Answer :
To determine the wavelength of a photon with an energy of [tex]\( 3.38 \times 10^{-19} \)[/tex] joules, we can use the relationship between energy [tex]\(E\)[/tex], Planck's constant [tex]\(h\)[/tex], and the speed of light [tex]\(c\)[/tex]. The formula to find the wavelength [tex]\(\lambda\)[/tex] of a photon is given by:
[tex]\[ \lambda = \frac{h \cdot c}{E} \][/tex]
where:
- [tex]\(E\)[/tex] is the energy of the photon,
- [tex]\(h\)[/tex] is Planck's constant ([tex]\(6.626 \times 10^{-34}\)[/tex] joule seconds),
- [tex]\(c\)[/tex] is the speed of light ([tex]\(3.00 \times 10^8\)[/tex] meters per second).
Step-by-Step Solution:
1. Identify the given values:
- Energy [tex]\(E = 3.38 \times 10^{-19}\)[/tex] joules
- Planck's constant [tex]\(h = 6.626 \times 10^{-34}\)[/tex] joule seconds
- Speed of light [tex]\(c = 3.00 \times 10^8\)[/tex] meters per second
2. Plug the values into the wavelength formula:
[tex]\[ \lambda = \frac{6.626 \times 10^{-34} \, \text{Js} \times 3.00 \times 10^8 \, \text{m/s}}{3.38 \times 10^{-19} \, \text{J}} \][/tex]
3. Calculate the wavelength [tex]\(\lambda\)[/tex]:
[tex]\[ \lambda = \frac{1.9878 \times 10^{-25} \, \text{Js} \cdot \text{m/s}}{3.38 \times 10^{-19} \, \text{J}} \][/tex]
[tex]\[ \lambda = 5.881 \times 10^{-7} \, \text{m} \][/tex]
4. Convert the wavelength from meters to nanometers:
[tex]\[ 1 \, \text{m} = 10^9 \, \text{nm} \][/tex]
[tex]\[ \lambda = 5.881 \times 10^{-7} \, \text{m} \times 10^9 \, \text{nm/m} \][/tex]
[tex]\[ \lambda = 588.1 \, \text{nm} \][/tex]
5. Compare the calculated wavelength to the given choices:
- A. [tex]\(510 \, \text{nm}\)[/tex]
- B. [tex]\(460 \, \text{nm}\)[/tex]
- C. [tex]\(588 \, \text{nm}\)[/tex]
- D. [tex]\(416 \, \text{nm}\)[/tex]
The wavelength most closely matches option C: [tex]\(588 \, \text{nm}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(588 \, \text{nm}\)[/tex]
[tex]\[ \lambda = \frac{h \cdot c}{E} \][/tex]
where:
- [tex]\(E\)[/tex] is the energy of the photon,
- [tex]\(h\)[/tex] is Planck's constant ([tex]\(6.626 \times 10^{-34}\)[/tex] joule seconds),
- [tex]\(c\)[/tex] is the speed of light ([tex]\(3.00 \times 10^8\)[/tex] meters per second).
Step-by-Step Solution:
1. Identify the given values:
- Energy [tex]\(E = 3.38 \times 10^{-19}\)[/tex] joules
- Planck's constant [tex]\(h = 6.626 \times 10^{-34}\)[/tex] joule seconds
- Speed of light [tex]\(c = 3.00 \times 10^8\)[/tex] meters per second
2. Plug the values into the wavelength formula:
[tex]\[ \lambda = \frac{6.626 \times 10^{-34} \, \text{Js} \times 3.00 \times 10^8 \, \text{m/s}}{3.38 \times 10^{-19} \, \text{J}} \][/tex]
3. Calculate the wavelength [tex]\(\lambda\)[/tex]:
[tex]\[ \lambda = \frac{1.9878 \times 10^{-25} \, \text{Js} \cdot \text{m/s}}{3.38 \times 10^{-19} \, \text{J}} \][/tex]
[tex]\[ \lambda = 5.881 \times 10^{-7} \, \text{m} \][/tex]
4. Convert the wavelength from meters to nanometers:
[tex]\[ 1 \, \text{m} = 10^9 \, \text{nm} \][/tex]
[tex]\[ \lambda = 5.881 \times 10^{-7} \, \text{m} \times 10^9 \, \text{nm/m} \][/tex]
[tex]\[ \lambda = 588.1 \, \text{nm} \][/tex]
5. Compare the calculated wavelength to the given choices:
- A. [tex]\(510 \, \text{nm}\)[/tex]
- B. [tex]\(460 \, \text{nm}\)[/tex]
- C. [tex]\(588 \, \text{nm}\)[/tex]
- D. [tex]\(416 \, \text{nm}\)[/tex]
The wavelength most closely matches option C: [tex]\(588 \, \text{nm}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(588 \, \text{nm}\)[/tex]
