Step-by-step solution:
1. Write down the given values:
- Energy [tex]\( E = 4.56 \times 10^{-19} \)[/tex] Joules
- Planck's constant [tex]\( h = 6.626 \times 10^{-34} \)[/tex] Joules [tex]\(\cdot\)[/tex] seconds
- Speed of light [tex]\( c = 3.0 \times 10^8 \)[/tex] meters per second
2. Substitute these values into the formula:
[tex]\[
\lambda = \frac{6.626 \times 10^{-34} \text{ J} \cdot \text{s} \times 3.0 \times 10^8 \text{ m/s}}{4.56 \times 10^{-19} \text{ J}}
\][/tex]
3. Calculate the wavelength in meters:
[tex]\[
\lambda \approx 4.3592105263157895 \times 10^{-7} \text{ meters}
\][/tex]
4. Convert the wavelength from meters to nanometers:
Since [tex]\( 1 \text{ meter} = 10^9 \text{ nanometers} \)[/tex]:
[tex]\[
\lambda \approx 4.3592105263157895 \times 10^{-7} \text{ meters} \times 10^9 \text{ nanometers / meter} = 435.92105263157896 \text{ nm}
\][/tex]
Therefore, the wavelength of the photon is approximately 436 nm.
The correct answer is:
D. 436 nm
