Electron Configurations - IB Chemistry PDF

Structure 1.3 Electron congurations How c an we model the energy states of electrons in atoms? This question is complex with many layers. What are electrons? How do we know they exist in energy states? What various mode...

Structure 1.3 Electron congurations How c an we model the energy states of electrons in atoms? This question is complex with many layers. What are electrons? How do we know they exist in energy states? What various models about these energy states are there? According to modern views, electrons are quantum objects that behave as both particles and waves. Although such behaviour has no analogues in our everyday life, we c an visualize electrons in atoms as fuzzy clouds. The shapes and sizes of these clouds depend on the energies of electrons, which c an have only certain, predened v alues. Understandings Structure 1.3.1 — Emission spectra are produced by Structure 1.3.5 — E ach orbital has a dened energy atoms emitting photons when electrons in excited states state for a given electron conguration and chemic al return to lower energy levels. environment, and c an hold two electrons of opposite spin. Sublevels contain a xed number of orbitals, regions Structure 1.3.2 — The line emission spectrum of of space where there is a high probability of nding an hydrogen provides evidence for the existence of electron. electrons in discrete energy levels, which converge at LHA higher energies. Structure 1.3.6 — In an emission spectrum, the limit Structure 1.3.3 — The main energy level is given an of convergence at higher frequency corresponds to 2 integer number, n, and c an hold a maximum of 2n ionization. electrons. Structure 1.3.7 — Successive ionization energy data Structure 1.3.4 — A more detailed model of the atom for an element give information about its electron describes the division of the main energy level into s, p, d conguration. and f sublevels of successively higher energies. Emission spectra (Structure 1.3.1) Much of our understanding of electron congurations in atoms has come from studies involving interaction with light. In the 1600s, Sir Isaac Newton showed that sunlight c an be broken down into dierent coloured components using a prism. This generates a continuous spectrum (gure 1a). This type of spectrum contains light of all wavelengths, and appears as a continuous series of colours, in which each colour merges into the next, and no gaps are visible. The classic example of a continuous spectrum is the rainbow. The wavelength of visible light ranges from 400 nm to 700 nm. A pure gaseous element subjected to a high voltage under reduced pressure will glow — in other words, it will emit light. When this light passes through a prism, it produces a series of lines against a dark background. This is known as an emission spectrum (gure 1b) In contrast, when a cold gas is placed between the prism and a source of visible light of all wavelengths, a series of dark lines within a continuous spectrum will appear. This is known as an absorption spectrum (gure 1c) 34  Structure 1.3 Electron congurations a continuous spectrum emission spectrum b hot gas c cold gas absorption spectrum  Figure 1 The spectra generated from (a) visible light of all wavelengths (b) a heated gas (c) visible light of all wavelengths passing through a cold gas  Figure 2 The aurora borealis (Northern Lights) in Lapland, Sweden. Charged high-energy particles from the Sun are drawn by the E arth’s magnetic eld to the polar regions, where they excite atoms and molecules of atmospheric gases, c ausing them to emit light 35  Structure 1 Models of the particulate nature of matter Emission spectra Emission spectra c an be observed through a simple Method handheld spectroscope by holding it up to a light source. 1. Observe natural light through the spectroscope. Note Discharge lamps contain low-pressure gases which are down the details of the spectrum you observe. ionized when a voltage is applied. 2. O bs e r ve a r ti fi c i a l light f ro m a c o m pu te r s c re e n or LE D. No te d ow n t he details of th e s pe c tr u m yo u Relevant skills o bs e r ve. Tool 3: Construct graphs and draw lines of best t 3. Observe light from various discharge lamps. Note Inquiry 2: Identify and record relevant qualitative down the details of the emission lines you observe, observations and sucient relevant quantitative data. including colours, wavelengths and number of lines. Inquiry 2: Identify and describe patterns, trends and Q uestions relationships 1. Sketch the spectra you observed. Inquiry 2: Assess accuracy 2. Describe each as a continuous, emission or S afety absorption spectrum. Wear eye protection. 3. Look up the emission spectra of the elements in The discharge lamps will get very hot. Handle them the discharge lamps you observed. Compare the with c are. theoretic al and observed emission lines, commenting Further safety prec autions will be given by your on the number, colours and positions of the teacher, depending on the exact nature of the emissionlines. discharge lamps. 4. Next, you will compare the theoretic al and observed wavelengths of the emission lines. Construct a graph M aterials of theoretic al wavelength vs observed wavelength. Discharge lamps Draw a line of best t through your data. Handheld spectroscope 5. Comment on the relationship shown in your graph. 6. Comment on the accuracy of the observed wavelength data. E ach element has its own characteristic line spectrum, which c an be used to identify the element. For example, excited sodium atoms emit yellow- orange light with wavelengths of 589.0 and 589.6 nm (gure3, right). The same yellow-orange colour appe ars in a ame test of any sodium-containing substance. Like barcodes in a shop that c an be used to identify products, line emission spectra c an be used to identify chemic al elements.  Figure 3 Sodium streetlights (le) and the line emission spectrum of sodium (right) 36  Structure 1.3 Electron congurations Observations Chemists oen generate data from observing the What is the dierence between observing a natural properties of matter. Observations c an be made phenomenon directly and with the aid of an instrument? directly through the human senses (oen sight), or with instruments. Advancements in technology expand the boundaries of our observations, revealing otherwise imperceptible features or detail. Sodium vapour lamps emit orange-yellow light. As seen in gure 3, observing the light through a spectroscope reveals a strong emission in the yellow region of the spectrum. The light  Figure 4 Helium emission spectrum from helium lamps is also orange to the naked eye but the emission spectrum of helium is more complex (gure 4). Flame tests Flame testing is an analytical technique that can be used to Materials identify the presence of some metals. The principle behind Flame test wire (platinum or nichrome) ame tests is atomic emission. Electrons are promoted Small portion of dilute hydrochloric acid to a higher energy level by the heat of the ame. When Bunsen burner and heatproof mat they fall back to a lower energy level, photons of certain Small samples of various metal salts (e.g. LiCl, NaCl, wavelengths are emitted. Some of these photons are in the KCl, C aCl , SrCl , CuCl ) 2 2 2 visible region of the spectrum. Method 1. Clean the end of the ame test wire by dipping it into the HCl solution and placing it in a non-luminous Bunsen burner ame. Repeat until no ame colour is observed. 2. Dip the end of the ame test wire into one of the salt samples, and place it in the edge of the non-luminous Bunsen burner ame, noting down the identity of the metal in the salt and the colour(s) you observe.  Figure 5 Flame test colours for dierent elements Relevant skills Inquiry 2: Identify and record relevant qualitative observations S afety Wear eye protection. 3. Clean the wire again and repeat with other salt samples. Take suitable prec autions around open ames. 4. Clear up as instructed by your teacher. Dilute hydrochloric acid is an irritant. A variety of dierent chloride salts will be used, some Q uestions: of which are irritants — avoid contact with the skin. 1. Look up the emission spectra of the metals you tested. Dispose of all substances appropriately. Compare these to the colours you observed. Comment Further safety prec autions will be given by your on any similarities and dierences. teacher, depending on the identity of the salts being 2. Explain why the dierent metals show dierent analysed. amecolours. 37  Structure 1 Models of the particulate nature of matter TOK One of the ways knowledge is developed is through reasoning. Reasoning c an be deductive or inductive. Inductive reasoning involves drawing conclusions from experimental observations. Inductive arguments are “bottom up”: they take specic observations and build general principles from them. inductive reasoning (“bottom-up” approach): 4. theory 3. hypothesis 2. pattern 1. observation For example, you might make the following observations about lithium salts: Lithium chloride gives a red ame test. Lithium sulfate gives a red ame test. Lithium iodide gives a red ame test. From these observations, you c an make the conclusion that all lithium salts give red ame tests. Deductive arguments are “top down”: they infer specic conclusions from general premises. You do this all the time when asked to apply your scientic knowledge in a new context. deductive reasoning (“top-down” approach): 1. theory 2. hypothesis 3. pattern 4. observation For example, suppose your scientic knowledge includes the following existing premises: Lithium bromide is a lithium salt. Lithium salts give red ame tests. From this, you could propose that lithium bromide gives a red ame test. What are the advantages and disadvantages of each type of reasoning? C an reasoning always be neatly classied into these two types? On what grounds might we doubt a claim reached through inductive reasoning? On what grounds might we doubt a claim reached through deductive reasoning? Visible light is one type of electromagnetic (EM) radiation. In addition to visible light, microwaves, infrared radiation (IR), ultraviolet (UV), X-rays and gamma rays are all part of the electromagnetic spectrum. The energy of the radiation is inversely proportional to the wavelength, λ: 1 E ∝ λ 38  Structure 1.3 Electron congurations Electromagnetic waves all travel at the speed of light, c, in a vacuum. The speed 8 –1 of light is approximately equal to 3.00 × 10 m s. Wavelength is related to the frequency of the radiation, f, by the following equation: c = f × λ High energy EM waves, such as gamma rays, have short wavelengths and high frequencies while low energy waves, such as microwaves, have long wavelengths and low frequencies. f λm 1 10 4 10 gamma rays 14 10  10 (γ rays) 1 λnm 10 0 10 400 10 X-rays 10 1 10  ultraviolet 10 00 1 10 elbisiv (UV) ygrene  10 14 10 infrared 00 (IR) 4 10 1 10  10 Activity microwaves 10 10 00 0 Compare the colours red and 10  10 green in gure 6. Determine which  colour has: 10  10 radio waves a. the highest wavelength 4 10 4 10 b. the highest frequency  Figure 6 The wavelength (λ) of electromagnetic radiation is inversely c. the highest energy proportional to both frequency and energy of that radiation Data-based questions Look at the spectra below. Explain how we know that stars are partly composed of hydrogen. 3900 7600 4000 4500 5000 5500 6000 6500 7000 7500  Figure 7 The hydrogen emission spectrum (top) and the absorption spectrum generated from the Sun (bottom) 39  Structure 1 Models of the particulate nature of matter The line emission spectrum of hydrogen (Structure 1.3.2 and 1.3.3) E ach line in the emission spectrum of an element has a specic wavelength, which corresponds to a specic amount of energy. This is c alled quantization: the idea that electromagnetic radiation comes in discrete packets, or quanta. A photon is a quantum of energy, which is proportional to the frequency of the radiation as follows: E = h × f Where E = the specic energy possessed by the photon, expressed in joules, J –34 h = Planck’s constant, 6.63 × 10 J s f = frequency of the radiation, expressed in hertz, Hz, or inverse –1 seconds, s In 1913, Niels Bohr proposed a model of the hydrogen atom based on its emission spectra. The main postulates of his theory were: 1. The electron c an exist only in certain stationary orbits around the nucleus. These orbits are associated with discrete energy levels 2. When an electron in the orbit with the lowest energy level absorbs a photon of the right amount of energy, it moves to a higher energy level and remains at that level for a short time. 3. When the electron returns to a lower energy level, it emits a photon of light. This photon represents the energy dierence between the two levels. Bohr ’s theory was the rst attempt to overcome the main problem of the Rutherford model of the atom (Structure 1.2). Classic al electrodynamics predicted that orbiting electrons would radiate energy and quickly fall into the nucleus, making any prolonged existence of atoms impossible. Bohr postulated that electrons did not radiate energy when staying in stationary orbits. Since electrons in the Bohr model of the atom could have only certain, well- dened energies, their transitions between stationary orbits could absorb or emit photons of specic wavelengths, producing characteristic lines in the atomic spectra. By measuring the wavelengths of these lines, it was possible to c alculate the energies of electrons in stationary orbits. For a hydrogen atom, the electron energy (E ) in joules could be related to the n energy level number (n) by a simple equation: 1 E = –R n H 2 n –18 where R ≈ 2.18 × 10 J is the Rydberg constant. This equation clearly represents H the quantum nature of the atom, where the energy of an electron c an have only discrete, quantized values. These values are characterized by integer or half-integer parameters, known as quantum numbers. The principal quantum number (n) c an take only positive integer values (1, 2, 3, …), where greater numbers mean higher energy. 40  Structure 1.3 Electron congurations The most stable state of the hydrogen atom is the state at n = 1, where the electron has the lowest possible energy. This energy level is known as the ground state of the atom. In contrast, the energy levels with n = 2, 3, … are c alled excited states. Atoms in excited states are unstable and spontaneously return to the ground state by emitting photons of specic wavelengths (gure8). +energy e e + + + p p p excitation dec ay hf  Figure 8 Electrons returning to lower energy levels emit a photon of light, hf Energy levels in atoms resemble ladders with varying distances between the rungs. Electrons c annot exist between energy levels, much like how you c annot stand between the rungs of a ladder. Jumping up each rung or level requires a specic, discrete amount of energy, and jumping down a rung or level releases the same amount of energy. An electron c an be excited to any energy level, n, and return to any lower energy level. Electrons returning to n = 2 will produce distinct lines in the visible spectrum of hydrogen (gure 9). Note that the red line has a longer wavelength and lower frequency than the violet line. The energy of the photon released is lower when an electron falls from n = 3 to n = 2, than from n = 6 to n = 2. In both cases, it represents the dierence between two of the allowable energy states of the electron in the hydrogen atom. colour violet blue cyan red wavelength / nm 410 434 486 656 transition from n = 6 n = 5 n = 4 n = 3 n = 6 n = 5 n = 4 n = 3 n = 2 n = 1 ◂ Figure 9 The visible lines in the emission spectrum of hydrogen show electrons returning from higher energy levels to energy level n = 2 41  Structure 1 Models of the particulate nature of matter n = 7 Electron transitions to the ground state, n = 1, release higher energy, shorter n = 6 wavelength ultraviolet light, while electrons returning to n = 3 produce lines in n = 5 the infrared region of the electromagnetic spectrum (gure 10). n = 4 It is important to note that electrons will absorb or rele ase only the exact energy n = 3 required to move between allowable energy states. Any excess will not be IR radiation absorbed, and if an insucient amount of energy is supplied the electrons will not move. n = 2 visible light Energy levels closer to the nucleus hold fewer electrons. The maximum number 2 of electrons in any energy level, n, is 2n. For example, the energy level with n = 1 holds up to two electrons, at n = 2 there could be a maximum of eight electrons, n = 3 has a maximum of 18 electrons, and n = 4 has a maximum of 32 electrons. n = 1 UV radiation  Figure 10 Electron transitions for the ATL Communic ation skills hydrogen atom. Notice how the allowable energy levels get closer together when When explaining concepts, we sometimes use diagrams, graphs or images to the electron moves further away from the help us convey our ideas more clearly. nucleus. The energy dierence between Prepare a written explanation of atomic emission that does not include n = 3 and n = 2 is much smaller than that any diagrams. Exchange it with a partner. Give each other feedback, between n = 2 and n = 1 concentrating on: Use of scientic voc abulary Order in which ideas are given Whether any important concepts are missing from the explanation. When you have shared each other ’s feedback, spend some time using the feedback to make improvements to your work. Finally, choose a graph, image or diagram to accompany your explanation. Discuss why you chose it and whether or not it adds to the explanation. Linking questions What qualitative and quantitative data c an be collected from instruments such as gas discharge tubes and prisms in the study of emission spectra from gaseous elements and from light? (Inquiry 2) How do emission spectra provide evidence for the existence of different elements? (Structure 1.2) How does an element’s highest occupied main energy level relate to its period number in the periodic table? (Structure 3.1) 42  Structure 1.3 Electron congurations The quantum mechanic al model of the atom (Structure 1.3.4) The Bohr model was an attempt to explain the energy states of electrons in atoms. It was based on quantization: the idea that electrons existed in discrete energy levels. According to Bohr, the emission spectra of hydrogen consisted of narrow lines bec ause the wavelengths of these lines corresponded to the dierences in allowable energy levels. However, this model was limited by several problems and incorrect assumptions: 1. The model could not predict the emission spectra of elements containing more than one electron. It was only successful with the hydrogen atom. 2. It assumed the electron was a subatomic particle in a xed orbit about the nucleus. 3. It could not account for the eect of electric and magnetic elds on the spectral lines of atoms and ions. 4. It could not explain molecular bonding and geometry. The principles behind molecular 5. Heisenberg’s uncertainty principle states that it is impossible to precisely bonding and geometry are know the loc ation and momentum of an electron simultaneously. Bohr ’s explained in Structure 2.2 model stated that electrons exhibited xed momentum in specic circularorbits. Bec ause of these limitations, the Bohr theory has been eventually superseded by the modern quantum mechanic al model of the atom. TOK The modern quantum mechanics combines the idea of momentum and the tendency to be absorbed or released quantization with the following key principles. as discrete entities suggest their particulate nature. However, photons, electrons, and even whole atoms and Heisenberg’s uncertainty principle states that it is small molecules, are c apable of interference (combination impossible to determine accurately both the momentum of waveforms), diraction (bending around obstacles) and and the position of a particle simultaneously. This means tunnelling (passing through obstacles), all of which are that the more we know about the position of an electron, characteristic to waves. the less we know about its momentum, and vice versa. Although it is not possible to pinpoint the loc ation or We have two contradictory pictures of reality; predict the trajectory of an electron in an atom, we c an separately neither of them fully explains the c alculate the probability of nding an electron in each phenomena of light, but together they do. region of space. Albert Einstein (1879–1955) One aim of the physical sciences has been to give an The wave–particle duality of the electron is quantitatively exact picture of the material world. One achievement … described by the Schrödinger equation, which was has been to prove that this aim is unattainable. formulated in 1926 by the Austrian physicist Erwin Jacob Bronowski (1908–1974) Schrödinger (1887–1961). Solutions to the Schrödinger equation give a series of three-dimensional mathematic al What are the implic ations of this uncertainty principle on functions, known as wave functions, which describe the the boundaries of knowledge? possible states and energies of electrons in atoms. What are the limits of human knowledge? The concept of wave–particle duality illustrates the fact Wave–particle duality is the ability of electrons and other that objects of study do not always fall neatly into the subatomic species to behave as both particles and waves. discrete c ategories we have developed. What is the role Certain characteristics of these species, such as mass, of c ategorisation in the construction of knowledge? 43  Structure 1 Models of the particulate nature of matter Schrödinger ’s wave functions describe the electrons in atoms in terms of their probability density, using Heisenberg’s idea that the momentum and position of electrons are uncertain. Instead of saying that electrons follow a dened travel path, this theory gives the probability that an electron will be found in a specic region of space at a certain distance from the nucleus. An atomic orbital is a region in space where there is a high probability of nding an electron. There are several types of atomic orbitals, and each orbital c an hold a maximum of two electrons. E ach orbital has a characteristic shape and energy. The rst four atomic orbitals, in order of increasing energy are labelled s, p, d, and f. Subsequent orbitals are theoretic al, and these are labelled alphabetic ally (g, h, i, k and so on). The principal quantum number, n, introduced by the Bohr model represents the main energy levels. These energy levels are split into sublevels comprised of  Figure 11 An s orbital is spheric al. The atomic orbitals. For example, for n = 1, 2 and 3, the s atomic orbitals are 1s, 2s sphere represents the boundary space and 3s. As n increases, the s orbitals are further distanced from the nucleus. where there is a 99% probability of nding an electron. The s orbital c an hold two Figure 12 shows that, for 1s, there is a high probability of nding electrons close electrons to the nucleus and this probability never reaches zero when we move further away from the nucleus. For 2s, the highest probability is somewhat further away, although there is a small probability that an electron could be found closer to the nucleus. There is zero probability of nding the electron between the two peaks. The same is true for 3s, with the highest probability at an even greater distance from the nucleus and two regions of zero probability. 1s 0 50 pm 2s 0 50 100 pm average radius 3s 0 50 100 150 pm  Figure 12 The plots of the wavefunctions for the rst three s orbitals 44  Structure 1.3 Electron congurations Imagine that you are a student waiting for your DP chemistry lesson to begin at 8.00am. At 8.15am, there is still no sign of your teacher, so you wonder where they could be. Some students from your class suggest that the teacher: is possibly in the sta room, the chemistry laboratory, or the library could be in the school principal’s oce or in the school c ar park may be at their house in the town centre might perhaps be at the airport might even have gone to the North Pole! Although the exact loc ation of the teacher is unknown, it is possible to draw a three-dimensional cluster of dots showing areas where there is a high probability of nding the teacher. A boundary surface could be drawn around this cluster to dene a region of space where there is a 99% chance of nding them. This might be the school perimeter, or the town where your school is loc ated, or a certain region around the town that includes the airport. Similarly, an atomic orbital represents the region of space with a high probability of nding an electron (gure13). t Figure 13 Representation of a 1s atomic orbital as y y a cluster of dots (le) and a sphere that encloses 99% of the dots (right) x x z z A p orbital is dumbbell shaped There are three p orbitals, each described with orientations parallel to the x, y and z axes (gure 14). These are labelled p , p and x y p. These shapes all describe boundaries with the highest probability of nding z electrons in these orbitals. t Figure 14 The three p atomic orbitals are dumbbell shaped, aligned along the z z z x, y and z axes. There is zero probability of nding the electron at the intersection of the axes between the two lobes of the x x x dumbbell. E ach of the p orbitals c an hold two electrons y y y p orbital p orbital x y p orbital z 45  Structure 1 Models of the particulate nature of matter Theories and models Current atomic theory evolved from previous models, each superseding the one that c ame before. Theories are comprehensive systems of ideas that model and explain an aspect of the natural world. Contrary to the use of the word “theory” in everyday language, scientic theories are substantiated by vast amounts of observations and tested hypotheses, which are amassed, documented and communic ated by a large number of scientists. + + + + + + + + + + + + + 800–400 BCE 1897 1913 1930 Āruņi’s kana Thomson’s “plum ohr model uantum mechanic al Democritus’ atomos pudding” model model 1803 1912 1926 D alton’s “billiard ball” Rutherford’s model Heisenberg’s uncertaint model and regions of probabilit model  Figure 15 The atomic theory has seen the idea of atoms evolve from indestructible spheres to the quantum mechanic al model where electrons have specic energies and are found in regions of high probability What other examples of theories c an you think of ? Linking question What is the relationship between energy sublevels and the block nature of the periodic table? (Structure 3.1) 46  Structure 1.3 Electron congurations Electron congurations (Structure 1.3.5) E ach atomic orbital type has a characteristic shape and energy. The s orbital is spheric al and it has the lowest possible energy. There are three p orbitals, each oriented dierently. There are ve d orbitals and seven f orbitals, and these are higher in energy than s or p. z y x s z z z y y y x x x p p p –1 0 1 z z z z z y y y y y x x x x x d d d d d –2 –1 0 1 2 z z z z z z z y y y y y y y x x x x x x x f f f f f f f –3 –2 –1 0 1 2 3  Figure 16 The shapes of the s, p, d and f orbitals. Only the shapes of s and p orbitals need to be known E ach energy level dened by the principal quantum number, n, c an hold n types of orbitals (table 1). For n = 1, only the s orbital exists. For n = 2, there are two types of orbital: s and p. For n = 3, there are three types: s, p, and d. For n = 4, there are four types: s, p, d, and f. Total number M aximum Principal Number Type of of orbitals number of quantum of orbitals sublevel per energy electrons within number (n) per type 2 2 level (n ) energy level (2n ) 1 s 1 1 2 s 1 2 4 8 p 3 s 1 3 p 3 9 18 t Table 1 Each energy level, dened by 2 d 5 n, c an hold 2n electrons. The number of sublevels, or atomic orbital types, is equal s 1 to n. For n = 4 there are four types of orbitals p 3 (s, p, d, and f ) with 16 atomic orbitals in total 4 16 32 2 d 5 occupied by a maximum of 2(4) = 32 total electrons f 7 47  Structure 1 Models of the particulate nature of matter Activity State the following for the energy level with n = 5: a. the sublevel types b. the number of atomic orbitals in each sublevel c. the total number of atomic orbitals d. the maximum number of electrons at that energy level. Orbital diagrams For convention, an “arrow in box” notation c alled an orbital diagram is used to represent how electrons are arranged in atomic orbitals (gure 17). The arrangement of electrons in orbitals is c alled electron conguration u Figure 17 In orbital diagrams, each box s sublevel (one box representing an s orbital) represents an orbital. This diagram shows the number of orbitals for each sublevel. Arrows are drawn in the boxes to represent electrons. A maximum of two electrons c an occupy each orbital, so each box has a maximum of two “arrows” p sublevel (three boxes representing the three p orbitals p , p , and p ) x y z d sublevel (five boxes representing the five d orbitals) f sublevel (seven boxes representing the seven f orbitals) Atomic orbitals are regions of space where there is a high probability of nding electrons. Electrons are charged negatively, and like charges repel each other, so two electrons should not be able to occupy the same region of space. Quantum mechanics solves this problem by using a ± spin notation for each electron. A pair of electrons with opposite spins behave like magnets facing in opposite directions. Hence each orbital box is shown with an upwards half-arrow, , and one downwards half-arrow, (gure 18). This is known as the Pauli exclusion principle: Only two electrons c an occupy the same atomic orbital and those electrons must have opposite spins. 48  Structure 1.3 Electron congurations t Figure 18 Electron spin is represented N S by an arrow pointing up (positive spin) or down (negative spin) S N N S magnet analogy S N half-arrows representing 3d electrons of opposite spin in an orbital degenerate 3p TOK ygrene degenerate 3s Electron spin is oen interpreted as the rotation of the electron around its own axis. However, this interpretation has no physic al basis: electrons in 2p atoms behave like waves, and a wave c annot rotate. Unfortunately, neither the spin nor the wave-like behaviour of electrons c an be visualized in any way, 2s as they have no analogues in our everyday life and c an be expressed only in degenerate mathematic al form. This lack of visualization does not undermine the quantum 1s theory but rather shows the limits of human perception and, at the same time, the power of mathematics as the language of science. 1 2 3 To what extent does mathematics support knowledge development in the n natural sciences?  Figure 19 The three 2p orbitals are degenerate as they have the same energy. E ach of the atomic orbitals of the same type in one sublevel are of equal energy. These three degenerate atomic orbitals Orbitals with the same energy are referred to as degenerate orbitals (?