Lesson 7.2 The Bohr Theory of the Hydrogen Atom

Suggested Reading
·  Zundahl Chapter 7 Sections 7.3 & 7.4
Essential Question
·  How does the Bohr model account for the stability of the hydrogen atom and the line spectrum of the atom?
Learning Objectives
·  Distinguish between a continuous spectrum and a line spectrum.
·  Explain how the lines in the emission spectrum of hydrogen are related to electron energy levels.
·  Calculate the energy associated with electron transition in the hydrogen atom.
Introduction

According to Rutherford's nuclear model, the atom consists of a nucleus which has most of the mass of the atom and a positive charge. Enough electrons to make the atom electrically neutral orbit around the nucleus. However, using Rutherford's theory, scientists showed that the electrons would lose energy as they orbited the nucleus causing them to eventually spiral into the nucleus. Thus, Rutherford's model could not explain the stability of the atom. The atoms described by Rutherford's model fell apart!
A colleague of Rutherford, Neils Bohr came up with an alternative model that was based upon the work of Planck and Einstein that you learned about in the last lesson.Bohr applied quantum theory to the simplest atom, hydrogen. This model still applies to hydrogen, but deteriorates for more complicated atoms. Therefore, when we talk about Bohr's model, we must remember that it only applies to hydrogen.
Before we can look at Bohr's model we need to discuss the line spectra of atoms, so lets go.
Atomic Line Spectra

You learned when you carried out flame tests in Honors Chemistry that heated chemicals can emit light. A heated tungsten filament in an old fashioned light bulb is a familiar example of this. With a prism, we can spread out the white light emitted by solids to give a continuous spectrum. A continuous spectrum is a spectrum containing light of all visible wavelengths, like that of a rainbow.
The light emitted by a gas gives a different result. Rather than a continuous spectrum, with all of the colors of the rainbow, we obtain a line spectrum. A line spectrum is a spectrum showing only certain colors or specific wavelengths of light. When light from heated gaseous hydrogen is separated by a prism, it gives a spectrum of lines. Each line corresponds to a different wavelength of light. It has been determined that each element has a characteristic line spectrum. Thus, spectra can be used to identify elements. The spectra also tell us something about the structure of atoms, and if you know something about the structures of atoms, you can explain the formation of ions and molecules.

The line spectrum of hydrogen is especially simple. It consists of only four lines with wavelengths corresponding to the visible region of the electromagnetic spectrum (see figure to the left). In 1885, a scientist named J.J. Balmer showed that the wavelengths in the line spectrum of hydrogen could be reproduced by the following formula

where n is some whole number greater than 2, such as 3, 4, 5. Substituting these values into the equation above gives wavelengths that correspond to wavelengths found in the line spectrum of hydrogen. Although you will not apply this equation directly, it is important because Bohr used Balmer's equation in his model of the atom.
Bohr's Model
Bohr set down the following postulates to account for 1) the stability of the hydrogen atom (that the atom exists) and 2) the line spectrum of hydrogen.

1) Energy-Level Postulate: An electron can have only specific energy values in an atom. Bohr based this idea ofquantization of energy fromPlanck. Bohr was able to derive a mathematical equation that could be used to calculate the energy of the electron in the hydrogen atom.

where RH is a constant withe the value 2.179 x 10-18 J. Substituting gives


Different values of the possible energies of the electron are calculated by plugging in different values of n, which can have only the values 1, 2, 3,... Here n is referred to as the principal quantum number, and the values1, 2, 3,... are referred to as energy levels.The negative sign is used to indicate that the energy of an electron bound in an atom is lower than the energy of the unbound electron.
2) Transitions Between Energy Levels: An electron in an atom can change energy only by going from one energy level to another energy level. By

doing so, the electron undergoes a transition. The emission of light by atoms to give a line spectrum is explained as follows. An electron in a higher energy level (initial energy Ei) undergoes a transition to a lower energy level (final energy Ef). In this process the electron loses energy, which is emitted as a photon. This means that the initial energy of the electron is equal to the final energy plus the energy of the photon. Apply the law of conservation of energy to this gives, Ef +hv = Ei. This equation is rearranged to give the energy of the emitted photon.
Energy of emitted photon = hv =Ei-Ef or ∆E = hv
Here, Bohr used Einstein's photon concept to explain the line spectra of atoms. Bohr also combined Balmer's equation for wavelengths with his equation for the energy levels to derive an equation that could be used to calculate the change in energy associated with the transition of an electron from one energy level to another. This equation is as follows.

However, this equation is not given the AP Exam while Bohr's equation for the energy of the electron is. Recall that this is equation is as follows.

Because of this, you may want to get accustomed to using this equation instead. The change in energy for a transition can be determined by calculating the energy of the electron at each level given in the problem and then subtracting, where
∆E = Efinal - Einitial
Example: Determining the Wavelength or Frequency of Hydrogen Atom Transition
What is the wavelength of light emitted when the electron in a hydrogen atom undergoes a transition from energy level n = 4 to level n =2?
Solution:
From the formula for the energy levels, you can determine the change in energy for the transition as follows.

We can then use the relationship ∆E = hv to related the change in energy to the wavelength of light emitted as follows.

The the relationship c = ⅄v can be used to find the wavelength where⅄ = c/v. Thus,
⅄ = 3.00 x 108 m/s ÷ 6.16 x 1014 /s = 4.86x10-7 m or 486 nm.
There are a lot of steps to a problem like this, but these types of problems are pretty formulaic and vary little. If you master this problem, you should be able to complete other similar problems..
According to the Bohr model, the emission of light from an atom occurs when an electron undergoes a transition form an upper energy level to a lower one. But how does an electron get to an upper level in the first place? Well if energy is lost when electrons move to a lower energy level then energy must be gained in order for an electron to transition to a higher energy levle. Normally, the electron in a hydrogen atom exits in its lowest, or n = 1 energy level. To get to a higher level, the electron must gain energy, or be excited. One way this can happen is through the collision of two hydrogen atoms. During this collision, some of the kinetic energy of one atom can be gained by the electron of another atom. When this occurs, the electron can be boosted from the n = 1 energy level to a higher energy level.
Postulates 1 and 2 hold for atoms other than hydrogen except that the energy levels can not be obtained by a simple formula. Wave functions, which are beyond the scope of this course are used for these atoms.

Homework Problems:

Practice exercises 6.4-6.5

Book questions pg. 322 questions 45, 47, & 49