To determine which color of light corresponds to the photon with the given energy, we will follow these steps:
1. Calculate the Frequency of the Photon:
- The energy of a photon [tex]\(E\)[/tex] is given as [tex]\(3.38 \times 10^{-19} \, \text{J}\)[/tex].
- Planck's constant [tex]\(h\)[/tex] is given as [tex]\(6.63 \times 10^{-34} \, \text{J} \cdot \text{s}\)[/tex].
The relationship between the energy of a photon and its frequency [tex]\(f\)[/tex] is given by the formula:
[tex]\[
E = h \cdot f
\][/tex]
Rearranging this formula to solve for frequency [tex]\(f\)[/tex]:
[tex]\[
f = \frac{E}{h} = \frac{3.38 \times 10^{-19} \, \text{J}}{6.63 \times 10^{-34} \, \text{J} \cdot \text{s}} = 5.098 \times 10^{14} \, \text{Hz}
\][/tex]
2. Compare with the Provided Frequencies:
- Violet: [tex]\(7.59 \times 10^{14} \, \text{Hz}\)[/tex]
- Green: [tex]\(5.10 \times 10^{14} \, \text{Hz}\)[/tex]
- Yellow: [tex]\(5.01 \times 10^{14} \, \text{Hz}\)[/tex]
- Red: [tex]\(4.72 \times 10^{14} \, \text{Hz}\)[/tex]
3. Calculate the Absolute Differences:
- Difference with Violet:
[tex]\[
\left| 7.59 \times 10^{14} - 5.098 \times 10^{14} \right| = 2.492 \times 10^{14} \, \text{Hz}
\][/tex]
- Difference with Green:
[tex]\[
\left| 5.10 \times 10^{14} - 5.098 \times 10^{14} \right| = 0.002 \times 10^{14} \, \text{Hz} = 2.0 \times 10^{12} \, \text{Hz}
\][/tex]
- Difference with Yellow:
[tex]\[
\left| 5.01 \times 10^{14} - 5.098 \times 10^{14} \right| = 0.088 \times 10^{14} \, \text{Hz} = 8.8 \times 10^{12} \, \text{Hz}
\][/tex]
- Difference with Red:
[tex]\[
\left| 4.72 \times 10^{14} - 5.098 \times 10^{14} \right| = 0.378 \times 10^{14} \, \text{Hz} = 37.8 \times 10^{12} \, \text{Hz}
\][/tex]
4. Find the Minimum Difference:
- The smallest difference is with the frequency of Green light:
[tex]\[
2.0 \times 10^{12} \, \text{Hz}
\][/tex]
Therefore, the color of light that corresponds to the given photon is Green.
