Book Cover Chemistry 8th Edition / Chang
Student Study Guide

Chapter 7: Quantum Theory and the Electronic Structure of Atoms


Index | 7.1 7.2 | 7.3 | 7.4 | 7.5 7.7 | 7.8 7.9 |

BOHR'S MODEL OF THE HYDROGEN ATOM (7.3)

STUDY OBJECTIVES

  1. Distinguish between a line spectrum and a continuous spectrum.
  2. List the two assumptions made in Bohr's model of the hydrogen atom.
  3. Calculate the wavelength of radiation emitted by a specific electron transition in the hydrogen atom.

Emission Spectra. Elements in the gaseous state that are excited by an electrical current give off electromagnetic radiation in a process called emission. By passing emitted light through a prism, an emission spectrum consisting of the wavelengths of emitted electromagnetic radiation is produced. A continuous spectrum contains all the wavelengths between the longest and shortest emitted by that substance. "White" light results from the emission of all wavelengths in the visible part of the spectrum.

The light emitted from excited atoms in the gaseous state produces a line spectrum rather than continuous spectrum. A line spectrum consists of several discrete wavelengths. The observed wavelengths are characteristic of the element and can serve to identify the presence of the element in a sample.

Emission Spectrum of Hydrogen. A portion of the line spectrum of hydrogen falls in the visible region, and is called the Balmer series. visual aid Figure 7.6(b) of the text shows that the prominent visible lines are at 656 nm, 486 nm, and 434 nm. In all, five spectral series are known for hydrogen. The Lyman series is in the ultraviolet region, and the Paschen, Brackett, and Pfund series are in the infrared region

Bohr's Model. In 1913 Niels Bohr produced his famous model of the hydrogen atom. The great achievement of this model was that it explained the source and the observed wavelengths of lines in the spectrum of the H atom. Bohr postulated a "solar system" model for the H atom in which the electron traveled in circular orbits about the proton. The basic assumptions of Bohr's model were:

  1. The electron moves in a circular orbit about the nucleus. Of the infinite number of possible orbits only certain orbits with distinct radii are allowed. As long as the electron remains in an orbit, its energy remains constant.
  2. The energy of the electron increases the farther its orbit lies from the nucleus. When energy is absorbed by the atom, the electron must jump to a higher energy orbit. When an electron transition occurs from a higher to a lower energy orbit, radiation is emitted by the atom.

Bohr was able to calculate the radii of the allowed orbits and their energies. The energies of the H atom are given by:

where RH is the Rydberg constant, which in units of joules has the value 2.18 x 1018 J. The minus sign in the equation simply means that the energy of the hydrogen atom is lower than that of a completely separated proton and electron for which the force of attraction is zero. It does not signify negative energy. The quantity n is called the principal quantum number, and is an integer; n = 1, 2, 3,. Inserting values for n into the energy equation gives the energies of all allowed orbits shown in visual aid Figure 7.11 of the text.

The radius of each electron orbit in the hydrogen atom is proportional to n2. As n increases, the orbit radius increases rapidly. The farther the electron is from the nucleus, the higher its energy.

Bohr explained that the emission of light, when an electrical current passes through a gas, is due to the formation of excited atoms. He proposed that in the emission process an electron drops from a higher to a lower orbit. During this transition the atom emits radiation. For energy to be conserved, the energy lost as radiation must equal the difference in energy between the initial energy level Ei and the final energy level Ef. For the emission process, ni represents the higher energy level and nf lower level. Then the change in energy of the atom E is

E = Ef Ei

where, according to Bohr's equation, the energy change for the atom is

Bohr's stroke of genius was to equate Eatom to the energy of a photon of emitted light.

Ephoton = Eatom

Since, Ephoton = hn
=
hc
l

Then, hn = Eatom

and the frequency of light emitted should be given by

n =
Eatom
h

The wavelengths of emitted light can be calculated from

l =
c
=
hc
n
Eatom

With this model of the hydrogen atom, Bohr was able to predict the wavelengths of all the literally hundreds of observed lines in the spectrum of hydrogen!

However, it should be noted that the Bohr model of the atom is not an accurate representation. The field of quantum mechanics has provided us with tools to describe the atom, but these are harder to picture.


EXAMPLE The Hydrogen Atom

a. What amount of energy in joules is lost by a hydrogen atom when an electron transition from n = 3 to n = 2 occurs in the hydrogen atom?
E = x 10^ J

b. What is the wavelength of the light emitted when the electron transition n = 3 n = 2 occurs?
l = nm


OBJECTIVE CHECK

Complete the following questions to check your understanding of the material. Select the check button to see if you answered correctly.

  1. Consider the following energy levels for the hydrogen atom.
    __________ n = 4
    __________ n = 3
    __________ n = 2
    __________ n = 1
    Considering only these levels:
    1. How many emission lines are possible?
    2. Which transition produces photons of the greatest energy?
    3. Which transition for the H atom produces the emission line with the longest wavelength?
  2. What wavelength of radiation will be emitted when an electron in a hydrogen atom jumps from the n = 5 to the n = 1 principal energy level? Name the region of the electromagnetic spectrum corresponding to this wavelength.
  3. A hydrogen emission line in the ultraviolet region of the spectrum at 95.2 nm corresponds to a transition from a higher energy level n to the n = 1 level. What is the value of n for the higher energy level?
  4. The second line in the Balmer series occurs at 486.1 nm. What is the energy difference between the initial and final energy levels involved in the electron transition?



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