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Chapter Review

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In this chapter we discuss the ideas that make up our modern understanding of the atoms. We focus primarily on the simplest atom, Hydrogen. The ideas discussed in this chapter are at the foundation of what is called quantum physics. The development of quantum physics (or quantum mechanics) revolutionized physics (and all physical sciences) in the 1900s.

31-1   Early Models of the Atom

The concept of the atom was originally proposed as the indivisible, fundamental, quantity out of which things are made. The 1897 discovery of the electron, by J.J. Thomson, changed this belief by revealing that atoms must have internal structure. Thomson proposed what is called the "plum-pudding model" of the atom in which negatively charged electrons are embedded in a nearly uniform distribution of positively charged matter.

A new picture of atomic structure emerged after Ernest Rutherford scattered positively charged alpha particles from a thin gold foil. The results of Rutherford's experiments suggested that the structure of atoms is more like a miniature solar system rather than plum pudding. Rutherford's model placed all the positive charge in an atom, and most of its mass, at a small, central location, called the nucleus, with the negatively charged electrons moving around the nucleus in orbits, similar to the planets orbiting the sun. While Rutherford's model seemed more reasonable based on the scattering experiments, it was not consistant with experiments on the light given off by atoms nor was it a stable structure according to Maxwell's electromagnetic theory.

31-2   The Spectrum of Atomic Hydrogen

By applying a large potential difference across a tube containing a low pressure atomic gas, the atoms of the gas can be made to give off light. Passing this light through a diffraction grating separates the light into different wavelengths producing a line spectrum. The process just described produces an emission spectrum. However, when light consisting of different wavelengths is passed through a gas, some of the wavelengths from this light will be absorbed by the gas and the resulting light can then produce a line spectrum that is an absorption spectrum. Each atom has its own unique emission/absorption spectrum.

For hydrogen the wavelengths that make up its emission spectrum are given by the formula.

The constant R, called the Rydberg constant, is, R = 1.097 x 10⁷ m^-1. The best known wavelengths of this spectrum are those from the Lyman series (n' = 1), the Balmer series (n' = 2), and the Paschen series (n' = 3). However, there are an infinite number of series because there is no upper limit on n'.

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Exercise 31.1   The Paschen Series:   Determine the range of wavelengths that make up the Paschen series of the hydrogen spectrum.

Solution:

The Paschen series has n' = 3. Therefore, the values of n range from 4 to . For n = 4 we have

,

For n = we have 1/n = 0. Therefore,

.

Can you determine to which parts of the electromagnetic spectrum these wavelength's belong?

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Physlet Illustration: The Spectrum of Atomic Hydrogen
Balmer Paschen Lyman
The spectrum of atomic hydrogen can be considered in several pieces: the Balmer series, which lies mostly in the visible portion of the spectrum; the Paschen series, which lies in the infrared; and the Lyman series in the ultraviolet.  The Balmer series can be described by a mathematical formula.  Can the Paschen and Lyman series by described by a similar mathematical relationship?
Hints: Can you verify that the mathematical formula works for the Balmer series?   Whereas the leading term in the Balmer formula contains 22, the Paschen and Lyman series will contain a different integer squared.  What might the integer be? Try a different integer in the leading term and calculate the wavelengths of a few lines.  Do they match any series? Does the same Rydberg constant work for all of the series? What combinations of integers produce each of the series?

Practice Quiz

Which of the following is not a wavelength in the Balmer series? 656.3 nm 486.2 nm 364.6 nm 434.1 nm 541.4 nm

31-3 - 31-4   Bohr's Model of the Hydrogen Atom and de Broglie Waves

To deal with the inconsistency between Rutherford's model and the experiments on spectrum of hydrogen, Bohr refined Rutherford's model with four assumptions:

(1) The electron in a hydrogen atom moves in a circular orbit around the nucleus.

(2) The only circular orbits that are allowed are those for which the angular momentum of the electron is given by for n = 1, 2, 3, ..., where h is Planck's constant.

(3) Electrons do not give off electromagnetic radiation while in an allowed orbit.

(4) Electromagnetic radiation is given off, or absorbed, by an atom only when an electron changes from one allowed orbit to another. The frequency of the single photon that is emitted or absorbed is DE = hf, where DE is the energy difference between the two allowed orbits.

Taking the centripetal force for circular motion from Coulomb's law leads to an expression for the radii of the allowed orbits in a hydrogen atom, often called Bohr orbits,

,

where m is the mass of an electron, e is the elementary charge, and from Coulomb's law. The speed with which an electron moves in a Bohr orbit is given by

The above results can be extended to any single-electron atom, such as singly ionized helium and doubly ionized lithium, by accounting for the charge of the additional protons in the nucleus. In general, we take a nucleus to contain Z protons giving it a charge of +Ze; the quantity Z is called the atomic number of the nucleus. The above results for r_n and v_n, then become valid for single-electron atoms with heavier nuclei if we replace e² by Ze².

Each Bohr orbit has a certain amount of energy given by the sum of the kinetic and electric potential energies of the electron. For single-electron atoms the total energies of the orbits are given by

The lowest possible energy of the atom corresponds to n = 1 and is called the ground state of the atom. Orbits with energies higher than the ground state (n > 1) are called excited states. Using the above energy states, Bohr's model predicts an emission spectrum for hydrogen (Z = 1) that is described by the equation

,

where c is the speed of light. This result is consistent with the empirical result quoted previously, and in addition, it tells us from what quantities the Rydberg constant R is derived.

In 1923 deBroglie provided some physical insight into Bohr's model by showing that Bohr's condition on the allowed angular momenta of the orbiting electrons was equivalent to a standing wave condition for the electrons' matter waves: nl = 2pr_n. This success helped the idea of matter waves to be taken more seriously. The properties of matter waves are determined by what is called Schrodinger's equation; this equation is the fundamental equation of quantum mechanics. Today, the most common view of matter waves is that amplitude of a particle's matter wave, at a given time, is related to the probability of the particle being located at that point at that time.

Physlet Illustration: Emission in the Bohr Model
Starting Level=  Ending Level=
In this simulation, an electron is shown orbiting a proton in the Bohr model for hydrogen.  The user may change the ending level (1 to 3) and the starting level (ending level + 1 to ending level +4).  After a short time, the electron "drops" to a lower level, giving off a photon.  The line spectrum is shown below.  How does the wavelength of the emitted photon depend on the starting and ending orbital levels?
Hints: Keeping the ending level at 2, try different starting orbits.  What happens to the wavelength of the photon when the difference is small?  When it is large? Set the ending level to 1 and try different starting orbits.  What is different now? Try setting the ending level to 3.  What is different in this case? Which of these corresponds to the Paschen series?  The Balmer series?

Physlet Illustration: Absorption in the Bohr Model
Ending Level=
In this simulation, an electron is shown orbiting a proton in the Bohr model for hydrogen in level 2.  Full spectrum light is incident upon the atom.  The user may change the final level of the electron between 3 and 6.  After a short time, a photon of the appropriate wavelength is absorbed and the electron "jumps" to the chosen level.  The absorption spectrum is shown.  How does the wavelength of the absorbed photon depend on the starting and ending orbital levels?
Hints: Try different ending levels for the electron.  How does the choice of level affect the absorption spectrum? How are the wavelengths related to those in the emission case for the same levels? Why does the spectral line appear black?

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Example 31.2   Ionized Helium:   What is the energy of the 1^st excited state of singly ionized helium?

Picture the Problem   The picture is a sketch of the first two Bohr orbits around the nucleus of a singly ionized helium atom (not to scale).

Strategy   We use the more general expression for the allowed energies of single-electron atoms.

Solution

1. The atomic number of helium is:	Z = 2
2. The 1st excited state is:	n = 2
3. The energy then is:

Insight   This is the same energy as the ground state of hydrogen, but for a very different situation. The energy is negative because, as with planets orbiting the sun, the electron and proton form a bound system so that the (negative) potential energy "over powers" the kinetic energy.

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Practice Quiz

According to Bohr's model of hydrogen, what is the energy of an electron in the n = 3 excited state? -6.04 eV -1.51 eV -13.6 eV -0.85 eV -4.53 eV

According to Bohr's model of hydrogen, what is the radius of the orbit of an electron in the n = 3 excited state? 5.29 x 10-11 m 1.59 x 10-10 m 4.76 x 10-10 m 1.06 x 10-10 m 2.38 x 10-10 m

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