To find the wavelength of a photon given its energy, you can use the relationship between energy and wavelength given by the equation:
\[ E = \frac{hc}{\lambda} \]
Where:
- \( E \) is the energy of the photon (in joules),
- \( h \) is the Planck constant (\(6.626 \times 10^{-34} , \text{Js}\)),
- \( c \) is the speed of light (\(3.00 \times 10^8 , \text{m/s}\)),
- \( \lambda \) is the wavelength (in meters).
First, convert the energy from electronvolts to joules. The conversion factor is:
\[ 1 , \text{eV} = 1.602 \times 10^{-19} , \text{J} \]
Thus,
\[ E = 12.4 , \text{eV} \times 1.602 \times 10^{-19} , \text{J/eV} = 1.987 \times 10^{-18} , \text{J} \]
Next, rearrange the equation to solve for the wavelength \( \lambda \):
\[ \lambda = \frac{hc}{E} \]
Substituting in the values:
\[ \lambda = \frac{(6.626 \times 10^{-34} , \text{Js})(3.00 \times 10^8 , \text{m/s})}{1.987 \times 10^{-18} , \text{J}} \]
Calculating this gives:
\[ \lambda = \frac{1.9878 \times 10^{-25} , \text{J m}}{1.987 \times 10^{-18} , \text{J}} \approx 1.001 \times 10^{-7} , \text{m} \]
Converting this into nanometers (1 nm = \( 10^{-9} , \text{m} \)):
\[ \lambda \approx 100.1 , \text{nm} \]
Thus, the wavelength of the photon with an energy of 12.4 eV is approximately 100 nm.