Chapter 28 Atomic Physics GOALS After you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms and use it in an operational definition: atomic number energy level Bohr radius spectra quantum number laser Bohr Model State the assumptions and predictions of the Bohr model of the hydrogen atom. Deduce the concept of ionization potential. Pauli Exclusion Principle Use the Pauli exclusion principle and the four quantum numbers; n, l, m l, and m s in the periodic chart of the elements Bohr Model Problems Solve problems using the Bohr equations for hydrogen. Energy Level Diagrams Use energy level diagrams to explain the emission and absorption of radiation by atomic systems. Spectrometry and Lambert's Law Sketch a simple spectrometer, and discuss how it can be used in measuring the concentration of an element in a solution. Laser Explain the principle of laser action. PREREQUISITES Before you begin this chapter you should have achieved the goals in Chapter 4, Forces and Newton's Laws, Chapter 5, Energy, Chapter 21, Electrical Properties of Matters, and Chapter 27, Quantum and Relativistic Physics. 233
Chapter 28 Atomic Physics OVERVIEW The introduction of Bohr's model for the atom was a significant advancement for science. Although this model does not explain much of the fine detail of more complicated atoms, this model was successful in representing the hydrogen atom and explaining how electrons behave in and around the nucleus. In this chapter you will find the derivation of the Bohr model and will study the atomic effects which are examples of its success. SUGGESTED STUDY PROCEDURE Before you begin to study this chapter, be familiar with the following Chapter Goals: Definitions, Bohr Model, Bohr Model Problems, Energy Level Diagrams, and Spectronomy and Lambert's Law. More complete discussion of the terms listed under Definitions can be found in the first section of this Study Guide Chapter. Next, read text sections 28.1-28.8 and 28.10. As you read the derivation of the Bohr model in section 28.3, be sure to note the major hypotheses (Postulate 1 and 2) and follow carefully through the derivation on page 632. The Example on page 633 will help you see how the model can be used in a problem situation. At the end of the chapter, read the Chapter Summary and complete Summary Exercises 1-3 and 5-12. Now do Algorithmic Problems 1-4 and complete Exercises and Problems 3, 5, 8, 9, and 10. For more practice with the concepts introduced in this chapter, see the Examples section of this Study Guide. Now you should be prepared to attempt the Practice Test provided at the end of this Study Guide chapter. If you have difficulties with a part of the test, refer back to the appropriate text section or to this study procedure. This study procedure is outlined below. --------------------------------------------------------------------------------------------------------------------- Chapter Goals Suggested Summary Algorithmic Exercises Text Readings Exercises Problems & Problems --------------------------------------------------------------------------------------------------------------------- Definitions 28.1,28.2,28.4,28.10 Bohr Model 28.3 1,2,3 Bohr Model 28.3 5,6,7 1,2 3 Problems Energy Level 28.5 5,6,8,9 5 Diagrams Spectrometry 28.6,28.7,28.8 10,11,12 3,4 8-10 & Lambert's Law - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Pauli Exclusion 28.4 4 7,11 Principle Laser 28.10 234
DEFINITIONS ATOMIC NUMBER - The number of protons (or electrons) in an atom. The symbol Z represents the atomic number. The total electric charge of the nucleus of an atom is equal to +Z. BOHR RADIUS - The radius of the first Bohr orbit for the hydrogen atom 0.53 x 10-10 m. This represents the distance from the nucleus (proton) to the electron when the electron has a minimum energy (ground state). QUANTUM NUMBER - The discrete nature of quantum theory is characterized by four quantum numbers (n, l, m l, m s ) that determine the energy structure and quantum state of atoms. Notice that quantum numbers can have only certain discrete values, n, l, and m l are always integers; ms is either +1/2 or -1/2. ENERGY LEVEL - The allowed energy values predicted for an atom by quantum theory. In the original work of Bohr the condition of an atom in which, at least for an instant, it had a definite value for its energy was called a stationary state. Since each stationary state has a characteristic energy we also call them energy levels. SPECTRA - Each atom has a unique set of emission lines, these spectra are the "fingerprints" of atoms. The discovery by Sir Isaac Newton that white light consisted of a continuous mixture of all the colors lead to a study of the light emitted by many light sources. The study of spectra was highly developed before the formulation of the modern quantum theory of matter. LASER - An acronym for light amplifier by stimulated emission of radiation. A highly monochromatic coherent light source. ANSWERS TO QUESTIONS FOUND IN THE TEXT SECTION 28.1 Introduction Perhaps a laser is the most common, visible device that illustrates quantum properties of matter. However, since the quantum model is the best model we have there is a sense in which every phenomenon is an example of the quantum model. The excitation of atoms in the gases in a fluorescent light results in the emission of electromagnetic radiation. SECTION 28.3 Bohr's Model for the Hydrogen Atom The negative values for the energy of the hydrogen atom indicate that the electron states are bound, or finite, states for the electrons. The energy difference between the n=3 and n=1 states of hydrogen is given by E 3 - E 1 = -13.6 (1/3 2-1/1 2 ) ev = -13.6 (-8/9) ev = 12.1 ev The transition from n=5 to n=2 permits the emission of energy given by E 2 - E 5 = -13.6 (1/2 2-1/5 2 ) = -2.86 ev 235
EXAMPLES BOHR MODEL PROBLEMS 1. Calculate the wavelengths of the three lines of the Paschen (n=3) series of spectra for hydrogen. What data are given? Principle quantum number = 3. What data are implied? The usual Bohr model postulates are adequate for this system. What physics principles are involved? The basic concepts of the Bohr model for the hydrogen atom. What equations are to be used? E m - E n = -13.6 (1/m 2-1/n 2 ) ev (28.5) E = hc/λ (27.1) Note: λ E = 1240 ev/mn (30.2) Solutions The Paschen series is formed by electron transitions from higher n values to n=3. The first three Paschen lines will thus be for transitions from the n=4, 5, and 6 levels down to n=3. E 4 - E 3 = -13.6 (1/4 2-1/3 2 ) ev = 0.661eV E 5 - E 3 = -13.6 (1/5 2-1/3 2 ) ev = 0.967eV E 6 - E 3 = -13.6 (1/6 2-1/3 2 ) ev = 1.13eV Thinking about the answers None of the Paschen lines are visible but they lie in the infrared portion of the electromagnetic spectra. 2. What is the approximate quantum number n for a hydrogen orbit large enough to be observed under a microscope? What data are given? A hydrogen atom is being inspected under a microscope. What data are implied? Typical quality microscope can resolve objects about 0.0005 millimeters or 5 x10-7 meters in size. What physics principles are involved? The Bohr model for the hydrogen atom. What equations are to be used? r n = (0.53 x 10-10 ) n 2 (28.3) Solution Let us take the diameter of the orbit to be 5 x10-7 m then r = 2.5 x10-7 m 2.5 x 10-7 m = (0.53 x 10-10 ) n 2 4.7 x 10 3 = n 2 69 n Thinking about the answer The energy of an electron in the n=69 state is about 3 x 10-3 ev. The energy of a visible photon that you would need to reflect off of the electron in order to see it has an energy ~ 3eV. What other reasons can you give to explain why no one has seen an electron? 236
SPECTROMETRY AND LAMBERT'S LAW 3. A technician is mixing silver into molten glass to make partially reflecting mirrors. She knows that an 8% silver glass only transmits 88% of the incident light for a given thickness of glass. For twice the thickness of glass she wishes to have a glass transmit only 50% of the light. What percentage silver glass does she need for this glass? What data are given? For a thickness of glass x the transmission is 88% when the silver concentration is 8%. What data are implied? It is assumed the conditions for Beer's Law to be valid are met. What physics principles are involved? The basic concept of absorption of energy as it passes through matter, as expressed in Lambert's Law. What equations are to be used? Lambert's Law (28.9) I = I o exp(-µx) Beer's Law µ = αc (1) Algebraic solution Let x o be the initial thickness, T o the initial percent transmission, x 1 the second thickness, and T 1 the final transmission, C o the initial concentration and C 1 the final concentration. T o /100 = I/I o = exp( -α Cx) = exp( -α C o x o ) (2) T 1 /100 = exp( -α C 1 x ) (3) Take the logarithms to the base e of both sides of Equations (2) and (3). ln (T o /100) = -αc o x o (4) ln (T 1 /100) = -αc 1 x 1 (5) Then (ln(t o /100)) / (ln (T 1 /100)) = (C o x o ) / (C 1 x 1 ) (6) So by knowing T o, T 1, C o, x o, and x 1 you can calculate C 1. Numerical solution T o = 88%; T 1 = 50%; x o = x o ; x 1 = 2x o ; C o = 8% (ln (0.88))/(ln (0.50)) = (8% x o )/(2C 1 x o ) 0.184 = 8%/2C 1 C 1 = 8/(2(.184)) = 21.7% silver glass Thinking about the answer Notice the logarithmic dependence on concentration makes it difficult to estimate correct answers by linear extrapolation. 237
PRACTICE TEST 1. The remarkable Bohr model was designed to explain the structure and stability of the atom, and to explain the characteristic line spectra produced when the atom was excited. a. Briefly state the two postulates of the Bohr model for the atom. b. Does the Bohr model successfully predict the observed spectra of most atoms? Explain your answer. 2. The first four energy levels for the Hydrogen Atom are: -13.6 ev (ground state), -3.40 ev, -1.51 ev, and -.85 ev. a. Predict the wavelength of the radiation produced when an excited atom in the n = 4 state returns to the ground state. b. What wavelength of incident radiation will be absorbed by the hydrogen atom and produce an atom from the ground state to jump into the n = 3 state? c. Predict the energy of the n = 7 state. 3. A lab technician uses a colorimeter to conduct blood sugar studies. She uses the following calibration data: Solution Readout Current (ma) Distilled Water 300. 20.0 ppm sugar in water 30 a. When a patient's morning sample is placed in the colorimeter a readout current of 20.0 ma is obtained. What is the blood sugar concentration in this patient's morning sample? b. When the same patient's evening sample is placed in the colorimeter, the scale reading is 150 ma. What is his evening concentration of blood sugar? ANSWERS: 1. a) *Electrons are contained in stationary states whose energy level is predicted by quantum conditions. *Radiation is emitted or absorbed when an electron makes a transition from one stationary state to another b) For the relatively uncomplicated hydrogen atom, the theory is remarkably accurate. With more complex atoms containing many orbiting electrons, the theory does not predict the nature of the fine structure found in spectral analysis. Therefore the use of the theory is limited. 2. 98 nm, 100 nm, -.28 ev's 3. 23.4 ppm, 6.0 ppm 238