To determine the wavelength of radiation given its frequency, we use the relationship between the speed of light, frequency, and wavelength, which is given by the equation:
[tex]\[ c = \lambda \nu \][/tex]
where:
- [tex]\( c \)[/tex] is the speed of light in a vacuum (~ [tex]\( 3 \times 10^8 \)[/tex] m/s)
- [tex]\( \lambda \)[/tex] is the wavelength of the radiation in meters
- [tex]\( \nu \)[/tex] is the frequency of the radiation in s[tex]\(^{-1}\)[/tex]
Given:
- Frequency ([tex]\( \nu \)[/tex]) = [tex]\( 5.39 \times 10^{14} \)[/tex] s[tex]\(^{-1} \)[/tex]
- Speed of light ([tex]\( c \)[/tex]) = [tex]\( 3 \times 10^8 \)[/tex] m/s
We need to solve for the wavelength ([tex]\( \lambda \)[/tex]). Rearranging the formula to solve for [tex]\( \lambda \)[/tex]:
[tex]\[ \lambda = \frac{c}{\nu} \][/tex]
Substituting the given values:
[tex]\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{5.39 \times 10^{14} \text{ s}^{-1}} \][/tex]
Upon performing the division:
[tex]\[ \lambda \approx 5.565862708719852 \times 10^{-7} \text{ meters} \][/tex]
To convert this wavelength into nanometers (nm), we recall that:
[tex]\[ 1 \text{ meter} = 10^9 \text{ nanometers} \][/tex]
Thus:
[tex]\[ \lambda \approx 5.565862708719852 \times 10^{-7} \text{ meters} \times 10^9 \text{ nm/m} \][/tex]
[tex]\[ \lambda \approx 556.5862708719852 \text{ nanometers} \][/tex]
Rounding to a reasonable precision, the wavelength is approximately:
[tex]\[ \lambda \approx 557 \text{ nm} \][/tex]
Therefore, the wavelength of the radiation that has a frequency of [tex]\( 5.39 \times 10^{14} \)[/tex] s[tex]\(^{-1}\)[/tex] is approximately [tex]\( 557 \)[/tex] nm.
