Let us assume that the de-Broglie wave asssociated with an electron froms a standing wave between the atoms arranged in a one-dimensional array with nodes at each of the atomic sites. If is found that one such standing wave is formed, if the distance between the atoms of the array is `3 Å`. A similar standing wave is again formed, if d is increased to `3.5 Å` but not for any intermediate value of d. Find the energy of the electrons in electron volts and the least value of d for which the standing wave of the type described above can from.

Explanation

To solve the problem step by step, we will follow the reasoning and calculations outlined in the video transcript.

### Step 1: Understand the standing wave condition
The standing wave condition for electrons in a one-dimensional atomic array states that the distance between the atomic sites must be an integer multiple of the de Broglie wavelength, λ.

For the first case where the distance between the atoms is \(d_1 = 3 \, \text{Å}\), we can express this as:
\[ d_1 = n \lambda \]
where \(n\) is an integer.
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