Question

For hydrogen atom, \(λ_1\) and \(λ_2\) are the wavelengths corresponding to the transitions \(1\) and \(2\) respectively as shown in figure. The ratio of \(λ_1\) and \(λ_2\) is \(x/32\). The value of \(x\) is

• A. \(13\)
• B. \(27\)
• C. \(29\)
• D. \(35\)

Hint:

Use the Rydberg formula for 3 to 1 and 2 to 1 transitions and divide the two wavelengths.

Answer 1

The Correct Option is B

Step 1: Use energy instead of wavelength:
Since $\lambda$ is inversely proportional to the photon energy, $\lambda_1/\lambda_2 = E_2/E_1$.

Step 2: Energies:
With $E_n = -13.6/n^2$ eV: transition 1 (3 to 1) releases $13.6(1-1/9) = 13.6\times 8/9$ eV and transition 2 (2 to 1) releases $13.6(1-1/4) = 13.6\times 3/4$ eV.

Step 3: Ratio:
$\frac{\lambda_1}{\lambda_2} = \frac{3/4}{8/9} = \frac{3}{4}\times\frac{9}{8} = \frac{27}{32}$.

Step 4: Result:
Matching with $x/32$ gives $x = 27$. The other options (13, 29, 35) cannot come from this ratio.

Final Answer:
x is 27. \[ \boxed{27} \]