Question

The wave number of an yellow radiation with wavelength 580 nm is

• A. \(1.724 \times 10^2 \text{ cm}^{-1}\)
• B. \(1.724 \times 10^4 \text{ cm}\)
• C. \(1.724 \times 10^3 \text{ cm}^{-1}\)
• D. \(1.724 \times 10^3 \text{ cm}\)
• E. \(1.724 \times 10^4 \text{ cm}^{-1}\)

Hint:

Check the units carefully.
Wave number must have units of \(\text{length}^{-1}\) (like \(\text{cm}^{-1}\)).
Options (B) and (D) are in \(\text{cm}\), so they can be eliminated immediately.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Wave number (\(\bar{\nu}\)) is defined as the number of wave cycles per unit length.
It is the reciprocal of the wavelength (\(\lambda\)).

Step 2: Key Formula or Approach:
The formula is:
\[ \bar{\nu} = \frac{1}{\lambda} \]
Ensure that the units of wavelength match the units requested in the options (centimeters).

Step 3: Detailed Explanation:
Given:
Wavelength (\(\lambda\)) = \(580 \text{ nm}\).
Convert nanometers to centimeters:
\(1 \text{ nm} = 10^{-9} \text{ m} = 10^{-7} \text{ cm}\).
Therefore, \(\lambda = 580 \times 10^{-7} \text{ cm}\) or \(5.8 \times 10^{-5} \text{ cm}\).
Calculating the wave number:
\[ \bar{\nu} = \frac{1}{5.8 \times 10^{-5} \text{ cm}} \]
\[ \bar{\nu} = \frac{10^5}{5.8} \text{ cm}^{-1} \]
\[ \bar{\nu} \approx 17241.37 \text{ cm}^{-1} = 1.724 \times 10^4 \text{ cm}^{-1} \]

Step 4: Final Answer:
The wave number is \(1.724 \times 10^4 \text{ cm}^{-1}\).