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Chapter 26: Problem 19

If \(\lambda_{1}\) and \(\lambda_{2}\) are the wavelengths of the first members of the Lyman and Paschen series respectively, then \(\lambda_{1}: \lambda_{2}\) is (a) \(1: 3\) (b) \(1: 30\) (c) \(7: 50\) (d) \(7: 108\)

Short Answer

Expert verified
The ratio \( \lambda_1 : \lambda_2 \) is \( 7:108 \), answer is (d).

Step by step solution

01

Understanding the Problem

We need to find the ratio of wavelengths of the first members of the Lyman series and the Paschen series in the hydrogen spectrum. This involves understanding the formula for wavelength in terms of series transitions.
02

Identifying Series Transitions

The Lyman series corresponds to transitions from higher energy levels to n=1, and the first member (longest wavelength) of this series is from n=2 to n=1. The Paschen series corresponds to transitions to n=3, and the first member is from n=4 to n=3.
03

Applying the Wavelength Formula

The wavelength formula for hydrogen's emission spectrum is given by \[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]where \( R \) is the Rydberg constant. For Lyman, \( n_1=1 \), \( n_2=2 \); for Paschen, \( n_1=3 \), \( n_2=4 \).
04

Calculating Wavelengths

For the Lyman series:\[ \frac{1}{\lambda_1} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = \frac{3R}{4} \]Thus, \( \lambda_1 = \frac{4}{3R} \).For the Paschen series:\[ \frac{1}{\lambda_2} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = R \left( \frac{1}{9} - \frac{1}{16} \right) \]The calculation simplifies as follows:\[ = R \left( \frac{16 - 9}{144} \right) = \frac{7R}{144} \]Thus, \( \lambda_2 = \frac{144}{7R} \).
05

Finding the Ratio of Wavelengths

We calculate the ratio \( \lambda_1 : \lambda_2 \):\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{4}{3R}}{\frac{144}{7R}} = \frac{4 \times 7R}{3R \times 144} = \frac{28}{432} = \frac{7}{108} \]
06

Choosing the Correct Answer

The calculated ratio of wavelengths \( \lambda_1 : \lambda_2 \) is \( 7:108 \), which matches option (d).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lyman Series
The Lyman series is part of the hydrogen atom's emission spectrum and is historically significant in the study of atomic physics.
  • It corresponds to electron transitions from higher energy levels (n 鈮� 2) down to the first energy level (n=1).
  • These transitions result in the emission of ultraviolet light, as the energy difference is quite large.
  • The first member, or the transition with the longest wavelength, involves an electron moving from n=2 to n=1.
When the electron transitions downward, it releases energy in the form of light waves, with each specific transition corresponding to a particular wavelength of this light. The Lyman series is crucial for understanding the quantization of energy levels in hydrogen and is used extensively in spectroscopy.
Paschen Series
The Paschen series is another group within the hydrogen atom's emission spectrum. Unlike the Lyman series, it deals with transitions down to the third energy level:
  • This series comprises electron transitions from higher energy levels (n 鈮� 4) to n=3.
  • The emitted radiation falls within the infrared range, as these transitions release less energy compared to the Lyman series.
  • The first member of the Paschen series is the transition from n=4 to n=3, resulting in the longest wavelength for this series.
The significance of the Paschen series lies in its application in studying stellar and interstellar gases. Understanding these transitions helps in comprehending the interaction of infrared radiation with matter.
Rydberg Formula
The Rydberg Formula is a fundamental equation used to calculate the wavelengths of spectral lines in many chemical elements. It is given by the equation:\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]where:
  • \( \lambda \) is the wavelength of the emitted or absorbed light.
  • \( R \) is the Rydberg constant, a fundamental constant of nature.
  • \( n_1 \) and \( n_2 \) are the principal quantum numbers of the electron's initial and final energy levels, with \( n_2 > n_1 \).
The Rydberg Formula is instrumental in predicting the spectral line positions of the hydrogen atom. It forms the mathematical basis for calculating the wavelengths of different series, such as the Lyman and Paschen series.
Wavelength Ratio
When evaluating the ratio of wavelengths between different spectral series, we often want to compare specific transitions. This exercise involves finding the ratio of the first members of the Lyman and Paschen series.
  • The wavelength of the first member of the Lyman series (\( \lambda_1 \)) is calculated from the transition n=2 to n=1.
  • For the Paschen series (\( \lambda_2 \)), the transition is from n=4 to n=3.
  • Using the Rydberg Formula, both wavelengths can be expressed in terms of the Rydberg constant \( R \).
Calculating and simplifying these allows for determining the wavelength ratio:\[ \lambda_1 : \lambda_2 = \frac{7}{108} \]Understanding this ratio requires using the concepts of spectral series and quantum transitions.

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