F94 fundamentals T combustion were 1.811 g of CG2 and 0.3172 g of H20. (a) Determine the empirical formula of the compound, (b) The molar mass was found to be 170.12 g-moP1. What is the molecular formula of gallic acid? M.17 An industrial by-product consists of C, H, O, and CI. When 0.100 g of the compound was analyzed by combustion analysis, 0.0682 g of C02 and 0.0140 g of H20 were produced. The mass percentage of CI in the compound way found to be 55.0%. "What are the empirical and molecular formulas of the compound? M.18 A mixture of 4.94 g of 85.0% pure phosphine, PI I3, and 0.110 kg of CuSCy5H20 (of molar mass 249.68 gmol"1) is placed in a reaction vessel, (a) Balance the chemical equation for the reaction that takes place, given the skeletal form CuS04-5H20(s) + PIT3(g) - Cu,P2(s) + H2S04(aq) -l- H20(1). (b) Name each reactant and product, (c) Determine the limiting reactant. (d) Calculate the mass (in grams) of Cu3P2 (of molar mass 252.56 g-mol-1} produced, given that the percentage yield of the reaction is 6.31%. ATOMS: The Quantum World What Are the Key Ideas? Matter is composed of atoms. The structures of atoms can be understood in terms of the theory of matter known as quantum mechanics, in which the properties of particles and waves merge together. Why Do We Need to Know This Material? Atoms are the fundamental building blocks of matter. They arc the currency of chemistry in the sense that almost all the explanations of chemical phenomena are expressed in terms of atoms. This chapter explores the structures and properties of atoms and the periodic variation of their properties. Quantum mechanics is central to this discussion because it accounts for the structures and therefore the properties of atoms. What Do We Need to Know Already? We need to be familiar with the nuclear model of the atom and the general layout of the periodic table (Fundamentals Section B). Wc also need the concepts of kinetic and potential energy (Section A). % A #e need insight to think like a chemist. Chemical insight means that, \g\g when we look at an everyday object or a sample of a chemical, we can ▼ T imagine the atoms that make it up. Not only that, we need to be able to plunge, in our mind's eye, deep into the center of matter and discover the internal structure of atoms. To be able to understand this structure and how it relates to the chemical properties of elements, we need to understand the electronic structure of an atom, the description of how electrons arc arranged around its nucleus. As soon as we start this journey into the atom, we encounter an extraordinary feature of our world. When Rutherford proposed the nuclear model of the atom in the early twentieth century (Section B), he expected to he able to use classical mechanics, the laws of motion proposed by Newton in the seventeenth century, to describe its electronic structure. After all, classical mechanics had been tremendously successful for describing the motion of visible objects such as balls and planets. However, it soon became clear that classical mechanics fails when applied to electrons in atoms. New laws, which came to be known as quantum mechanics, had to be developed. The opening sections of this chapter follow the development of quantum mechanics and the modern nuclear model of the atom. First, we see how experiments have led to our modern concepts of the nature of matter and radiation. Next, we see how additional experiments led to a profound refinement of Dal-ton's model of an atom as an uncuttable sphere and have clarified the structure of Rutherford's nuclear model of the atom (Section B). OBSERVING ATOMS To investigate the internal structures of objects as small as atoms, we observe them indirectly through the properties of the electromagnetic radiation they emit. We then propose a model of the structure of the atom that accounts for these properties. The analysis of the electromagnetic radiation emitted or absorbed by substances is a branch of chemistry called spectroscopy. We shall see how to use atomic spectroscopy—spectroscopy applied to atoms—to determine their structures. OBSERVING ATOMS 1.1 The Characteristics of Electromagnetic Radiation 1.2 Radiation, Quanta, and Photons 1.3 The Wave-Particle Duality of Matter 1.4 The Uncertainty Principle 1.5 Wavefunctions and Energy Levels 1.6 Atomic Spectra and Energy Levels MODELS OF ATOMS 1.7 The Principal Quantum Number 1.8 Atomic Orbitals 1.9 Electron Spin 1.10 The Electronic Structure of Hydrogen THE STRUCTURES OF MANY-ELECTRON ATOMS 1.11 Orbital Energies 1.12 The Building-Up Principle 1.13 Electronic Structure and the Periodic Table THE PERIODICITY OF ATOMIC PROPERTIES 1.14 Atomic Radius 1.15 Ionic Radius 1.16 Ionization Energy 1.17 Electron Affinity 1.18 The Inert-Pair Effect 1.19 Diagonal Relationships THE IMPACT ON MATERIALS 1.20 The Main-Group Elements 1.21 The Transition Metals chapter 1 atoms: the quantum world 1.1 The Characteristics of Electromagnetic Radiation r As we saw in Section A, we can think of a field as a region of influence, like the gravitational field of the Earth. Values of the fundamental constants are listed inside the back cover of the book. A more precise value for the speed of light is 2.998 X 10B . Wavelength X . N--—>l T Amplitude FIGURE 1.1 The electric field of electromagnetic radiation oscillates in space and time. This diagram represents a "snapshot" of an electromagnetic wave at a given instant. The length of an arrow at any point represents the strength of the force that the field exerts on a charged particle at that point. The distance between the peaks Is the wavelength of the radiation, and the height of the wave above the center line is the amplitude. A ray of electromagnetic radiation consists of oscillating (time-varying) electric and magnetic fields that travel through empty space at 3.00 X 10s m-s , or at just over 670 million miles per hour. This speed is denoted c and called the ""speed of light." Visible light is a form of electromagnetic radiation, as are radio waves, microwaves, and x-rays. All these forms of radiation transfer energy from one region of space to another. 1'he warmth we feel from the Sun is a tiny proportion of the total energy it emits, and k is carried to us through space as electromagnetic radiation. One reason why electromagnetic radiation is a good tool for the study of atoms is that an electric field pushes on charged particles such as electrons. As a light ray passes an electron, its electric field pushes the electron first in one direction and then in the opposite direction, over and over again. That is, the field oscillates in both direction and strength (Fig. 1.1). The number of cycles (complete reversals of direction away from and back to the initial strength and direction) per second is called the frequency, v (the Greek letter nu), of the radiation. The unit of frequency, 1 hertz (1 Hz), is defined as 1 cycle per second: 1 Hz - 1 s"1 Electromagnetic radiation of frequency 1 Hz pushes a charge in one direction, then the opposite direction, and returns to the original direction once per second. The frequency of electromagnetic radiation that we see as visible light is close to 1015 Hz, and so its electric field changes direction at about a thousand trillion (10 ) times a second as it travels past a given point. An instantaneous snapshot of a wave of electromagnetic radiation spread through space would look like Fig. 1.1. The wave is characterized by its amplitude and wavelength. The amplitude is the height of the wave above the center line. The square of the amplitude determines the intensity, or brightness, of the radiation. The wavelength, X (the Greek letter lambda), is the peak-to-peak distance. The wavelengths of visible light arc close to 500 nm. Although 500 nm is only half of one-thousandth of a millimeter (so you might just be able to imagine it), it is much longer than the diameters of atoms, which are typically close to 0.2 nm. Different wavelengths of electromagnetic radiation correspond to different regions of the spectrum (sec Table 1.1). Our eyes detect electromagnetic radiation with wavelengths in the range from 700 nm (red light) to 400 nm (violet light). Radiation in this range is called visible light, and the frequency of visible light determines its color. "White light, which includes sunlight, is a mixture of all wavelengths of visible light. TABLE 1.1 Color, Frequency, and Wavelength of Electromagnetic Radiation Radiation type x-rays and -y-rays ultraviolet visible light violet blue green yellow orange red infrared microwaves and radio waves ':The abbreviation sf denotes the number ol significant figures in the data. The frequencies, wavelengths, and energies are typical values; they should nor be regarded as precise. Frequency Wavelength Energy per (1014 Hzj (nm, 2 sjf)* photon (io-"j) all)3 8.6 350 5.7 7.1 420 4.7 6.4 470 4.2 5.7 530 3.8 .5.2 580 3.4 4.8 620 3.2 4.3 700 2.8 3.0 1000 2.0 SlO-3 >3 X 106 S10 3 1.1 the characteristics of electromagnetic radiation FIGURE 1.2 (a) ShorUwavelength radiation: the vertical arrow shows how the electric field changes markedly at five successive instants, (b) For the same five instants, the electric field of the long-wavelength radiation changes much less. The horizontal arrows in the uppermost images show that in each case the wave has traveled the same distance. Short-wavelength radiation has a high frequency, whereas long-wavclcngth radiation has a low frequency. (a) Shore wavelength, high frequency fb) Long wavelength, low frequency Now imagine the wave in Fig. 1.1 zooming along at its actual speed, the speed of light, c. If the wavelength of the light is very short, very many complete oscillations pass a given point in a second (Fig. 1.2a). If the wavelength is long, the light still travels at the speed c, but fewer complete oscillations pass the point in a second (Fig. 1.2b). A short wavelength therefore corresponds to high-frequency radiation and a long wavelength corresponds to low-frequency radiation. The precise relation is Wavelength X frequency = speed of light, or kv — c For instance, suppose we want to find the wavelength of blue light of frequency 6.4 x 1014Hz. Then, From \ ■ 2.998 X 10s m-s"1 2.998 X 1 08 6.4 X I01' 6.4 X 1014 m = 4.7 X 10" or about 470 nm. Blue light, which has a relatively high frequency, has a shorter wavelength than that of red light, which has a frequency near 700 nm. Self-Test 1.1 A Calculate the wavelengths of the light from traffic signals as they change. Assume that the lights emit the following frequencies: green, 5.75 X I ()14 Hz; yellow, 5.15 X 1014 Hz.; red, 4.27 X 1014 Hz. [Answer: Green, 521 nm; yellow, 582 nm; red, 702 nmj Self-Tfst 1.1B What is the wavelength of a radio station transmitting at 98.4 MHz? As far as we know, there is neither an upper limit nor a lower limit to the wavelength of electromagnetic radiation (Fig. 1.3). Ultraviolet radiation is radiation at higher frequency than violet light; its wavelength is less than about 400 nm. This damaging component of radiation from the Sun is responsible for sunburn and tanning, but it is largely prevented from reaching the surface of the Farth by the ozone layer. Infrared radiation, the radiation we experience as heat, has a lower frequency and longer wavelength than red light; its wavelength is greater than about 800 nm. Microwaves, which are used in radar and microwave ovens, nave wavelengths in the millimeter-to-centimeter range. The color of light depends on its frequency and wavelength; long-wavelength radiation has a lower frequency than that of short-wavelength radiation. Cosmic rays FIGURE 1.3 The electromagnetic spectrum and the names of its regions. The region we call "visible light" occupies a very narrow range of wavelengths. The regions are not drawn to scale. Modern theories suggest that our concept of space breaks down on a scale of 10-34 m, so that may constitute a lower bound to electromagnetic radiation wavelength. chapter 1 atoms: the quantum world 2 4 6 Wavelength (jim) FIGURE 1.4 The intensity of radiation emitted by a heated black body as a function of wavelength. As the temperature increases, the total energy emitted (the area under the curve) increases sharply, and the maximum intensity of emission moves to shorter wavelengths. (To obtain the energy in a volume Vand at wavelengths A and \ + AX, multiply the energy density by Vand A\.) The name Stefan-Boltzmann law recognizes Ludwig Boltzmann's theoretical contribution. § 2 1.2 Radiation, Quanta, and Photons As an object is heated to high temperatures, it glows more brightly—the phenomenon of incandescence—and the color of light it gives off changes from red through orange and yellow toward white. Those are qualitative observations. To study the effect quantitatively, scientists had to measure the intensity of radiation at each wavelength and repeat the measurements at a variety of different temperatures. These experiments caused one of the greatest revolutions that has ever occurred, in science. Figure 1.4 shows some of the experimental results. The "hot object" is known as a black body (even though it might: be glowing white hot!). The name signifies that the object does not favor one wavelength over another in the sense of absorbing a particular wavelength preferentially or emitting one preferentially. The curves in Fig. 1.4 show the intensity of black-body radiation, the radiation emitted at different wavelengths by a heated black body, for a series of temperatures. Two crucial pieces of experimental information that had to be incorporated into any model of black-body radiation were discovered in the late nineteenth century. In 1879, Josef Stefan investigated the increasing brightness of a black body as it is heated and discovered that the total intensity of radiation emitted over all wavelengths increases as the fourth power of the temperature (Fig. 1.5). This quantitative relation is now called the Stefan-Boltzmann law and is usually written I vali 1.2 RADIATION, QUANTA, AND PHOTONS Power emitted Surface area constant X T4 (2) where T signifies an absolute temperature, one reported on the Kelvin scale (Appendix IB). The experimental value of the constant is 5.67 X 10~8 W-m_2-K , where W denotes watts (1W-1 J-s"1). A few years later, in 1893, Wilhelm Wien examined the shift in color of black-body radiation as the temperature increases and discovered that the wavelength corresponding to the maximum in the intensity, Xmax, is inversely proportional to the temperature, Xmax 1/T, and therefore that \max X T is a constant (Fig. 1.6). This quantitative result is now called Wien's law and is normally written 1 . (3) 0 200 400 600 800 1000 Temperature (K) FIGURE 1.5 The total intensity of radiation emitted by a heated black body increases as the fourth power of the temperature, so a body at 1000 i< emits more than 1 20 times as much energy as that emitted by the same body at 300 K. 0 0.005 0.01 lyTcmperature, 1/T (K_1) FIGURE 1.6 As the temperature is raised (1/T decreases), the wavelength of maximum emission shifts to smaller values. Inhere c2 is called the second radiation constant. The empirical (experimental) value ofc2is 1.44 X 10"2 Km. IfXAMPLE 1.1 Sample exercise: Determining the temperature of the surface of a star The maximum intensity of solar radiation occurs at 490. nm. "What is the temperature of the surface of the Sun? BpLUTION We can use Wien's law to determine the surface temperature of stars treated as hot black bodies: From TKmgx = jc2 written as T = 5km 1.44 x 10~lK-] 1,44 x 10"2 , 7-7^-= 5.88 X 103K 5 X 4.90 X 10 ' 5 x 4.90 x 10"7 That is, the surface temperature of the Sun is about 6000 K. Self-Test 1.2A In 1965, electromagnetic radiation with a maximum at 1.05 mm (in the microwave region) was discovered to pervade the universe. What is the temperature of "empty'" space? [Answer: 2.74 K] Self-Test 1.2B A red giant is a late stage in the evolution of a star. The average wavelength maximum at 700. nm shows that a red giant cools as it dies. What is the surface temperature of a red giant? For nineteenth-century scientists, the obvious way to account for the laws of black-body radiation was to set up a model of electromagnetic radiation in terms of waves and to use classical physics to derive its characteristics. However, much to their dismay, they found that the characteristics they deduced did not match their observations. Worst of all was the ultraviolet catastrophe: classical physics predicted that any black body at any nonzero temperature should emit intense ultraviolet radiation and even x-rays and 7-rays! According to classical physics, any warm object would devastate the countryside with high-frequency radiation. Even a human body at 37°C would glow in the dark. There would, in fact, be no darkness. The suggestion that resolved the problem came in 1900 from the German physicist Max Planck. He proposed that the exchange of energy between matter and radiation occurs in quanta, or packets of energy. Planck focused his attention on the hot, rapidly oscillating atoms of the black body. His central idea was that an atom oscillating at a frequency v can exchange energy with its surroundings only in packets of magnitude E = hv The constant b, now called Planck's constant, has the value 6.626 X 10~34 J-s. If the oscillating atom releases an energy E into the surroundings, radiation of frequency v = E/h will be detected. Planck's hypothesis implies that radiation of frequency v can be generated only if an oscillator of that frequency has acquired the minimum energy required to start oscillating. At low temperatures, there is not enough energy available to stimulate oscillations at very high frequencies, and so the object does not generate high-frequency, ultraviolet radiation. As a result, the intensity curves in Fig. 1.4 die away at high frequencies (short wavelengths} and the ultraviolet catastrophe is avoided. In contrast, in classical physics it was assumed that an oscillator could oscillate with any energy and therefore, even at low temperatures, high-frequency oscillators could contribute to the emitted radiation. The hypothesis is quantitatively successful, too, because Planck was able to use his proposal to derive the Stefan-Boltzmann and Wien laws. He was also able to calculate the variation of The word quantum comes from the Latin for amount-—literally, "How much?" chapter i atoms: the quantum world 1.2 radiation, quanta, and photons Ultraviolet radiation source FIGURE 1.7 When a metal is illuminated with ultraviolet radiation, electrons arc ejected, provided the frequency is above a threshold frequency that is characteristic of the metal. We say that a property y varies linearly with x if the relation between y and x can be written y = b + mx, where b and m are constants. See Appendix IE. / \ / \ Kinetic energy of Energy electron, | brought 7 V by Knergy nee photon, hv to remove electron, O FIGURE 1.8 In the photoelectric effect, a photon with energy hv strikes the surface of a metal and its energy is absorbed by an electron. If the energy of the photon is greater than the work function, (uppercase phi). If the energy of a photon is less than the energy required to remove an electron from the metal, then an electron will not be ejected, regardless of the intensity of the radiation. However, if the energy of the photon, hv, is greater than , then an electron is ejected with a kinetic energy, V,K = jmtv2, equal to the difference between the energy of the incoming photon and the work function: £K = hv — O (Fig. 1.8). It follows that kinetic energy ot ejecreri electron hv energy supplied energy required to eject electron (5) This expression tells us that a plot of the kinetic energy of the ejected electrons against the frequency of the radiation should be a straight line of slope h, the same for all metals, and an extrapolated intercept with the vertical axis at — , different FIGURE 1.9 When photons strike a metal, no electrons are ejected unless the incident radiation has a frequency above a value characteristic of the metal. The kinetic energy of the ejected electrons varies linearly with the frequency of the incident radiation. The inset shows the relation of the slope and the two intercepts to the parameters in Eq. 5. Frequency of incident radiation, v for each metal. The intercept with the horizontal axis (corresponding to zero kinetic energy of the ejected electron) is at $*lh in each case. We can now interpret the experimental observations of the photoelectric effect in light of Einstein's theory: 1 An electron can be driven out of the metal only if it receives at least a certain minimum energy, <£, from the photon during the collision. Therefore, the frequency of the radiation must have a certain minimum value if electrons are to be ejected. This minimum frequency depends on the work function and hence on the identity of the metal (Fig. 1.9). 2 Provided a photon has enough energy, a collision results in the immediate ejection of an electron. 3 The kinetic energy of the electron ejected from the metal increases linearly with the frequency of the incident radiation according to Eq. 5. The photoelectric effect strongly supports the view that electromagnetic radiation consists of photons that behave like particles. However, there is plenty of evidence to show that electromagnetic radiation behaves like waves! The most compelling evidence is the observation of diffraction, the pattern of high and low intensities generated by an object in the path of a ray of light {Fig. 1.10). A diffraction pattern results when the peaks and troughs of waves traveling along one path interfere with the peaks and troughs of waves traveling along another path. If the peaks coincide, the amplitude of the wave (its height) is enhanced; we call this enhancement constructive interference (Fig. 1,11a). If the peaks of one wave coincide with the troughs of another wave, the amplitude of the wave is diminished by destructive interference (Fig. 1.11b). This effect is the basis of a useful technique for studying matter. Eor example, x-ray diffraction is one of the most important tools for studying the structures of molecules (see Major Technique 3, following Chapter 5). Let's review the evidence for the nature of electromagnetic radiation. The results of some experiments (the photoelectric effect) compel us to the view that electromagnetic radiation is particlelike. Those of other experiments (diffraction) compel us equally firmly to the view that electromagnetic radiation is wavelike. Thus we are brought to the heart of modern physics. Experiments oblige us to accept the wave-particle duality of electromagnetic radiation, in which the concepts of waves and particles blend together. In the wave model, the intensity of the radiation is proportional to the square of the amplitude of the wave. In the particle model, intensity is proportional to the number of photons present at each instant. Incident light FIGURE 1.10 In this illustration, the peaks of the waves of electromagnetic radiation are represented by orange lines. When radiation coming from the left (the vertical linos) passes through a pair of closely spaced slits, circular waves are generated at each slit. These waves interfere with each other. Where they interfere constructively (as indicated by the positions of the dotted lines), a bright line is seen on the screen behind the slits; where the interference is destructive, the screen is dark. 8 chapter 1 atoms: the quantum world 1.3 the wave-particle duality of ma iter 9 FIGURE 1.11 (a) Constructive interference. The two component waves (left) are "in phase" in the sense that their peaks and troughs coincide. The resultant (right) has an amplitude that is the sum of the amplitudes of the components. The wavelength of the radiation is not changed by interference, only the amplitude is changed, (b) Destructive interference, the two component waves are "out of phase" in the sense that the troughs of one coincide with the peaks of the other. The resultant has a much lower amplitude than either component. Studies of black-body radiation led to Planck's hypothesis of the quantization of electromagnetic radiation. The photoelectric effect provides evidence of the particulate nature of electromagnetic radiation; diffraction provides evidence of its wave nature. il.3 The Wave-Particle Duality of Matter If electromagnetic radiation, which for a long time had been regarded as wavelike, has a dual character, could it be that matter, which since Dalton's day had been regarded as consisting of particles, also has wavelike properties? In 1925, the French scientist Louis de Broglie proposed that all particles should be regarded as having wavelike properties. He went on to suggest that the wavelength associated with the "matter wave" is inversely proportional to the particle's mass, m, and speed, v, and that X - — (6a) gth EXAMPLE 1.2 Interpreting a light ray in terms of photons A discharge lamp rated at 25 W (1 W = 1 J's-1) emits yellow light of wavel 580 run. How many photons of yellow light does the lamp generate in 1.0 s? STRATEGY We should expect a large number of photons because the energy of a single photon is very small. The number of photons generated in a given time interval is the total energy emitted in that interval divided by the energy of a single photon. Wc calculate the energy of a photon from its wavelength by using Eq. 1 to convert the given wavelength into a frequency and Eq, 4 to convert frequency into energy. To reduce rounding errors, it is good practice to work through a calculation symbolically as far as possible and to introduce numerical data at the last possible stage. SOLUTION Step 1 Find the frequency of the light from Eq. 1 From \v = c, Step 2 Calculate the energy, Ephoton> erf eacn photon of wavelength A. From F. - hv, Ep] Step 3 Divide the total energy, Etotal, by the energy of one photon to find the number of photons, N, that account for the total energy. Step 4 The total energy emitted by the lamp is its power (P in watts, or joules per second) multiplied by the time, t, for which it is turned on. Step 5 Express the wavelength in meters and insert the data. From N - Substitute Etata, = PX t: he X (hc/X) N US* he \Pt (.5.80 x l(T7m) x (25J-s"1) x (1.0s) (6.626 x l(r34Js) x (3.00 x 10» m-s ') 5.80 x 10~7 x 25 x 1.0 ~ 6.626 x 10"34 x 3.00 x 10s = 7.3 x 10" This result means that, when you turn on the lamp, it generates about 1()20 photons of yellow light each second. As predicted, this is a very large number. SELF-test 1.4A Another discharge lamp produces 5.0 J of energy per second in the blue region of the spectrum. How many photons of blue (470 nm) light would the lamp generate if it were left on for 8.5 s? [Answer: 1.0 x 10201 Self-Test 1.4B In 1.0 s, a lamp that produces 25 J of energy per second in a certain region of the spectrum emirs 5.5 x 1019 photons of light in that region. What is the wavelength of the emitted light? The product of mass and speed is called the linear momentum, p, of a particle, and so this expression is more simply written as the de Broglie relation: \ = - (6b)* P It is easy to see why the wavelike properties of particles had not been noticed. According to the de Broglie relation, a particle of mass 1 g traveling at 1 m-s_l has a wavelength of h 6.626 X 10-34 J-s p mv (ix 10-3 kg) x (1 m-s"1) 7 x 10~31 m 7 x 10" kg-m-s This wavelength is undetectably small; the same is true for any macroscopic (visible) object traveling at normal speeds. The wavelike character of electrons was detected by showing that they could be diffracted. The experiment was first performed in 1.925 by two American scientists, Clinton Davisson and Lester Germer, who directed a beam of fast electrons at a single crystal of nickel. The regular array of atoms in the crystal, with centers separated by 250 pm, acts as a grid that diffracts waves; and a diffraction pattern was observed (Fig. 1.12). Electron diffraction is now an important technique for determining the structures of molecules and exploring the structures of solid surfaces. [tXAMPLE 1.3 Estimating the wavelength of a particle Estimate the wavelength of (a) a proton moving at 1/100 the speed of light; (b) a mar-He of mass 5.00 g traveling at 1.00 nvs-1. strategy Use the de Broglie relation, Eq. 6a. The mass of a proton and the speed of light are given inside the back cover. Expect the macroscopic particle, the marble, to have a very short wavelength. Remember to express all quantities in kilograms, pleters, and seconds, and use I J = 1 kg-m2-s~2. SOLUTION (a) The mass of a proton is 1.673 x 10~27 kg, and the speed of light is 2.998 x 108 m-s '; therefore, from X : 6.626 x 10~34 J-; (1.673 x 10_27kg) x (2.998 x 10s "M/lOO 1.32 x 10-"^g,m;S"= 1.32 x 10-kg x (m-s 1) FIGURE 1.12 Davisson and Germer showed that electrons produce a diffraction pattern when reflected from a Crystal. G. P.Thomson, working in Aberdeen, Scotland, showed that they also produce a diffraction pattern when they pass through a very thin gold foil. The latter is shown here. C. P. Thomson was the son of J. J. Thomson, who identified the electron [Section B). Both received Nobel prizes: j. J. for showing that the electron is a particle and C. P. for showing that it is a wave. 10 chapter 1 atoms: the quantum world 1.5 wavefunctions and enf.rcy levels i! [I (a) (b) FIGURE 1.13 A representation of the uncertainty principle, (a) The location of the particle is ill defined, and so the momentum of the particle (represented by the arrow) can be specified reasonably precisely, (b) The location of the particle is well defined, and so the momentum cannot be specified very precisely. This wavelength corresponds ro 0.132 pm. (b) The mass of the marble is 5.00 X 10 3 kg; so __6.626 X 10"34J-s_ _9J X~ (5.00 X tO-3 kg) X (1.00 m-s-1) ~ ^ X 10 m As expected, the wavelength of the marble is undetectably small. However, the wavelength of the rapidly moving proton is not very much smaller than the diameter of an atom (about 200 pm). SELF-TEST 1.5A Calculate the wavelength of an electron traveling at 1/1000 the speed of light. [Answer: 2.43 ran] Self-Test 1k5B Calculate the wavelength of a rifle bullet of mass 5.0 g traveling at twice the speed of sound (the speed of sound is 331 m-s-1). Electrons (and matter in general) have both wavelike and particlelike properties. 1.4 The Uncertainty Principle The wave-particle duality not only changes our understanding of electromagnetic radiation and matter, but also sweeps away the foundations of classical physics. In classical mechanics, a particle has a definite trajectory, or path on which location and linear momentum are specified at each instant. However, we cannot specify the precise location of a particle if it behaves like a wave: think of a wave in a guitar string, which is spread out all along the string, not localized at a precise point. A particle with a precise linear momentum has a precise wavelength; but, because it is meaningless to speak of the location of a wave, it follows that we cannot specify the location of a particle that has a precise linear momentum. The difficulty will not go away. Wave-particle duality denies the possibility of specifying the location if the linear momentum is known, and so we cannot specify the trajectory of particles. If we know that a particle is here at one instant, we can say nothing about where it will be an instant later! The impossibility of knowing the position with arbitrarily great precision if the linear momentum is known precisely is an aspect of the complementarity of location and momentum, that if one property is known the other cannot be known. The Hciscnberg uncertainty principle expresses this complementarity quantitatively: it states that, if the location of a particle is known to within an uncertainty Ax, then the linear momentum parallel to the x-axis can be known only to within an uncertainty Ap, where ApAx > jfi (7)* The symbol h, which is read "h bar," means h/lir, a useful combination that is found widely in quantum mechanics. From inside the back cover, we see that h = 1.054 57 X 10~j4 J-s. Equation 7 tells us that, if the uncertainty in position is very small (Ax very small), then the uncertainty in linear momentum must be large, and vice versa (Fig. 1.13). The uncertainty principle has negligible practical consequences for macroscopic objects, but it is of profound importance for electrons in atoms and for a scientific understanding of the nature of the world. EXAMPLE 1.4 Using the uncertainty principle Estimate the minimum uncertainty in (a) the position of a marble of mass 1.0 g given that its speed is known to within ±1.0 nun's and (h) the speed of an electron confined to within the diameter of a typical atom (200. pm). STRATEGY We expect the uncertainty in the position of an object as heavy as a marble to be very small but the uncertainty in the speed of an electron, which is very light and confined to a small region, to be very large. In each case, be sure to use units that Latch the units of h, (a) We calculate the uncertainty Ap from mAv, where Av is the Hrticertainty in the speed; then we use Eq. 7 to estimate the minimum uncertainty in . sition, fcX) along the direction of its travel from ApAx = \h (the minimum value of {foe product of uncertainties). Because the reported uncertainty in speed is given to plus at minus a stated value, the value of Av is equal to twice that value, (b) We assume Ax mbbe the diameter of the atom and use Eq. 7 to estimate Ap; by using the mass of the electron in Table B.l, we find Av from mAv = Ap. SOLUTION (a) First we convert mass and speed into SI base units. The mass, m, is 10 X ]0'3 kg, and the uncertainty in the speed, Av, is 2 X (1.0 X 10~3 m-s-1). The minimum uncertainty in position, Ax, is i ft from ApAx ^ jA, Ax = -—— ■ 2m Av Substitution of the data then gives 1.05457 X 10"^ J-s_ 2 X (1.0 X 10_3kg) x (2.0 X I0_3m-s~1) 1.05457 X 10-34 J-s , M =-—-=-—-r -—--7 = 2.6 X 10 23 m 2 X 1.0 X 10~3 X 2.0 X 10"3 kg-m-s"1 Note that in the last step we have used 1 J = 1 kg-m2-s~2. As predicted, this uncertainty is very small. (b) The mass of the electron in Table B.1 is .9.109 X 10~31 kg; the diameter of the atom is 200. X 10~12 m, or 2.00 X 10~]0 m. The uncertainty in the speed, At', is equal to Ap/m: Ap _ h From ApAx > jh, Av ■ Av ■ 2m Ax _1.0.5457 X 10~34J-s 2 X (9.109 X 10 31 kg) X (2.00 X 10"10 m) 1.05457 X 10" 2 X 9.109 X 10"31 X 2.00 X 10 10 kg-m ' = 2.89 X 105m-s_1 As predicted, the uncertainty in the speed of the electron is very large, nearly + 150 km-s"1. pELF-TEST 1.6A A proton is accelerated in a cyclotron to a very high speed that is , known to within 3.0 X 102 km-s~1. What is the minimum uncertainty in its position? \Answer: 0.10 pm] ÍSelf-Test 1.6B The police are monitoring an automobile of mass 2.0 t (1 t = 103 kg) speeding along a highway. They are certain of the location of the vehicle only to within I m. What is the minimum uncertainty in the speed of the vehicle? The location and momentum of a particle are complementary; that is, both the location and the momentum cannot be known simultaneously with arbitrary precision. The quantitative relation between the precision of each measurement is described by the Heisenberg uncertainty principle, Eq, 7, 1.5 Wavefunctions and Energy Levels Twentieth-century scientists had to revise the nineteenth-century description of matter to take into account wave-particle duality. One of the first people to formulate a successful theory (in 1927) was the Austrian scientist Erwin Schrodmger. Because particles have wavelike properties, we cannot expect them to behave like pointlike objects moving along precise trajectories. Schrodinger's approach was to replace the precise trajectory of a particle by a wavefunction, v|/ (the Greek letter psi), a mathematical function with values that vary with position. There is