Quantum Behavior of Atoms Assignment

Quantum Behavior of Atoms Emission spectra and photons In Chapter 4, we discussed how light (specifically infrared light) can be absorbed by molecules such as greenhouse gases. Compared to absorption, emission is the opposite process wherein energy is released from matter in the form of light. This occurs when matter that is oscillating (has energy) at a From Nothing to Everythi given frequency releases that energy (it changes oscillation frequency or stops moving). In molecules, this is seen through the change in oscillation in the bond as discussed in Chapter 4. For individual atoms, however, there are aroundcontinuous which to spectrum oscillate.the Despite this difference, whenthe atoms that, rather thanno thebonds completely eye perceives in a rainbow, sun’s spectrum release energy, they do so in the form of light at very specific frequencies (and therefore specific (Fig. 6-2) actually contains many very thin, dark lines superimposed on the continuous spectrum. Ten wavelengths).years Thislater, wasthe discovered by ofenergizing (gases that contain wavelengths more thanlow-density 600 of these elemental lines were gases measured by Joseph Fraunhofer. single elements) to the point of glowing. When the light is observed, it was seen that different elements emitted different frequencies of light in a spectrum of bright lines. Only specific frequencies were spectrum Figure 6-2. The visible of the sun. It and is filled observed instead of a continuous spectrum (as you would see with white light, rainbows, the with dark (“Fraunhofer”) lines superposed blackbody emission of Earth). on the continuous blackbody spectrum. spectrum of hydrogen (see [Source: hesperia.gsfc.nasa.gov] In the 1880’s, Johann Balmer recognized that the visible light emission Figure 7.0), known by then to be the Wavelength lightest andλprobably (nm) → simplest element, follows a pattern in which the lines are more closely spaced at shorter wavelengths (i.e., from red to violet). Moreover, each Later has in that century, physicists thatemission a low-density, hot gas7.0), (in aaglass emits light with a chemical element a distinctive pattern offound colored lines (Figure sort tube) of “fingerprint” spectrum of bright lines, with no at all.phenomenon They were able study – single of the element. For early spectroscopists thiscontinuous proved tospectrum be a difficult to to explain why isolated Each chemical element has elements in this way, determining their emission-line spectra. would an atom only emit a few, very distinct wavelengths of light? While it may have made sense thata distinctive of colored emission lines (see Fig.for 6-3), a sorttime of “fingerprint” of the element. In the 1880’s, each elementpattern would have a unique emission spectrum, a long the exact frequencies of emitted Johann Balmer found that the spectrum of hydrogen, known by then to be the lightest and probably light from each element proved impossible to predict or explain. These observations would be the basis th th simplestin element, follows a pattern in which the into linesthe areearly more 20 closely spacedwould at shorter wavelengths (i.e century century) propose on which physicists the coming years (late 19 red to violet; see Fig. atomic model physicists of the early 20th century their atomic from theories. They realized that6-3). any The successful of the atom must explain therealized patternsthat of any successful model of the atom must explain this pattern of wavelengths. wavelengths emitted by different elements. Hydrogen (H) Sodium (Na) Helium (He) Neon (Ne) Mercury (Hg) | | | | | | | 650 600 550 500 450 400 350 Figure 6-3. Emissionline spectra of five elements. [Note: the eye sees wavelengths between 380 and 430 nm to be violet, not the dark blue of this reproduction.] Note that wavelength decreases to the right, opposite to Fig. 6-2; there is no standard convention. ← Wavelength λ (nm) Figure 7.0: spectra for variousAtom elements The Bohr Model ofEmission the Hydrogen In 1912, Niels Bohr, a young Danish theorist, endeavored to figure out how atoms generate emission 1 lines. The best guess at the time was that negatively charged electrons orbit the positively charged nucleus. The energies involved in the orbits would naturally depend on the strength of the attractive electric force between each electron and the nucleus, which in turn would depend on the number of It is important to note that when light is absorbed, the wavelength, frequency, and energy of that light are not changed. In actuality, it is the intensity of the light that is decreases during absorption, which makes it dimmer. Lowering the intensity of a light wave results in a lower amplitude, but the frequency and wavelength of that wave remains the same. Every time light transfers energy to matter (read: is absorbed), it is transferring exactly hνlight. Similarly, every time light is released from matter (read: is emitted), it is released in multiples of hνlight. This is one of ways in which we ascribe “particle-like” nature to light – despite the fact that light is an electromagnetic wave (definitely not a particle), it does have the particle-like property of transferring fixed, discrete quantities of energy known as photons. The energy of a photon of light with given frequency ν is Ephoton = hνphoton. A photon is a quantum of light, meaning that it represents the smallest possible discrete “unit” of light. As a result, photons cannot be subdivided into smaller units and have discrete energies (Ephoton = hνphoton). To be clear, a photon is not an object; rather, it is the total amount of energy (a quantum of energy) that is transferred between light and matter when light is absorbed by the matter or emitted from matter. This quantity is always the same (quantized) for light of a given frequency, 𝜈"#$%$& . As we saw in Chapter 4, it is the resonance between the oscillating frequency of the light and the matter (resonance) that leads to energy being transferred, and it takes many oscillations of the electric field for this energy to be transferred. Quantum mechanics is the branch of physics that describes the nature and behavior of objects that, like photons, have quantized energy values. This is distinct from macroscopic objects and the world of classical physics, which have no limitations on the possible amount of energy they can carry. As we will see, the behavior of photons and subatomic particles (like electrons) differs in many fundamental ways from the behavior of the objects we interact with on a daily basis. For example, the interactions of light and matter may strike you as somewhat odd. Why would light only be absorbed by matter at certain resonant frequencies? Why do atoms only emit specific energies (and wavelengths) of light? Why is there such a big difference between the classical and quantized worlds? The first glimpse at answering these questions came from a set of experiments conducted between 1801 and 1927, and a phenomenon known as interference. Wave interference When rigid objects collide, energy is transferred between them, and often they bounce back after the collision. Waves, on the other hand, do not collide with other waves they way rigid objects do. Instead, waves interact through a phenomenon called interference. When two waves pass through each other, they interact through constructive and destructive interference. Constructive interference occurs when two waves are in Figure 7.1: Waves that are in phase constructively interfering phase (both have either positive or negative (top) and out of phase destructively interfering (bottom) amplitudes). In such an occasion, the amplitudes of the two waves combine to create a wave with greater amplitude – either more positive or more negative (Figure 7.1 top). Destructive interference occurs when waves are in opposite phases and the amplitudes are subtracted from each other, resulting in a lesser amplitude wave (Figure 7.1 bottom). 2 Double slit experiment for light Light Source The question of the exact nature of light – whether it behaved as a particle or a wave – dates back at least 2000 years to the atomist Democritus. A few decades before Maxwell provided strong evidence in support the wave-like nature of light with his famous equations for electromagnetism, a scientist named Thomas Young conducted an experiment designed to test this question. In 1801, well before the conversation of quantum mechanics started in the scientific world, Young directed a light source at a screen that had two small slits cut into it (Figure 7.2). A screen then detected the pattern of the light after it passed through the slits. If light behaved like a particle, then the expected results would be two bright bands on the screen: one bright band for each of the slits. Instead, the results showed a diffraction pattern – a phenomenon that occurs when waves interact. Each bright spot on the screen (far right side of the figure) showed a peak in the amplitude due to constructive interference between the two radiating waves. Each dark spot represented destructive interference, where the amplitudes of the interfering waves cancelled (zero amplitude = zero intensity = dark). As such, Young concluded that the light acted as would be expected from a wave, not as particles. Figure 7.2: Young’s Double-Slit Experiment schematic with resulting diffraction pattern of red laser light. (http://cronodon.com/Atomic/Photon.html; hyperphysics.phy-astr.gsu.edu) Double slit experiment for electrons In 1961, 160 years after Young’s initial experiment, the double slit experiment was repeated using a beam of electrons instead of light. The expectation was that if electrons acted like particles, then the resulting screen would show two groups of electrons – those that passed through one slit or the other. Consider what would happen if you threw Ping-Pong balls at a screen with an opening just wide enough for a ball to pass through – the few that made it through would fly through and hit approximately the same location (Figure 7.3) The results of the experiment using electrons, however, 3 Figure 7.3: Double slit experiment results with classical particles. (http://scienceblogs.com/startswithabang/200 9/06/01/a-tale-of-two-slits/) Figure 7.4: Electron diffraction pattern from double slit experiment. Source:http://iopscience.iop.org/article/10.1088/13672630/15/3/033018/pdf were the same as what was seen with the diffraction of light: wave-like diffraction patterns emerged, with multiple “bright” and “dark” bands (Figure 7.4). Initially, it was thought that the results were inaccurate, and that electrons might have been “hitting” or scattering each other as they passed through the slits, causing them to spread out. To test this concern, the experiment was repeated in 1974 with individual electrons being emitted one at a time to prevent any possible scattering. What they saw, however, was another diffraction pattern. The only way this pattern could be created was if individual electrons had wave-like properties similar those of light. Note that each dot captured on the screen represents a single electron signal after having passed through the slits. As time continues, these dots collect into groups, resulting in a diffraction pattern similar to that of light. One of the greatest implications of these findings was that the mass of an electron is spread out energetically in a wave, not localized in one point like a classical object. This is why we say that atoms have “electron waves”. Observing the wave-like properties of electrons is the key to the quantized nature of light and light-matter interactions. Let’s take a look at how a classical example of waves gives us information about the quantized nature of matter. The images in Figure 7.5 show an example of a classic particle (a baseball) that is localized, and a classic wave (sound) that is delocalized. Figure 7. 5: Localized versus delocalized behavior In the localized example it is possible to circle the exact location of the baseball. That is because a classical object, or “particle”, is localized in space. With the sound example, however, it is not possible point to the exact location of the sound. This is because waves are delocalized, and are spread out in space. Based on the results of the double slit experiments, we can relate both light and electrons to the delocalized example because of their wave-like properties. 4 Remember, analogies are a good way to understand abstract topics, but they are limited. The analogies above do not give exact representations of electrons, but instead help to explain some of their more abstract behavior. The purpose of this example is to illustrate that the behavior of electrons in an atom is similar to that of delocalized waves spread out in space, not localized particles. Matter waves and the de Broglie wavelength The finding that electrons can behave as waves was a surprising one, and it raised new questions about the true nature of electrons. Louis de Broglie, a French physicist, wondered if electrons behaved as waves, does that mean one could calculate their wavelengths? It was already clear that electrons had momentum in the classical particle sense – i.e., the momentum of an electron was equal to its mass times its velocity (p = mv). The equation for the momentum of light, however, is different than for classical objects. For light, the momentum is defined as: 𝑝light = ℎ [7.1] 𝜆light where h is Planck’s constant (6.626 x 10–34 J⋅s) and λ is the wavelength of the light. This now provides an interesting comparison between photons and electrons. Because electrons also have wave-like behavior, it stands to reason that they would also have a wavelength. De Broglie, having observed a diffraction pattern for electrons passing through crystal, made a logical leap that, if electrons were behaving in a wave-like manner similar to light, he could combine Einstein’s formula for the momentum of light with the equation for momentum of a particle (p = mv) to find the wavelength of an electron (λelectron): 𝑝electron = 𝑚- 𝑣 = ℎ 𝜆electron = ℎ 𝜆db [7.2] where me is the mass of the electron (9.109 × 10–31 kg), and λelectron = λdb is the de Broglie wavelength of the electron wave. Rearranging this equation we can solve for the de Broglie wavelength: 𝜆db = ℎ 𝑝electron = ℎ 𝑚- 𝑣 [7.2] Worked Example 7.1: What is the de Broglie wavelength (in m) of an electron (m = 9.109 × 10–31 kg) moving at a speed of 5 × 105 m/s? Solution: the de Broglie wavelength of an electron (9.109 × 10–31 kg) moving with speed of 5 × 105 m/s is computed with equation 7.2: 𝜆db ℎ 6.626 × 10678 J ⋅ s = = = 1.5 × 106B m Careful: units can be a problem here; the mass must be in kg (not g) because Planck’s constant contains J. 5 Based on this equation, an electron traveling at a speed of 500 km/s would have a de Broglie wavelength of about 1.5 nm (1.5 × 10–9 m). While this seems small, it is significant compared to the size of an atom, which has a radius on the order of 1 Å (= 10–10 m). However, what would happen if we applied formula [7.2] to everyday objects? After all, according to de Broglie’s equation, anything with a mass and a velocity would also have a wavelength. To answer this question, consider a baseball in worked example 7.2. Worked Example 7.2: What is the de Broglie wavelength (in m) of a baseball (m = 0.15 kg) thrown at 90 miles per hour (40 m/s)? Does this baseball have a de Broglie wavelength? Why or why not? Solution: the de Broglie wavelength of a baseball (0.15 kg) moving with speed 40 m/s is computed with equation 7.2: 𝜆db = ℎ 6.626 × 10678 J ⋅ s = = 1.1 × 10678 m 𝑝 (0.15 kg)(40 m/s) It is clear that we can calculate a wavelength for this heavy and slow classical object, such as a baseball thrown at 90 mph, but there is no real meaning to the value. Because the de Broglie wavelength is much smaller than the baseball, there are no wave-like properties ascribed to this object. However, objects with very small masses (like electrons) will often have measureable and meaningful de Broglie wavelengths. An important implication here is that the boundary between classical and quantum objects is perhaps fuzzier than we might imagine. At what point do we decide that a particle is acting like a wave? Particle-in-a-box model Up until now, we have considered electrons moving freely (outside of an atom). But what happens to an electron when it is bound to an atom? Does it remain a wave? To help us answer this question, let’s start with a more familiar scenario that illustrates a wave between two boundaries. Suppose two people were to hold a spring or a jump rope at both ends and one person started shaking one of the ends. At first, waves would likely travel back and forth rather chaotically. But with continued shaking (and adjusting of the amount of energy supplied to the spring or rope), the two people would be able to create standing waves. The more energy put into the spring or rope, the more peaks the standing wave would have. If the two people moved farther away from each other, the spring or rope would be stretched out to fill the space, and the amplitude of the waves created would decrease. The activity below provides a CDF player to help you better understand this relationship with standing waves. For classical standing waves (such as our example of the spring or jump rope), the number of loops is quantized. This means that there can only be specific values of wavelengths within these types of standing waves (i.e., whole number multiples of ½ λ). As we have seen, if the wavelength is quantized, then the energy of the wave is also quantized. This is an appropriate analogy to an electron inside an atom. Because the electron is attracted to the positive charge of the nucleus, it becomes bound to the atom. The electron wave disperses around the nucleus and forms a standing wave much like a spring or jump rope. The boundaries in the atom are therefore created by the energy of this attraction. 6 Activity: Standing Waves Simulation When this first opens, you will see two sliders at the top. The first, labeled Barrier Separation (L), indicates the separation between the two barriers. The second, labeled loops (n), indicates the number of loops (1 loop = ½ of a waveform; one full wavelength is two loops) in the wave. Exercise 1: Set the separation of the barriers (L) to 5 nm. Draw the wave for: n = 1, n = 2, n = 3, n = 4, and n = 5. Answer the following questions based on your drawings. a. How does the number of loops affect the wave? b. Will you ever have only part of a loop between the two barriers? What does this mean for the number of wavelengths you can have inside the barriers? Exercise 2: Set n = 1. Change L and answer the following questions based on your observations. a. What happens to the amplitude of the wave as you change the separation of the barriers? What happens to the wavelength as you change the separation of the barriers? To simplify and help visualize this phenomenon, a one-dimensional electron wave can be trapped in a “box” of energy barriers. The result is a one-dimensional standing wave, much like the analogy to a spring fixed at both ends, called the “Particle in a Box” model (Figure 7.6). The separation of these barriers (L) can be equated to the size of the atom. For bonding electrons in a molecule, this same concept applies. In this case, the separation between the barriers (L) represents the size of the molecule. Figure 7.6: Particle-in-a-Box Model The model is based on the following assumptions: (1) the particle – usually an electron – is confined to a one-dimensional box with length, L, which has a uniform and flat bottom (lower energy than the wave) inside the box; (2) the walls of the box will be infinite (infinite energy), which means that the particle/wave cannot escape and will have zero amplitude at the walls (creating nodes); and (3) only whole numbers of loops (1/2 wavelengths) can be found in the box. Understanding the different energies (or “energy states”) of these standing waves gives insight into the quantized properties of electrons. We can demonstrate this by doing a little algebra. First, we can calculate the average energy of the particle trapped in the box by using the formula for kinetic energy and substituting p/m for v (from p = mv): 𝐸= 1 𝑝E 𝑚𝑣 E = 2 2𝑚 [7.3] 7 where m is the mass of the particle (usually an electron) and p is the momentum. Recall from de Broglie that the momentum of a matter wave is equal to h/λ for the wave (equation [7.2]). Substituting equation [7.2] into equation [7.3] gives us the energy of the matter wave in terms of the wavelength: 𝐸= ℎE . 2𝑚𝜆E [7.4] As you have seen in the standing wave activity above, the wave contained in the box must have a whole number of loops (n, or half-wavelengths). Therefore, the total length of the box, L, can be written as: 𝜆 𝐿 = 𝑛 H I. 2 [7.5] Rearranging equation [7.5] and substituting it into equation [7.4] gives the formula for the energy of a matter wave with mass, m, with n loops that is confined to a box of length, L: 𝑛E ℎE 𝐸& = . 8𝑚- 𝐿E [7.6] Because h and me (mass of an electron = 9.109 × 10–31 kg) are constants, we can further simplify this equation to: 𝐸& = K ℎE 𝑛E 𝑛E L E=𝐾 E 8𝑚- 𝐿 𝐿 where K is a constant equal to 6.02 × 10–38 J2 s2/kg. Notice that equation [7.6] (the particle-in-a-box model) indicates that matter waves that are confined to one-dimensional boxes will have finite, discrete energies that are proportional to n2. That is, the two lowest energy wave states (n = 1 and n = 2) will have energies E1 = K/L2 and E2 = 22K/L2 = 4E1, but there will be no states possible between these two! (Because standing waves cannot have fractions of nodes/loops – n must be a positive integer). This result constitutes a huge leap in our understanding of the nature of electrons in atoms – specifically their quantized behavior. Of course, unlike our particle in a box model, the electron will be a 3-dimensional wave, which will require a little more investigation. Worked Example 7.3: 𝛽 -carotene (right) is a strongly colored orange pigment found in plants and fruits (e.g., carrots). The molecule is long and straight, so the electron waves of the molecule can be modeled using the particlein-a-box model. 𝛽-carotene absorbs light most strongly around 470 nm (blue and green light), which means that the rest of the visible light shines through (is transmitted) or is reflected. As a result, you eye picks up the colors of visible light that are not absorbed, making it look orange-red. The absorption of blue light (470 nm) corresponds to a change in the electron waves from n = 11 to n = 12 due to the absorption of energy. Based on these observations, estimate the length of the molecule (the “box”) in nm. 8 Solution: Changes in electron waves (such as the absorption of light) involve two energy states. In this case, we are studying the change from n = 11 to n = 12, from the absorption of one photon of 470 nm light. First, we must calculate the energy of one photon with a wavelength of 470 nm using equations c = λν and Elight = hνlight. 𝛥𝐸 = ℎ𝜈 = ℎ𝑐 (6.626 × 10678 J ⋅ s)(3 × 10R m/s) = 𝜆 (4.7 × 106W m) Using equation [7.6] for the energy of each state, we get: (12)E ℎE (11)E ℎE ℎE E E) ( 𝛥𝐸 = − = 12 − 11 8𝑚- 𝐿E 8𝑚- 𝐿E 8𝑚- 𝐿E Solving for the length of the box (L), we get: E E 𝐿 = (12 − 11 𝐿E = (12E − 11E ) E) ℎE 8𝑚- 𝛥𝐸 (6.626 × 10678 J ⋅ s)E 𝐿 = 1.81 × 106B m = 1.81 nm If we assume that the electron wave that we are modeling is delocalized over the length of the entire molecule (it is), then we estimate that the molecule is approximately 1.8 nm long. Standing waves on a spring are a good analogy for electron waves As we saw, standing waves with specific energies and wavelengths can be produced on a spring (or rope) by applying the right amount of energy. Because we are producing standing waves, only specific waves with are possible: 1 loop (half-wavelength), 2 loops, 3 loops, etc. In the language of quantum mechanics, we would call the spring itself the “field” and the waves that are produced on this spring are the “excitations.” Electrons are also excitations of a field: the quantum field. All space is permeated with the quantum field, and electrons are specific excitations of that field. Similar to the excitations of waves on a spring, the electron excitations of the quantum field are also limited to being 1 loop, 2 loops, 3 loops, …, n loops. These are what we will refer to as electron waves. While it helps to think about quantized electron waves as being similar to the one-dimensional (1D) standing waves on a spring, they are actually three dimensional (3D) inside an atom. The 3D electron wave is often referred to as an “electron cloud” around the nucleus. While “cloud” is a helpful name for visualizing the electron, remember that it is just an analogy – an electron is a single delocalized wave in three dimensions. Just like the one-dimensional particle-in-a-box model, there are a finite number of energy states that a three-dimensional electron can occupy in an atom. In the next section we will explore the nature of the 3D electron waves. 9 Three-dimensional electron waves In the 1920s, the Austrian physicist Erwin Schrödinger developed an equation that can be used to find the energy states of electron clouds. While the mathematics of the Schrödinger equation are far beyond the scope of this course, we can discuss some of the different ways that electrons can exist in threedimensions. Just like adding more energy to a jump rope will increase the number of loops in one dimension, adding more energy to an electron will increase the number of “loops” that it has in three dimensions. When describing the energy states of electrons in three dimensions, we will run into two types of loops called radial loops and nodal planes. As we will see, the number of radial loops and/or nodal planes influences the shape of the electron cloud. Radial loops Radial loops form concentric spheres around the nucleus and are the reason why it is most appropriate to visualize atoms as being, for the most part, spherical. Let’s investigate these radial loops and learn how to identify the number of radial loops (3D loops) an electron cloud has by looking at it. Moving forward, we will indicate the number of radial loops by the letter “j”. Figure 7.7 shows the lowest energy electron wave that has only a single radial loop (j = 1). Electron waves with this shape (spherical) are called “s” orbitals. An orbital is the electron cloud energy states. Note: the term “orbital” is somewhat problematic when it comes to describing the behavior of electrons, as it suggests an “orbit” or a planetary model of the atom. Instead, “orbitals” describe electrons as waves – it is important to keep in mind that nothing is actually “orbiting.” Since the electron wave in Figure 7.7 only has a single loop, we call it the 1s orbital. Figure 7.8 shows the second lowest energy electron wave that has two radial loops (j = 2). You can see there is a node (black ring) between the two loops, much like where a standing wave passes through a zero point. This electron wave is still spherical, but because of the two loops we call it the 2s orbital. As the number of radial loops increases, so does the total number of loops in the electron wave. As we have seen, the more loops present in a wave, the smaller the wavelength and the greater the energy. Therefore, the 2s electron wave is higher in energy (more loops, more energy) than the 1s electron wave. Figure 7.7: 2D representation of a 1s electron cloud Figure 7.8: 2D representation of a 2s electron cloud Nodal planes In addition to having multiple radial loops, many electron clouds are broken up by nodal planes. Much like a node in a standing wave (where the energy is zero), nodal planes will result in two-dimensional areas where the electron density of an orbital is zero. In this section we will investigate nodal planes in an electron cloud. The example provided in Figure 7.9 shows the difference between an electron orbital with one radial loop and no nodal planes (Figure 7.9 on the left), versus an electron orbital with 10 CC212: Reality and the Modern World one radial loop and one nodal plane (Figure 7.9 on the right). Note that the addition of a nodal plane has changed the shape of the orbital from a sphere into more of a “dumbbell” shape. If an electron orbital with one radial loop is cut by two perpendicular nodal Radial Nodal planes, the resulting shape Loop Plane is a 3d orbital (see table j=1 l=1 below). The number of nodal planes in an electron wave are indicated using Figure 7.9: Bisecting a 1s orbital (j = 1) with a nodal plane (l = 1) makes an orbital the letter “l”, known as the with 2 loops total (n = 2) called 2p. angular quantum number. The 1s orbital has an l value of 0, meaning there are no nodal planes in this orbital, whereas the 3d orbitals have an l value of 2, meaning that there are two nodal planes in each of these orbitals. A summary of the different types of orbitals and their shapes is presented in Table 7.1. Table 7.1: Shapes and names of atomic orbitals with various values of l General shape of the electron wave l Name 0 s Spherical, no nodal planes 1 p Dumbbell shaped, 1 nodal plane 2 d Cloverleaf shaped, 2 nodal planes 3 f More complex shapes (see visualization activity) Image As the number of nodal planes in a wave increases, the energy of the electron wave also increases in a manner similar to the increase in energy with increasing radial loops. The total energy of an electron wave is thus calculated as the sum of the radial loops (j) and nodal planes (l): j+l=n where “n” is the principal quantum number. For example, the electron orbital with two radial loops and no nodal planes has j = 2, l = 0, and n = 2, and is called the 2s orbital. ©2019, Binyomin Abrams, Emily Allen, Robin Stevens 11 The two remaining quantum numbers: ml and ms In addition to the principal quantum number (n) and the angular momentum quantum number (l) that describe the energy and shape of electron clouds, there are two additional quantum numbers (variables) that describe electron cloud configurations. The first of these is called the magnetic quantum number (ml), which describes the orientation of a given electron cloud in three-dimensional space. For example, an “s” orbital (l = 0) can only have one configuration in three dimensions because it is a sphere. Therefore, there is only one value of ml for any “s” orbital (ml = 0). For a “p” orbital (l = 1), however, there are multiple orientations. Each of these orientations results in a different possible electron cloud. For example, the 2p electron clouds (Figure 5.9) have three possible orientations – the nodal plane can be located along the x-, y-, or z-axis. Therefore, the electron cloud 2p has three possible values for ml. You will notice that each of these orientations is an individual orbital. That is because while they each have the same energy value (because they have the same n and l values) each one is a separate orbital. This quantum number is written as subscripts on each orbital label to indicate the difference in orientation (e.g., 2px, 2py, 2pz). The final quantum number, ms, is called the “spin” quantum number. This title can be very confusing (and misleading), because there is nothing here that is actually spinning (remember, we are talking about a 3D wave, not a particle). What this term represents is an intrinsic magnetic property of electrons. Based on this property, electron clouds can have two different orientations of “spin”: “spin up” and “spin down”. The “spin up” magnetic property has a slightly lower energy than “spin down”. This property results in the following rules of electron configuration in an atom: 1. The two different spin orientations allow two electron clouds to exist in a single orbital (one “spin up” and one “spin down”) without cancelling each other out. 2. While electrons may repel each other through the electromagnetic force (two negative charges will repel), it is more energetically favorable for two electrons to occupy orbitals with a specific number of loops (n) before creating an orbital with a greater number of loops (because a higher n is a higher energy). 3. Within a single energy level (set of orbitals with the same value of n and l) electrons will spread out within the orbitals before pairing up due to their repulsion. Together, these form the Aufbau Principle for electron configurations. The box diagram on the right (Figure 5.10) shows an example of how electrons would fill energy levels by using boxes to represent orbitals. The single box on the left represents an s orbital (such as 2s), while the three boxes on the right represent three p orbitals (such as 2px, 2py, 2pz). The electrons are drawn as arrows to represent either “spin up” or “spin down”. Note that due to the Aufbau Principle, the two 2p orbital electrons will have different orientations. Figure 5.10: Electron configuration box diagram Please take a moment to note that box diagrams are easy to misinterpret. Remember that electrons are not particles that are “in” orbitals – the electrons are the orbitals. 12