The Paschen series in the hydrogen emission spectrum is...

The Paschen series in the hydrogen emission spectrum is formed by electron transitions from $n_{\mathrm{i}}>3$ to $n_{\mathrm{f}}=3$. (a) What is the longest wavelength in the Paschen series? (b) What is the wavelength of the series limit (the lower bound of the wavelengths in the series)? (c) In what part or parts of the EM spectrum is the Paschen series found (IR, visible, UV, etc.)?

The Paschen series in the hydrogen emission spectrum is formed by electron transitions from $n_{\mathrm{i}}>3$ to $n_{\mathrm{f}}=3$.
(a) What is the longest wavelength in the Paschen series? (b) What is the wavelength of the series limit (the lower bound of the wavelengths in the series)?
(c) In what part or parts of the EM spectrum is the Paschen series found (IR, visible, UV, etc.)?

Key Concepts

Electromagnetic Spectrum Regions

The electromagnetic spectrum encompasses all types of electromagnetic radiation, which range from gamma rays to radio waves. Each spectral region, such as the infrared, visible, or ultraviolet, is defined based on wavelength or frequency. Understanding the location of specific spectral series, like the Paschen series, within the electromagnetic spectrum helps in identifying the nature of the light emitted and its potential applications.

Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of light emitted or absorbed when an electron in a hydrogen atom transitions between energy levels. It relates the inverse of the wavelength to the difference between the inverse squares of the initial and final quantum numbers. This formula is essential in spectroscopy for identifying and predicting the wavelengths associated with various spectral series.

Series Limit

The series limit is the wavelength (or frequency) corresponding to the transition where the electron starts from an infinitely high energy level and falls to the defined lower energy level of the series. It represents the boundary point beyond which the spectral series no longer produces discrete lines, marking a continuous spectrum transition.

Electron Transitions

Electron transitions refer to the movement of electrons between fixed energy levels (or orbitals) within an atom. When an electron drops from a higher energy level to a lower one, it emits a photon whose energy corresponds to the difference between the two states. This concept is fundamental to understanding how atoms emit or absorb light in quantized wavelengths.

Hydrogen Emission Spectrum

The hydrogen emission spectrum is a classic example of quantized energy emission, where electrons in a hydrogen atom transition between discrete energy levels. These transitions produce light at specific wavelengths, which appear as distinct lines in the spectrum. The pattern and spacing of these lines can be explained by quantum mechanics and are predictable using formulas such as the Rydberg formula.

Spectral Series

A spectral series is a group of spectral lines that arise from electron transitions terminating at a common lower energy level. Each series in the hydrogen spectrum, such as Lyman, Balmer, or Paschen, corresponds to transitions ending at a particular energy level. Studying these series provides insight into the energy level structure of hydrogen and the mechanisms behind light emission.

Transcript

00:01 For this exercise, we have to consider the poscen series of the hydrogen atom that takes an atom at an original energy level of ny, larger than 3, takes it down to the third energy level, so nf, the final energy level, is 3.

00:20 And in the first question, we're asked, what is the longest wavelength, lambda maximum, of this series? in order to calculate this, i got to remember that the energy of the energy level of the hydrogen atom is given by minus 13 .6 n squared electron volts.

00:44 And that from the conservation of energy, you have that the energy of the initial energy level is going to be the energy of the emitted photon, plus the energy of the final energy level.

00:58 Here, the final energy level is 3.

01:02 So you can calculate the energy of the photon as the energy of the final energy level, which is minus 13 .6 over n...

01:13 I'm sorry, the energy of the initial energy level.

01:16 It's minus 13 .6 over n .i squared, plus minus the energy of the third energy level, which is 13 .6 over 9.

01:34 So this is going to be 13 .619 minus 1 .i squared.

01:46 And notice that the energy of the photon equals hc over lambda, so that the maximum wavelength will happen when the energy is minimum.

02:01 And the minimum energy that can be achieved from this formula here will be achieved when n i is the smallest value possible.

02:13 And the smallest value possible for an atom to transition from some initial n to a final n equals 3 is n equals 4.

02:25 So we're going to have n y equals 4.

02:28 And we can calculate the smallest energy now...

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