Question

If the difference between \((n+1)^{th}\) Bohr radius and \(n^{th}\) Bohr radius is given by \((n-1)^{th}\) Bohr radius, then the value of n is

• A. \(4\)
• B. \(3\)
• C. \(2\)
• D. \(1\)

Hint:

The Bohr radius of orbit n is proportional to n squared. Set (n+1)^2 - n^2 = (n-1)^2.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The radius of the \(n\)th Bohr orbit of hydrogen is \(r_n = n^2r_1\), where \(r_1\) is the first Bohr radius.

Step 2: Key Formula or Approach:
The condition is \(r_{n+1} - r_n = r_{n-1}\).

Step 3: Detailed Explanation:
\[ (n+1)^2 - n^2 = (n-1)^2 \]
Expand the left side: \(n^2 + 2n + 1 - n^2 = 2n + 1\).
Expand the right side: \(n^2 - 2n + 1\).
\[ 2n + 1 = n^2 - 2n + 1 \Rightarrow n^2 - 4n = 0 \Rightarrow n(n - 4) = 0 \]
The root \(n = 0\) is not allowed, so \(n = 4\).
Check: \(r_5 - r_4 = 25 - 16 = 9 = r_3\). It works.

Final Answer:
The value of \(n\) is 4, option (A). \[ \boxed{4 \text{ (A)}} \]