Question

For the hydrogen spectrum,the wavelength in Balmer series is given by \(\frac{1}{λ}\)=R(\(\frac{1}{n_{1}^{2}}\)-\(\frac{1}{n_{2}^{2}}\)) where λ= wavelength and R is Rydberg constant. What are the values of n₁ and n₂,for the longest wavelength in the Balmer series?

• A. n₁=2,n₂=3
• B. n₁=2,n₂=4
• C. n₁=1,n₂=2
• D. n₁=2,n₂=∞
• E. n₁=3,n₂=∞

Answer 1

The Correct Option is A

Given:

• Balmer series formula for hydrogen spectrum: \[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
• We need to find \( n_1 \) and \( n_2 \) for the longest wavelength in the Balmer series.

Step 1: Understand the Balmer Series

The Balmer series corresponds to transitions where the electron falls to the \( n_1 = 2 \) level. Thus, \( n_1 \) is fixed as 2.

Step 2: Determine \( n_2 \) for Longest Wavelength

The longest wavelength occurs when the energy difference is smallest, i.e., for the smallest possible transition:

\[ n_2 = n_1 + 1 = 3 \]

Substituting \( n_1 = 2 \) and \( n_2 = 3 \):

\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{5}{36} \right) \]

This gives the smallest \( \frac{1}{\lambda} \), and thus the largest \( \lambda \).

Conclusion:

The values of \( n_1 \) and \( n_2 \) for the longest wavelength in the Balmer series are \( n_1 = 2 \) and \( n_2 = 3 \).

Answer: \(\boxed{A}\)

Answer 2

Step 1: Recall the Balmer series formula and conditions for the longest wavelength.

The Balmer series corresponds to transitions of electrons in hydrogen atoms where the final energy level is \( n_1 = 2 \). The formula for the reciprocal of the wavelength is:

\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), \]

where:

• \( \lambda \) is the wavelength,
• \( R \) is the Rydberg constant,
• \( n_1 = 2 \) (for the Balmer series), and
• \( n_2 > n_1 \) (the initial energy level).

The wavelength is inversely proportional to \( \frac{1}{\lambda} \). For the longest wavelength, \( \frac{1}{\lambda} \) must be minimized. This occurs when \( n_2 \) is the smallest integer greater than \( n_1 \), i.e., \( n_2 = 3 \).

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Step 2: Identify the values of \( n_1 \) and \( n_2 \).

For the Balmer series:

• \( n_1 = 2 \) (fixed for the Balmer series),
• \( n_2 = 3 \) (smallest value greater than \( n_1 \) to minimize \( \frac{1}{\lambda} \)).

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Final Answer: The values of \( n_1 \) and \( n_2 \) for the longest wavelength in the Balmer series are \( \mathbf{n_1 = 2, n_2 = 3} \), which corresponds to option \( \mathbf{(A)} \).