To find the frequency of a radiation given its wavenumber, we can use the relationship between wavenumber and frequency. The wavenumber ([tex]\(\bar{\nu}\)[/tex]) is defined as the number of wavelengths per unit distance, and is typically measured in [tex]\( \text{cm}^{-1} \)[/tex]. The formula to convert wavenumber to frequency ([tex]\(\nu\)[/tex]) is given by:
[tex]\[ \nu = \bar{\nu} \cdot c \][/tex]
where:
- [tex]\(\nu\)[/tex] is the frequency.
- [tex]\(\bar{\nu}\)[/tex] is the wavenumber.
- [tex]\(c\)[/tex] is the speed of light.
The speed of light, [tex]\(c\)[/tex], is [tex]\(2.998 \times 10^{10} \, \text{cm/s}\)[/tex].
Given:
[tex]\[
\bar{\nu} = 97540 \, \text{cm}^{-1}
\][/tex]
Now, substituting the given values into the formula:
[tex]\[
\nu = 97540 \, \text{cm}^{-1} \times 2.998 \times 10^{10} \, \text{cm/s}
\][/tex]
When this calculation is performed (multiplying 97540 by 2.998 and the power of 10 terms), the result is:
[tex]\[
\nu \approx 2.9242492 \times 10^{15} \, \text{s}^{-1}
\][/tex]
Therefore, the correct answer is:
2) [tex]\(2.926 \times 10^{15} \, \text{s}^{-1} \)[/tex]
This matches the calculated frequency closely because the given answer choice is closest to the exact frequency calculated above.
