Question

excited hydrogen atom emits a Photon of wave length \(\lambda\) in returning to the ground state The quantum number \(\mathrm{n}\) of excited state is given by (A) \(\sqrt{[}(\lambda . \mathrm{R}-1) /(\lambda \mathrm{R})]\) (B) \(\sqrt{[}(\lambda \mathrm{R}) /(\lambda \mathrm{R}-1)]\) (C) \(\sqrt{[\lambda R}(\lambda \mathrm{R}-1)]\) (D) \(\lambda \mathrm{R}(\mathrm{R}-1)\)

Short answer

The correct expression for the quantum number n of an excited hydrogen atom that returns to its ground state by emitting a photon with a wavelength λ is (B) \(n = \sqrt{\frac{(\lambda R)}{(\lambda R - 1)}}\).

Step-by-step solution

Recall the Rydberg formula for hydrogen

The Rydberg formula for hydrogen can be expressed as: \[ \frac{1}{\lambda} = R \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \] where: - \(λ\) is the wavelength of the photon emitted, - \(R\) is the Rydberg constant for hydrogen, approximately equal to \(1.097 \times 10^7 \, \text{m}^{-1}\), - \(n_f\) is the final principal quantum number of the electron, - \(n_i\) is the initial principal quantum number of the electron. In this case, the hydrogen atom returns to its ground state (i.e., \(n_f = 1\)), and we need to find the expression for \(n_i\), referred to as \(n\).

Substitute the values in the Rydberg formula

Substitute \(n_f = 1\) into the Rydberg formula: \[ \frac{1}{\lambda} = R \left(\frac{1}{1^2} - \frac{1}{n^2}\right) \]

Simplify and solve for n

Now simplify the formula and solve for \(n\): \[ \frac{1}{\lambda} = R \left(1 - \frac{1}{n^2}\right) \] Divide both sides of the equation by R: \[ \frac{1}{\lambda R} = 1 - \frac{1}{n^2} \] Rearrange to isolate the term \(\frac{1}{n^2}\): \[ \frac{1}{n^2} = 1 - \frac{1}{\lambda R} \] Now, take the reciprocal of both sides: \[ n^2 = \frac{1}{1 - \frac{1}{\lambda R}} \] Finally, to obtain the expression for \(n\), take the square root of both sides: \[ n = \sqrt{\frac{1}{1 - \frac{1}{\lambda R}}} \] Comparing the given options, we find the correct expression is:

Answer

(B) \(n = \sqrt{\frac{(\lambda R)}{(\lambda R - 1)}}\)