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Using the concept of electron transitions, Bohr was able to reproduce Rydberg’s (Balmer’s) equation: Energy of initial energy level = Ei Energy of final energy level = Ef RH RH Ei = Ef = 2 ni nf2 ni = principal quantum number for the initial energy level nf = principal quantum number for the final energy level RH RH Energy of the emitted photon = h = Ei Ef = ( ) ( ) ni2 nf2 RH h = nf2 Recall: RH 1 1 = RH ( ) ni2 nf2 ni2 c= c = and therefore c 1 1 1 RH 1 1 h = RH ( ) ; By rearangement: = ( ) nf2 ni2 hc nf2 ni2 16 1 RH 1 1 = ( ) hc nf2 ni2 By substituting: RH = 2.179 x 1018 J h = 6.63 x 1034 J.s c = 3.00 x108 m/s RYDBERG’S GENERALIZED EQUATION IS OBTAINED. 1 1 1 = 1.097 x 107 m 1 nf2 ni2 17 1. SAMPLE PROBLEMS An electron in a hydrogen atom in the level n = 5 undergoes a transition to level n = 3. What is the frequency of the emitted radiation ? Energy of the emitted photon = h = Ei Ef = E5 – E3 RH RH E5 = = 52 25 RH E5 – E3 = 25 RH 9 RH RH E3 = = 32 9 25 RH - 9 RH 16 RH = = 225 225 16 RH E5 – E3 = h = 225 16 RH = 225 h 16 RH 16 2.179 x 10-18 J = = x = 2.34 x 1014 s1 225 h 225 6.63 x 1034 J . s 18 2. What is the difference in energy between the two levels responsible for the violet emission line of the calcium atom at 422.7 nm ? 1m ? m = 422.7 nm x = 4.227 x 107 m 109 nm c 3.00 x 108 m/s = = = 7.097 x 1014 s1 4.227 x 107 m E = h = (6.63 x 1034 J . s) (7.097 x1014 s1 ) = 4.71 x 1019 J 19 QUANTUM MECHANICS is a theory that applies to extremely small particles, such as electrons. DUAL NATURE OF MATTER Einstein: postulated that light has a dual nature: Wave Properties characterized by: frequency and wavelength c= Particle Properties a particle of light, called a photon has: Energy = E = h and Momentum = mass x speed = m c Louis de Broglie reasoned: - If Light (traditionally considered a Wave) exhibits Particle Properties then - Matter (traditionally considered made of Particles) exhibits Wave Properties This implies, that for a particle of matter: ENERGY of Particle of Matter = ENERGY of Wave of Matter 20 Eparticle of matter = Ewave of matter Eparticle = mc2 (Einstein) Ewave It follows: mc2 = hc For Light Particles h = mc hc =h = (Planck) h mc = For any kind of particles h = mv speed of light speed of particles NOTE: 1. Wave properties of common forms of matter are not observed because their relatively large mass results in a very short wavelengths, which cannot be detected. (in the range of 1034 m) 2. Electrons, with a very small mass produce longer wavelengths which can be detected (in the range of 109) CONCLUSION: THE ELECTRON HAS DUAL NATURE: The electron has both particle and wave properties 21 SAMPLE PROBLEMS 1. What is the de Broglie wavelength of a 145-g baseball travelling at 30.0 m/s ? m = 145 g v = 30.0 m/s =? h (6.63 x1034 J . s) = = mv m 145 g) (30.0 s Note that some units do not cancel! Recall: kg . m2 J = s This requires that the mass be expressed in kg = kg . m2 6.63 x1034 s m 0.145 kg) (30.0 s = 22 1.52 x 10 34 m cannot be detected! (too short) 2. At what speed must an electron travel to have a wavelength of 1.00 x 1011 m ? me = 9.11 x 1031kg h = mv kg . m2 h = 6.63x1034 s h v = m = 1.00 x 1011 m v=? kg . m2 6.63x1034 s v = (9.11 x 1031 kg)( 1.00 x 1011 m ) V = 7.27 x 107 m/s (about 4 times slower than the speed of light) 23 WAVE FUNCTIONS Erwin Schrodinger (1926) 1. Based on de Broglie’s work devised a theory that could be used to find the wave properties of electrons 2. Established the basis of quantum mechanics (the branch of physics that mathematically describes the wave properties of submicroscopic particles) Motion is viewed differently by Classical Mechanics and by Quantum Mechanics; Motion in Classical Mechanics: Motion in Quantum Mechanics: (for example: the path of a thrown ball) (for example: the motion of an electron in an atom) the path of the ball is given by - the electron is moving so fast and it has such a its position and its velocity at small mass, that its path cannot be predicted various times we think of the ball as moving along a continuous path Heisenberg’s Uncertainty Principle states that for particles of very small mass and moving at high speeds, it is impossible to predict: - the exact location of the particle at any particular time, - the direction in which the particle is moving In Bohr’s theory, the electron was thought of as orbiting around the nucleus, in the way the earth orbits the sun. Quantum Mechanics completely invalidates this view of the motion of the electron 24 MOTION OF THE ELECTRON AS VIEWED BY QUANTUM MECHANICS 1. We cannot describe the electron in an atom as moving in a definite orbit. 2. We can obtain the probability of finding the electron at a certain point in a H atom; we can say that the electron is likely (or not likely) to be at this position. 3. Information about the probability of finding the electron at a certain point is given by a mathematical expression called a wave function. 4. The wave function indicates that the probability of finding the electron at a certain position is high at some distance away from the nucleus 5. A wave function for an electron in an atom is called an atomic orbital. CONCLUSION AN ATOMIC ORBITAL IS A REGION IN SPACE WHERE THE PROBABILITY OF FINDING THE ELECTRON IS HIGH. (a region in space where the electron is most likely to be found) 25 25 26