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: Read the figure:
The figure is an energy level diagram of hydrogen. Arrow 1 goes from n = 3 down to n = 1. Arrow 2 goes from n = 2 down to n = 1.

Step 2: Use the Rydberg formula:
For a transition from \(n_i\) to \(n_f\), \(\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\).

Step 3: Transition 1 (3 to 1):
\[ \frac{1}{\lambda_1} = R\left(1 - \frac{1}{9}\right) = \frac{8R}{9} \Rightarrow \lambda_1 = \frac{9}{8R} \]

Step 4: Transition 2 (2 to 1):
\[ \frac{1}{\lambda_2} = R\left(1 - \frac{1}{4}\right) = \frac{3R}{4} \Rightarrow \lambda_2 = \frac{4}{3R} \]

Step 5: Take the ratio:
\[ \frac{\lambda_1}{\lambda_2} = \frac{9}{8R}\times\frac{3R}{4} = \frac{27}{32} \]
Comparing with \(x/32\) gives \(x = 27\).

Step 6: Why the others are wrong:
If someone took the ratio the wrong way round, they would get \(32/27\), which does not fit the form \(x/32\). Values 13, 29 and 35 do not come from any combination of \(8/9\) and \(3/4\).

Final Answer:
The ratio is 27/32, so \(x = 27\), option (B). \[ \boxed{x = 27} \]