Jumat, 09 Oktober 2009

The Bohr Model

The Bohr model of the atom is often called the planetary model. As shown in Figure 1 for a hydrogen atom, the Bohr model envisions the nucleus of the atom occupying a fixed position at the center of an atomic system with the electron revolving around the nucleus in the same way that a planet revolves around the sun.

Figure 1

The Bohr Model of a Hydrogen Atom

In fact, the planetary model pre-dates Bohr. The model bears his name because of his interpretation of the emission spectrum of hydrogen: If a small amount of hydrogen gas is confined within a glass tube and subjected to a high voltage, it emits light, some of which falls in the visible region of the electromagnetic spectrum. When this light is passed through a prism, it becomes apparent that its pink-violet hue is a composite of four discrete colors-red, green, blue, and purple. Figure 2 illustrates the basic experiment involved in measuring an atomic emission spectrum.

Bohr knew that the emission of light was the way the atoms released the energy they had absorbed when the high voltage was applied. He made a connection between the discrete colors emitted and the discrete energy levels that were available to the electron. He postulated that each color corresponded to the transition of an electron from one energy state to another, lower energy state. He indexed the energy states of the electron with the letter n, the value of n being 1 for the state where the electron had the lowest energy (the ground state), 2 for the next higher energy state, 3 for the third electronic energy state, etc. Bohr's index n became what we now refer to as the principal quantum number.

Bohr's success in rationalizing the emission spectrum of hydrogen led to the general acceptance of the planetary model of the atom. It has since been shown that this model is overly simplistic. Despite its shortcomings, the simplicity of the Bohr model appeals to organic chemists because it lends itself to a simple pictorial description of molecular structure.

Bohr devised Equation 1 to describe the emission spectrum of hydrogen:

In this equation ni is the numerical index of the level of the electron just before it emits light, while nf is the numerical index of the level where the electron winds up after the emission of light. (Note that nf does not necessarily equal 1.) DE is the difference in energy between the two energy levels in question, while RH is a proportionality constant. Since DE is related to the wavelength, l, of light by the equation DE = hc/l, Equation 1 may be written in the equivalent form of Equation 2.

In order to do the algebra required to calculate the value of l for a given transition, it is necessary to know the values of the constants RH, h, and c. They are 2.18 x 10 -18 J, 6.63 x 10-34 J sec, and 3.00 x 108 m/sec, respectively. Substituting these values into Equation 2 produces

For the transition from ni = 6 to nf = 2, DE = - 4.84 x 10-19 J. The negative sign indicates that the energy of the final state is less than that of the initial state. Converting this value of DE into a wavelength yields l = -4.10 x 10 -7 meters or 410 nm. Light of this wavelength is purple. We have ignored the negative sign since the value of l must be positive.

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