Question

In Bohr model of hydrogen atom, if the difference between the radii of \( n^{th} \) and \( (n+1)^{th} \) orbits is equal to the radius of the \( (n-1)^{th} \) orbit, then the value of \( n \) is:

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

Hint:

In Bohr's model, the radii of the orbits are proportional to \( n^2 \). Use this relationship to solve for the value of \( n \) when differences in radii are given.

Answer 1

The Correct Option is C

The radius of the \( n^{th} \) orbit in the Bohr model is given by: \[ r_n = \frac{n^2 h^2}{4 \pi^2 m e^2} = n^2 r_1 \] where \( r_1 \) is the radius of the first orbit. The difference between the radii of the \( n^{th} \) and \( (n+1)^{th} \) orbits is: \[ r_n - r_{n+1} = r_1 \left(n^2 - (n+1)^2\right) = r_1 \left(n^2 - (n^2 + 2n + 1)\right) = -2nr_1 - r_1 \] This should be equal to the radius of the \( (n-1)^{th} \) orbit: \[ r_{n-1} = (n-1)^2 r_1 \] Setting the two expressions equal and solving for \( n \), we get \( n = 4 \).