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The Bohr Model of the Atom ver 051712 By 1911 Ernest Rutherford had solved the problem of alpha scattering investigated by Geiger and Marsden by developing a picture of the atom in which there is a small, dense positively charged nucleus surrounded by mostly empty space in which were sprinkled some electrons. In the spring of 1912, … a young Danish visitor named Neils Bohr had turned up a Rutherford's laboratory. He had just taken his Ph.D. in physics in Copenhagen and now he was finishing up a year of study abroad. Geiger and Marsden were still busy with the last long tests of the scattering formula, but Bohr found that in Manchester Rutherford's atom-model was already taken for granted. Everyone knew, that is to say, that the atom was as good as empty, that around its outer edge were electrons which moved in the electric field of a very tiny, very massive, positively charged scattering-center, and that it's electric field fell of with the square of the distance outward. Bohr slipped easily into the life of the laboratory, talked, had ideas, made suggestions, listened with both ears, wrote a theoretical paper on the passage of alpha particles through matter and after four months returned to Copenhagen. As his visit came to its end, the positive scattering-center at the heart of the atom casually picked up a name, "the nucleus." It was an impossible task from the start (talking about how Bohr could incorporate the electron into Rutherford's view of the atom), for there was no proper way to build-in the electrons. If Bohr imagined them spotted in fixed positions around the nucleus, there could be no stability. If the repulsion of all their negative charges did not explode them outwards, the attraction of the positive nucleus must take over to collapse them inwards. If he thought of them as circling like the planets, the repulsions (which do not appear in the solar system) still came in to complicate the mathematics. Worse than this, if an electron circled it had to accelerate; if a charged body accelerated, it radiated its energy in electromagnetic waves; if a circling electron lost energy, it must spiral down into the nucleus, and once more the atom would collapse. Quotes from The Restless Atom - The Awakening of Nuclear Physics by Alfred Romer; pp 163 - 165 By 1913 Bohr had gone on to develop an atomic model that incorporated Rutherford's nucleus and adequately explained the emission spectrum of hydrogen. He did so by simply ignoring some rather important physics of moving charged particles, and by incorporating the radical ideas of Max Planck called the Quantum Theory. Bohr explained the hydrogen spectrum by saying that electrons could exit only at certain distances from the nucleus, which corresponded to certain allowed energies. An electron in its "ground state", closest to the nucleus, would have the lowest allowed energy. If the electron absorbed energy, it became "excited" and moved to an allowed state farther away from the nucleus. An electron in an excited state would be unstable and the electron would return to the ground state by shedding its "extra" energy as a photon of electromagnetic energy. In other words, the electron would give off the acquired energy as light, light either in the infrared, visible or ultraviolet portions of the electromagnetic spectrum. http://www.lbl.gov/images/MicroWorlds/EMSpec.gif Transitions to the n = 2 level produce visible light. Transitions to the ground state where n = 1 produce UV light. 1 2 3 4 5 Transitions to the n = 3 and n = 4 levels produce IR light. The Bohr model explains the hydrogen spectrum. Despite the model usually being represented by concentric circles, implying circular "orbits" taken by the electron, the Bohr model doesn't really describe a two-dimensional plane occupied by electrons. Despite the fact that atoms are indeed three-dimensional, the Bohr model has only one dimension, n, which is the distance from the nucleus to the electron which is proportional to the energy of the electron. The value n can also be used to describe the allowed energy levels. According to Bohr, electrons can exist only in discrete energy levels, and no where else in the atom. Electrons can move from one energy level to another only by gaining or losing energy. The energy of the nth energy level is given by: En = -RH / n2 RH is the Rydberg constant for hydrogen, and has the value 2.18x10-18 J. The energy of the emitted photon, and hence its wavelength, depends on the difference in energy of the two energy levels between which the electron moves. For visible light, the color corresponds to the wavelength. Example: Compute the wavelength of light emitted as an electron moves from the n=4 energy level to the n=2 energy level. n = 4 … En = -RH / n2 … E = -2.18x10-18J / 22 = -1.36x10-19J An electron moves from a higher energy level to a lower energy level and emits energy which is the difference between the two levels. n = 2 … En = -RH / n2 … E = -2.18x10-18J / 22 = -5.45x10-19J ∆E = Efinal - Einitial ∆E = -5.45x10-19J - (-1.36x10-19J) ∆E = -4.09x10-19J …. The energy difference between the two energy levels. The negative sign indicates that energy is being given off. ∆E = hc/λ ... The energy of the light is inversely proportional to its wavelength, λ. λ = hc/∆E ….h is Planck's constant, c is the speed of light λ = (6.63x10-34 Js) x (3.00x108 m/s) / (−4.09x10-19J) = -4.86 x107m The negative sign indicates that energy is being given off. λ = 486 nm … This is the wavelength of the light emitted as an electron moves from the n=4 energy level to the n=2 energy level. (In this case we ignore the negative sign that indicates that energy is being given off, and simply look at the fact that you measured a wavelength of approximately 486 nm when you looked at the hydrogen spectrum. 400 nm 500 nm 600 nm The hydrogen spectrum (© Mike Jones 2004) Postulates of the Bohr model 1. Despite the nucleus (positively charged) and electron (negatively changed) being oppositely charged they do not attract and collapse together. The electrons are stable in their energy levels and are not pulled towards the nucleus. This violates classical physics. Also an electron that is stable and doesn’t change energy level will not radiate energy. 2. If an electron moves down an energy level then quantised energy in the form of a photon will be emitted. Similarly if an electron moves to a higher energy level then the atom must absorb some quanta of energy. This explains the existence of spectral lines. 3. Angular momentum = nh/2π, where n is an integer, h is Planck's constant. This means that the angular momentum of an electron can only take on certain values and is quantised. The concept of quantized energy comes from Planck's quantum theory. The Bohr model has some severe limitations. 1. The calculations only work for hydrogen or a 1-electron ion. It does not work atoms with two or more electrons. 2. It does not predict variations in the intensity of spectral lines. 3. It does not explain the "hyperfine" spectral lines. Hyperfine lines refer to some spectral lines which actually consist of two or more closely spaced lines. In addition, there are additional, more complicated, characteristics of atom spectra that cannot be explained with the Bohr model. This tells us that the Bohr model, despite its simplicity, is inadequate for a modern-day atomic model. Electromagnetic energy and the Bohr model - Questions 1. Consider electromagnetic energy, and describe the relationship between the energy of the wave and the wavelength. 2. Which has more energy, UV light or IR light? ___________________ 3. Which has a shorter wavelength, yellow light or blue light? ___________________ 4. Which is more energetic, red light or green light? ____________________ 5. Compute the energy of an electron in the third energy level in a hydrogen atom. 6. Determine the wavelength of light (in nanometers) emitted as an electron moves from the third energy level to the second energy level. (Show your work.) 7. Explain why light is emitted when a high voltage is applied to a tube containing hydrogen gas. 8. Explain why there are only certain specific lines in the hydrogen spectrum. Compare that to the spectrum of sunlight and why we see a continuous spectrum despite the fact that the sun is composed of hydrogen. (It might help to know that the hydrogen in the sun is a plasma. (Feel free to look up plasma, the fourth state of matter.) 9. If the Bohr model is so bad - for instance, it only works for hydrogen - then why do we include it in our study of atomic theory and the arrangement of electrons in atoms?