An atom s energy levels

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Second First (ground state) Nucleus

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Figure 28-9 These decreasingheight steps are analogous to the allowed energy levels in an atom Note how the difference in energy between adjacent energy levels decreases as the energy level increases

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Section 281 The Bohr Model of the Atom

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Ephoton 1 Ephoton 2 Ephoton 3 Ephoton 1

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E1 E1 E2 Ephoton 3

When the atom makes the transition from its initial energy level, Ei, to its final energy level, Ef, the change in energy, Eatom, is given by the following equation Eatom Ef Ei

Ephoton 2

As shown in Figure 28-10, the change in energy of the atom equals the energy of the emitted photon Ephoton or Eatom Ef Ei

Ground state

Ephoton

Figure 28-10 The energy of the emitted photon is equal to the difference in energy between the initial and final energy levels of the atom

The following equations summarize the relationships between the change in energy states of an atom and the energy of the photon emitted Energy of an Emitted Photon Ephoton hf, or Ephoton Eatom

The energy of an emitted photon is equal to the product of Planck s constant and the emitted photon s frequency The energy of an emitted photon also is equal to the loss in the atom s energy

Predictions of the Bohr Model

A scientific theory must do more than present postulates; it must allow predictions to be made that can be checked against experimental data A good theory also can be applied to many different problems, and it ultimately provides a simple, unified explanation of some part of the physical world Bohr used his theory to calculate the wavelengths of light emitted by a hydrogen atom The calculations were in excellent agreement with the values measured by other scientists As a result, Bohr s model was widely accepted Unfortunately, the model only worked for the element hydrogen; it could not predict the spectrum of helium, the next-simplest element In addition, there was not a good explanation as to why the laws of electromagnetism should work everywhere but inside the atom Not even Bohr believed that his model was a complete theory of the structure of the atom Despite its shortcomings, however, the Bohr model describes the energy levels and wavelengths of light emitted and absorbed by hydrogen atoms remarkably well Development of Bohr s model Bohr developed his model by applying Newton s second law of motion, Fnet ma, to the electron The net force is described by Coulomb s law for the interaction between an electron of charge q that is a distance r from a proton of charge q That force is given by F Kq2/r2 The acceleration of the electron in a circular orbit about a much more massive proton is given by a v2/r, where the negative sign shows that the direction is inward Thus, Bohr obtained the following relationship:

Kq2 r2 mv 2 r

In the equation, K is the constant from Coulomb s law and has a value of 90 109 N m2/C2

28 The Atom

Next, Bohr considered the angular momentum of the orbiting electron, which is equal to the product of an electron s momentum and the radius of its circular orbit The angular momentum of the electron is thus given by mvr Bohr postulated that angular momentum also is quantized; that is, the angular momentum of an electron can have only certain values He claimed that the allowed values were multiples of h/2 , where h is Planck s constant Using n to represent an integer, Bohr proposed that mvr nh/2 Using Kq2/r2 mv 2/r and rearranging the angular momentum equation, v nh/2 mr, Bohr found that the orbital radii of the electrons in a hydrogen atom are given by the following equation Electron Orbital Radius in Hydrogen rn

h2n2 4 2Kmq2

The radius of an electron in orbit n is equal to the product of the square of Planck s constant and the square of the integer n divided by the quantity four times the square of , times the constant K, times the mass of an electron, times the square of the charge of an electron