To find the transition energy corresponding to an absorption line at 502 nm, we need to use the relationship between the energy of a photon and its wavelength. This relationship is given by the equation:
[tex]\[ E = \frac{hc}{\lambda} \][/tex]
where:
- [tex]\( E \)[/tex] is the energy of the photon,
- [tex]\( h \)[/tex] is Planck's constant ([tex]\(6.62607015 \times 10^{-34} \, \text{J} \cdot \text{s}\)[/tex]),
- [tex]\( c \)[/tex] is the speed of light ([tex]\(3.0 \times 10^8 \, \text{m/s}\)[/tex]),
- [tex]\( \lambda \)[/tex] is the wavelength of the photon in meters.
1. Convert Wavelength from Nanometers to Meters:
The given wavelength is 502 nm. We need to convert this to meters.
[tex]\[ \lambda = 502 \, \text{nm} = 502 \times 10^{-9} \, \text{m} \][/tex]
2. Calculate Energy:
Plug in the values for [tex]\( h \)[/tex], [tex]\( c \)[/tex], and [tex]\( \lambda \)[/tex] into the equation:
[tex]\[ E = \frac{6.62607015 \times 10^{-34} \, \text{J} \cdot \text{s} \times 3.0 \times 10^8 \, \text{m/s}}{502 \times 10^{-9} \, \text{m}} \][/tex]
Simplifying the units and the numerical values will yield:
[tex]\[
E = \frac{6.62607015 \times 10^{-34} \times 3.0 \times 10^8}{502 \times 10^{-9}}
\][/tex]
[tex]\[
E = 3.9598028784860553 \times 10^{-19} \, \text{J}
\][/tex]
The transition energy corresponding to an absorption line at 502 nm is:
[tex]\[ E = 3.96 \times 10^{-19} \, \text{J} \][/tex]
Thus, the correct answer is:
C. [tex]\( 3.96 \times 10^{-19} \, \text{J} \)[/tex]
