Question

A standing wave having 3 nodes and 2 antinodes is formed between two atoms having a distance $1.21$ $�$ between them. The wavelength of the standing wave is

• A. $6.05 $ $�$
• B. $2.42$ $�$
• C. $1.21$ $�$
• D. $3.63$ $�$

Answer 1

The Correct Option is C

 To determine the wavelength of a standing wave with 3 nodes and 2 antinodes, we need to understand the formation of standing waves in terms of nodes and antinodes.

In a standing wave system formed between two fixed points (such as two atoms), nodes are points of zero amplitude, and antinodes are points of maximum amplitude. The number of nodes (N) and antinodes (A) for a standing wave can be related to the number of half-wavelengths (m) as follows:

• A node is formed every half-wavelength, and therefore:
• Number of segments (half-wavelengths) between nodes = N - 1.
• Number of complete wavelengths is given by: \[ m = (N - 1) \]

In this problem:

• Given N = 3 nodes and A = 2 antinodes.
• The number of half-wavelengths (m) = 3 - 1 = 2.

If the distance between the two atoms is \(L = 1.21 \, \text{\AA}\), then the wavelength (\(\lambda\)) can be calculated as:

The wavelength \(\lambda\) is equal to twice the length divided by the number of half-wavelengths:

\[\lambda = \frac{2 \times L}{m}\]\[\lambda = \frac{2 \times 1.21 \, \text{\AA}}{2} = 1.21 \, \text{\AA}\]

Therefore, the wavelength of the standing wave is \(1.21 \, \text{\AA}\).

The correct answer is:

\(1.21\) \(�\)