The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed.

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Identify the initial and final energy levels for the Paschen series. The final energy level \( n_f \) is 3, and the initial energy level \( n_i \) is greater than 3.

Use the Rydberg formula for hydrogen to calculate the wavelength of the emitted light: \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \), where \( R_H \) is the Rydberg constant.

Calculate the wavelengths for transitions from higher energy levels (e.g., \( n_i = 4, 5, 6, \ldots \)) to \( n_f = 3 \).

Determine the range of wavelengths obtained from these calculations.

Identify the region of the electromagnetic spectrum corresponding to these wavelengths, which is the infrared region.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, arranged by wavelength or frequency. It includes gamma rays, X-rays, ultraviolet light, visible light, infrared radiation, microwaves, and radio waves. Each type of radiation has distinct properties and applications, with the visible spectrum being just a small part of the entire range.

Hydrogen Emission Spectrum

The hydrogen emission spectrum is the spectrum of light emitted by hydrogen atoms when they transition from higher energy levels to lower ones. Each transition corresponds to a specific wavelength of light, resulting in distinct emission lines. The Paschen series specifically refers to transitions where electrons fall to the n=3 energy level, producing infrared radiation.

Paschen Series

The Paschen series is a set of spectral lines in the hydrogen emission spectrum that occurs when electrons transition to the third energy level (n=3) from higher levels (n=4, 5, 6, etc.). These transitions emit photons in the infrared region of the electromagnetic spectrum, which is not visible to the human eye but can be detected with appropriate instruments.

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The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (b) Calculate the wavelengths of the first three lines in the Paschen series—those for which ni = 4, 5, and 6.

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Determine whether each of the following sets of quantum numbers for the hydrogen atom is valid. If a set is not valid, then indicate which of the quantum numbers has a value that is not valid: e. n = 2, l = 2, ml = 1, ms = +1/2