MHT CET · Physics · Atomic Physics

If \(^{\prime} \lambda_{1}\) ' and ' \(\lambda_{2}\) ' are the wavelengths of de-Broglie waves for electrons in first and second Bohr orbits in hydrogen atom, then \(\left(\frac{\lambda_{1}}{\lambda_{2}}\right)\) is equal to (Energy in \(1^{\text {st }}\) Bohr
orbit \(=-13.6 \mathrm{eV}\) )

A \(\frac{1}{5}\)
B \(\frac{1}{2}\)
C \(\frac{1}{4}\)
D \(\frac{1}{3}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{1}{2}\)

Step-by-step Solution

Detailed explanation

\(\lambda_n = \frac{2\pi r_n}{n}\) \(\lambda_n = \frac{2\pi (n^2 a_0)}{n} = 2\pi n a_0\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If \(^{\prime} \lambda_{1}\) ' and ' \(\lambda_{2}\) ' are the wavelengths of de-Broglie waves for electrons in first and second Bohr orbits in hydrogen atom, then \(\left(\frac{\lambda_{1}}{\lambda_{2}}\right)\) is equal to (Energy in \(1^{\text {st }}\) Bohrorbit \(=-13.6 \mathrm{eV}\) )

Step-by-step Solution

If \(^{\prime} \lambda_{1}\) ' and ' \(\lambda_{2}\) ' are the wavelengths of de-Broglie waves for electrons in first and second Bohr orbits in hydrogen atom, then \(\left(\frac{\lambda_{1}}{\lambda_{2}}\right)\) is equal to (Energy in \(1^{\text {st }}\) Bohr
orbit \(=-13.6 \mathrm{eV}\) )