Chapter 35 The Bohr Theory of Hydrogen

Why is the hydrogen atom "easy" to analyze compared to other atoms in the periodic table?

Imagine you are a physicist in the late 1800’s. The electron has just been discovered as a separate, negative, subatomic particle. You know that atoms are usually electrically neutral. The proton and neutron have not been discovered. What would you propose for a model of the atom?

The adopted model for the atom was called the "plum pudding" model. It described the atom as a diffuse ball of positive charge with embedded electrons (plums). Rutherford performed an experiment in 1912 that led to a very different model for the atom.

In the late 1800’s radioactivity was discovered. Three types of radiation had been discovered, alpha, beta, and gamma. The alpha particles were known to have 2 units of positive charge and a mass roughly equal to the mass of a helium atom. Rutherford used natural sources of alpha particles to bombard thin gold foil. What would you predict for the path of an energetic alpha particle as it approached the plum pudding atoms in the foil?

Rutherford and his assistants observed that most alpha particles went through the foil with very little deflection. But, some alpha particles were deflected to such a degree that they came back toward the source. What could cause the alpha particle to bounce back?

Rutherford realized that the atoms actually had a small positive nucleus. The electrical repelling force when the alpha particle was close to the nucleus (and headed directly towards the nucleus) caused some of the alphas to be sent back towards the source of the alphas.

Page 2 The Classical Hydrogen Atom

A hydrogen atom has one proton and one electron. It was known by 1905 that atoms have a certain size, roughly 10^-10 meters. It is now known that the proton and electron are much smaller, less than 10^-15 meters in size. Draw a picture of an atom with a proton near the left side of the page and an electron near the right side of the page. Perhaps you remember doing this activity in a previous chapter. It is difficult to draw the objects small enough to match the distance between the objects. Suppose both the proton and electron are initially at rest. Describe the future motion of the proton and electron.

If the electron was near the proton atoms would be much smaller than their observed size. How can the electron remain at its distance from the proton? Hint: Imagine that the earth and the moon are the only objects in the universe. Would the moon fall into the earth? Why not?

Bohr assumed that the electron moved in a circular orbit around the proton. The electrical attractive force between the proton and electron provides the centripetal force in Bohr’s model of the atom. This model conflicts with Classical Physics. Classical Physics correctly predicts that electromagnetic waves are generated when a charged particle accelerates. This would quickly cause the electron to lose energy and spiral into the nucleus. This would lead to very small atoms, something that is not observed. The electron in Bohr’s model is accelerating. Bohr made a hypothesis that the electron in an atom did not radiate, contrary to the prediction of Maxwell’s Equations.

The text shows how it is possible to derive the speed of the electron in it orbit (equation 4) and the total energy of the electron (equation 8). What is the meaning of the minus sign in equation 8? Answer: Objects that have negative total energy are "bound" in their orbits. The electron can be freed from the atom (ionized) by giving energy to the electron such that its total energy is positive, or zero.

Page 4 Energy Levels

All hydrogen atoms emit the same wavelengths of light. We saw these spectral lines of hydrogen. Each wavelength is associated with a certain frequency, and by hf, a certain energy. The energies of the first three Balmer series lines are 1.89 eV, 2.55 eV and 2.86 eV. Which of these three is the red line in the series?

Bohr postulated that the electrons resided in certain energy levels in the atom. He further hypothesized that the atom only emitted light when an electron moved from a higher energy level to a lower energy level. The difference in energies of the two levels is the energy of the photon that is emitted by the atom. The values of the energies possible in the Bohr model are E_n = -13.6ev/n² . Calculate the difference in energy between the n=4 and n=2 levels.

* Refer to Figure 3. Which line represents the red line of the Balmer series?

The text shows how Balmer’s empirical formula can be derived from Bohr’s theoretical model of the hydrogen atom. This is another way of showing that the experimental data supports Bohr’s model of the hydrogen atom. Before you get too excited about the Bohr model I need to tell you that it is an approximately accurate model. And, it only works approximately well for hydrogen, not for the other 91 naturally occurring elements in the periodic table. The development of Quantum Mechanics in the 1920’s provided a theory that works for all the elements in the periodic table. Bohr’s model was important in that in provided a transition from Classical Physics to Modern Physics.

Refer to Figure 3. Calculate the energy of a photon created when an electron moves from the n=3 level to the n=1 level. Calculate the energy of a photon created when an electron moves from the n=4 level to the n=3 level.

Page 6 Electron transitions in hydrogen atoms in which the electron ends at level 1 form the Lyman series. This series has wavelengths in the ultraviolet portion of the spectrum. Transitions that end at n=3 generate the Paschen series that is in the infrared.

Page 7 The Bohr Model

Figure 6 shows the circular electron orbits of the Bohr model. The radii are not drawn to scale. The radius of each orbit is equal to n²r_o. The value for r_o is given in the front cover of the text. Calculate the radii of the first three Bohr orbits.

Page 8 Angular Momentum in the Bohr Model

The angular momentum of an object is L = mvr. Bohr assumed that the value of the angular momentum of the electron in the hydrogen atom was quantized. (equation 19). This assumption leads to the correct values of the energies of the electron in the allowed orbits. The angular momentum of the electron in the n=1 orbit is equal to h/(2 pi) , where h is Planck’s constant.

Page 10 De Broglie’s Hypothesis

We have seen that light is described as either a wave or as something like a particle. In 1923 de Broglie hypothesized that matter also had this dual nature. The wavelength of an object is calculated wavelength = h/(mv) .

We will do some example calculations with this formula.

de Broglie calculated the wavelength of the electron in the hydrogen atom and found that the circumference of the Bohr orbits is equal to an integer times the wavelength. In effect, the electron is a standing wave in the atom. It constructively interferes with itself in the stable Bohr orbits.

The wave nature of moving matter was verified in experiments in 1925. In these experiments electrons behaved like waves when passing through a crystal. This is the equivalent of light passing through a diffraction grating and creating maxima and minima. What microscope relies on the wave nature of electrons to produce an image?

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