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1 The Quantum Mechanical Model of the Atom You have seen how Bohr s model of the atom eplains the emission spectrum of hdrogen. The emission spectra of other atoms, however, posed a problem. A mercur atom, for eample, has man more electrons than a hdrogen atom. As ou can see in Figure 3.12, mercur has more spectral lines than hdrogen does. The same is true for other man-electron atoms. Observations like these forced Bohr and other scientists to reconsider the nature of energ levels. The large spaces between the individual colours suggested that there are energ differences between individual energ levels, as stated in Bohr s model. The smaller spaces between coloured lines, however, suggested that there were smaller energ differences within energ levels. In other words, scientists hpothesied that there are sublevels within each energ level. Each of these sublevels has its own slightl different energ. Hg H nm nm Figure 3.12 The emission spectrum for mercur shows that it has more spectral lines than the emission spectrum for hdrogen. 3.2 Section Preview/ Specific Epectations In this section, ou will describe the quantum mechanical model of the atom and its historical development state the meaning and significance of the first three quantum numbers communicate our understanding of the following terms: quantum mechanical model of the atom, orbitals, ground state, principal quantum number (n), orbital-shape quantum number (l), magnetic quantum number (m l ) It was fairl straightforward to modif Bohr s model to include the idea of energ sublevels for the hdrogen spectrum and for atoms or ions with onl one electron. There was a more fundamental problem, however. The model still could not eplain the spectra produced b man-electron atoms. Therefore, a simple modification of Bohr s atomic model was not enough. The man-electron problem called for a new model to eplain spectra of all tpes of atoms. However, this was not possible until another important propert of matter was discovered. The Discover of Matter Waves B the earl 192s, it was standard knowledge that energ had matter-like properties. In 1924, a oung phsics student named Louis de Broglie stated a hpothesis that followed from this idea. What if, de Broglie wondered, matter has wave-like properties? He developed an equation that enabled him to calculate the wavelength associated with an object large, small, or microscopic. For eample, a baseball with a mass of 142 g and moving with a speed of 25. m/s has a wavelength of m. Objects that ou can see and interact with, such as a baseball, have wavelengths so small that the do not have an significant observable effect on the object s motion. However, for microscopic objects, such as electrons, the effect of wavelength on motion becomes ver significant. For eample, an electron moving at a speed of m/s has a wavelength of m. The sie of this wavelength is greater than the sie of the hdrogen atom to which it belongs. (The calculated atomic radius of hdrogen is m.) Chapter 3 Atoms, Electrons, and Periodic Trends MHR 131

2 De Broglie s hpothesis of matter waves received eperimental support in Researchers observed that streams of moving electrons produced diffraction patterns similar to those that are produced b waves of electromagnetic radiation. Since diffraction involves the transmission of waves through a material, the observation seemed to support the idea that electrons had wave-like properties. Figure 3.13 The model of the atom in The fu, spherical region that surrounds the nucleus represents the volume in which electrons are most likel to be found. The Quantum Mechanical Model of the Atom In 1926, an Austrian phsicist, Erwin Schrödinger, used mathematics and statistics to combine de Broglie s idea of matter waves and Einstein s idea of quantied energ particles (photons). Schrödinger s mathematical equations and their interpretations, together with another idea called Heisenberg s uncertaint principle (discussed below), resulted in the birth of the field of quantum mechanics. This is a branch of phsics that uses mathematical equations to describe the wave properties of sub-microscopic particles such as electrons, atoms, and molecules. Schrödinger used concepts from quantum mechanics to propose a new atomic model: the quantum mechanical model of the atom. This model describes atoms as having certain allowed quantities of energ because of the wave-like properties of their electrons. Figure 3.13 depicts the volume surrounding the nucleus of the atom as being indistinct or cloud-like because of a scientific principle called the uncertaint principle. The German phsicist Werner Heisenberg proposed the uncertaint principle in Using mathematics, Heisenberg showed that it is impossible to know both the position and the momentum of an object beond a certain measure of precision. (An object s momentum is a propert given b its mass multiplied b its velocit.) According to this principle, if ou can know an electron s precise position and path around the nucleus, as ou would b defining its orbit, ou cannot know with certaint its velocit. Similarl, if ou know its precise velocit, ou cannot know with certaint its position. Based on the uncertaint principle, Bohr s atomic model is flawed because ou cannot assign fied paths (orbits) to the motion of electrons. Clearl, however, electrons eist. And the must eist somewhere. To describe where that somewhere is, scientists used an idea from a branch of mathematics called statistics. Although ou cannot talk about electrons in terms of certainties, ou can talk about them in terms of probabilities. Schrödinger used a tpe of equation called a wave equation to define the probabilit of finding an atom s electrons at a particular point within the atom. There are man solutions to this wave equation, and each solution represents a particular wave function. Each wave function gives information about an electron s energ and location within an atom. Chemists call these wave functions orbitals. In further studies of chemistr and phsics, ou will learn that the wave functions that are solutions to the Schrödinger equation have no direct, phsical meaning. The are mathematical ideas. However, the square of a wave function does have a phsical meaning. It is a quantit that describes the probabilit that an electron is at a particular point within the atom at a particular time. The square of each wave function (orbital) can be used to plot three-dimensional probabilit distribution graphs for that orbital. These plots help chemists visualie the space in which electrons are most likel to be found around atoms. These plots are 132 MHR Unit 2 Structure and Properties

3 also referred to as electron probabilit densit graphs. Note: Although orbitals are wave functions without associated phsical characteristics like shape and sie, chemists often use the term orbitals when the mean three-dimensional probabilit distribution graphs. To simplif discussion, this tetbook will discuss the sie and shapes of orbitals. However, what is reall meant is their associated probabilit distribution graph, which is calculated using the square of the wave function. Each orbital has its own associated energ, and each represents information about where, inside the atom, the electrons would spend most of their time. Scientists cannot determine the actual paths of the moving electrons. However, orbitals indicate where there is a high probabilit of finding electrons. Figure 3.14A represents the probabilit of finding an electron at an point in space when the electron is at the lowest energ level (n = 1) of a hdrogen atom. Where the densit of the dots is greater, there is a higher probabilit of finding the electron. This graph is fu-looking because the probabilit of finding the electron anwhere in the n = 1 energ level of a hdrogen atom is never ero. Farther from the nucleus, the probabilit becomes ver small, but it will still never reach ero. Therefore, because the shape of the n = 1 orbital for hdrogen represents the level of probabilit of finding an electron, and since the probabilit never reaches ero, ou have to select a cut-off level of probabilit. A level of probabilit is usuall epressed as a percentage. Therefore, the contour line in Figure 3.14B defines an area that represents 95 percent of the probabilit graph. This two-dimensional shape is given three dimensions in Figure 3.14C. What this means is that, at an time, there is a 95 percent chance of finding the electron within the volume defined b the spherical contour. Distinguish clearl between an electron orbit, as depicted in Bohr s atomic model, and an electron orbital, as depicted in the quantum mechanical model of the atom. A B C Figure 3.14 Electron densit probabilit graphs for the lowest energ level in the hdrogen atom. These diagrams represent the probabilit of finding an electron at an point in this energ level. Quantum Numbers and Orbitals Figure 3.14 showed electron-densit probabilities for the lowest energ level of the hdrogen atom. This is the most stable energ state for hdrogen, and is called the ground state. The quantum number, n, for a hdrogen atom in its ground state is 1. When n = 1 in the hdrogen atom, its electron is associated with an orbital that has a characteristic energ and shape. In an ecited state, the electron is associated with a different orbital with its own characteristic energ and shape. This makes sense, because the electron has absorbed energ in its ecited state, so its total energ increases and its motion changes. Figure 3.15 on the net page compares the sies of hdrogen s atomic orbitals when the atom is in its ground state and when it is in an ecited state. Chapter 3 Atoms, Electrons, and Periodic Trends MHR 133

4 n = 1 Figure 3.15 The relationship between orbital sie and quantum number for the hdrogen atom. As n increases, the electron s energ increases and orbital sie increases. Orbitals have a variet of different possible shapes. Therefore, scientists use three quantum numbers to describe an atomic orbital. One quantum number, n, describes an orbital s energ level and sie. A second quantum number, l, describes an orbital s shape. A third quantum number, m l, describes an orbital s orientation in space. These three quantum numbers are described further below. The Concept Organier that follows afterward summaries this information. (In section 3.3, ou will learn about a fourth quantum number, m s, which is used to describe the electron inside an orbital.) The First Quantum Number: Describing Orbital Energ Level and Sie The principal quantum number (n) is a positive whole number that specifies the energ level of an atomic orbital and its relative sie. The value of n, therefore, ma be 1, 2, 3, and so on. A higher value for n indicates a higher energ level. A higher n value also means that the sie of the energ level is larger, with a higher probabilit of finding an electron farther from the nucleus. The greatest number of electrons that is possible in an energ level is 2n 2. The Second Quantum Number: Describing Orbital Shape The second quantum number describes an orbital s shape, and is a positive integer that ranges in value from to (n 1). Chemists use a variet of names for the second quantum number. For eample, ou ma see it referred to as the angular momentum quantum number, the aimuthal quantum number, the secondar quantum number, or the orbital-shape quantum number. Regardless of its name, the second quantum number refers to the energ sublevels within each principal energ level. The name that this book uses for the second quantum number is orbital-shape quantum number (l ), to help ou remember that the value of l determines orbital shape. (You will see eamples of orbital shapes near the end of this section.) The value of n places precise limits on the value of l. Recall that l has a maimum value of (n 1). So, if n = 1, l = (that is, 1 1). If, l ma be either or 1. If, l ma be either, 1, or 2. Notice that the number of possible values for l in a given energ level is the same as the value of n. In other words, if, then there are onl two possible sublevels (two tpes of orbital shapes) at this energ level. Each value for l is given a letter: s, p, d, or f. The l = orbital has the letter s. The l = 1 orbital has the letter p. The orbital has the letter d. The l = 3 orbital has the letter f. To identif an energ sublevel (tpe of orbital), ou combine the value of n with the letter of the orbital shape. For eample, the sublevel with and l = is called the 3s sublevel. The sublevel with and l = 1 is the 2p sublevel. 134 MHR Unit 2 Structure and Properties

5 There are, in fact, additional sublevels beond l = 3. However, for chemical sstems known at this time, onl the s, p, d, and f sublevels are required. The Third Quantum Number: Describing Orbital Orientation The magnetic quantum number (m l ) is an integer with values ranging from l to +l, including. This quantum number indicates the orientation of the orbital in the space around the nucleus. The value of m l is limited b the value of l. If l =, m l can be onl. In other words, for a given value of n, there is onl one orbital, of s tpe (l = ). If l = 1, m l ma have one of three values: 1,, or +1. In other words, for a given value of n, there are three orbitals of p tpe (l = 1). Each of these p orbitals has the same shape and energ, but a different orientation around the nucleus. Notice that for an given value of l, there are (2l + 1) values for m l. The total number of orbitals for an energ level n is given b n 2. For eample, if, it has a total of 4 (that is, 2 2 ) orbitals (an s orbital and three p orbitals). The Sample Problem below shows further use of this calculation. CHEM FACT The letters used to represent energ sublevels are abbreviations of names that nineteenth-centur chemists used to describe the coloured lines in emission spectra. These names are sharp, principal, diffuse, and fundamental. Concept Organier The Relationship Among the First Three Quantum Numbers Quantum Numbers principal quantum number, n describes sie and energ of an orbital allowed values: 1, 2, 3, orbital shape quantum number, l describes shape of an orbital allowed values: to (n 1) magnetic quantum number, m l describes orientation of orbital allowed values: l,, +l Sample Problem Determining Quantum Numbers Problem (a) If, what are the allowed values for l and m l, and what is the total number of orbitals in this energ level? (b) What are the possible values for m l if n = 5 and l = 1? What kind of orbital is described b these quantum numbers? How man orbitals can be described b these quantum numbers? Continued... Chapter 3 Atoms, Electrons, and Periodic Trends MHR 135

6 Continued... Solution (a) The allowed values for l are integers ranging from to (n 1). The allowed values for m l are integers ranging from l to +l including. Since each orbital has a single m l value, the total number of values for m l gives the number of orbitals. To find l from n: If, l ma be either, 1, or 2. To find m l from l: If l =, m l = If l = 1, m l ma be 1,, +1 If, m l ma be 2, 1,, +1, +2 Since there are a total of 9 possible values for m l, there are 9 orbitals when. (b) You determine the tpe of orbital b combining the value for n with the letter used to identif l. You can find possible values for m l from l, and the total of the m l values gives the number of orbitals. To name the tpe of orbital: l = 1, which describes a p orbital Since n = 5, the quantum numbers represent a 5p orbital. To find m l from l: If l = 1, m l ma be 1,, +1 Therefore, there are 3 possible 5p orbitals. Check Your Solution (a) Since the total number of orbitals for an given n is n 2, when, the number of orbitals must be 9 (that is, 3 2 ). (b) The number m l values is equivalent to 2l + 1 : 2(1) + 1 = 3. Since the number of orbitals equals the number of m l values, the answer of 3 must be correct. Practice Problems 1. What are the allowed values for l in each of the following cases? (a) n = 5 (b) n = 1 2. What are the allowed values for m l, for an electron with the following quantum numbers: (a) l = 4 (b) l = 3. What are the names, m l values, and total number of orbitals described b the following quantum numbers? (a), l = (b) n = 4, l = 3 4. Determine the n, l, and possible m l values for an electron in the 2p orbital. 5. Which of the following are allowable sets of quantum numbers for an atomic orbital? Eplain our answer in each case. (a) n = 4, l = 4, m l = (c), l =, m l = (b),, m l = 1 (d) n = 5, l = 3, m l = MHR Unit 2 Structure and Properties

7 Shapes of Orbitals An orbital is associated with a sie, a three-dimensional shape, and an orientation around the nucleus. Together, the sie, shape, and position of an orbital represent the probabilit of finding a specific electron around the nucleus of an atom. Figure 3.16 shows the probabilit shapes associated with the s, p, and d orbitals. (The f orbitals have been omitted due to their compleit. You ma stud these orbitals in post-secondar school chemistr courses.) Notice that the overall shape of an atom is a combination of all its orbitals. Thus, the overall shape of an atom is spherical. Be careful, however, to distinguish for ourself between the overall spherical shape of the atom, and the spherical shape that is characteristic of onl the s orbitals. Finall, it is important to be clear about what orbitals are when ou view diagrams such as those in Figure Orbitals, remember, are solutions to mathematical equations. Those solutions, when manipulated, describe the motion and position of the electron in terms of probabilities. Contour diagrams, such as those shown here and in numerous print and electronic resources, appear solid. It therefore becomes eas to begin thinking about orbitals as phsical containers that are occupied b electrons. In some was, this is unavoidable. Tr to remind ourself, now and then, of the following: Electrons have phsical substance. The have a mass that can be measured, and trajectories that can be photographed. The eist in the phsical universe. Orbitals are mathematical descriptions of electrons. The do not have measurable phsical properties such as mass or temperature. The eist in the imagination. s orbitals n = 1 l = l = l = p orbitals l = 1 l = 1 l = 1 d orbitals Figure 3.16 Shapes of the s, p, and d orbitals. Orbitals in the p and d sublevels are oriented along or between perpendicular,, and aes. Chapter 3 Atoms, Electrons, and Periodic Trends MHR 137

8 Section Summar In this section, ou saw how the ideas of quantum mechanics led to a new, revolutionar atomic model the quantum mechanical model of the atom. According to this model, electrons have both matter-like and wave-like properties. Their position and momentum cannot both be determined with certaint, so the must be described in terms of probabilities. An orbital represents a mathematical description of the volume of space in which an electron has a high probabilit of being found. You learned the first three quantum numbers that describe the sie, energ, shape, and orientation of an orbital. In the net section, ou will use quantum numbers to describe the total number of electrons in an atom and the energ levels in which the are most likel to be found in their ground state. You will also discover how the ideas of quantum mechanics eplain the structure and organiation of the periodic table. Section Review K/U Eplain how the quantum mechanical model of the atom differs from the atomic model that Bohr proposed. K/U List the first three quantum numbers, give their smbols, and identif the propert of orbitals that each describes. C Design a chart that shows all the possible values of l and m l for an electron with n = 4. C Agree or disagree with the following statement: The meaning of the quantum number n in Bohr s atomic model is identical to the meaning of the principal quantum number n in the quantum mechanical atomic model. Justif our opinion. I Identif an values that are incorrect in the following sets of quantum numbers. (a) n = 1, l = 1, m l = ; name: 1p (b) n = 4, l = 3, m l =+1; name: 4d (c), l = 1, m l = 2; name: 3p I Fill in the missing values in the following sets of quantum numbers. (a) n =?, l =?, m l = ; name: 4p (b), l = 1, m l = ; name:? (c),, m l = 2; name:? (d) n =?, l =?, m l =?; name: 2s 138 MHR Unit 2 Structure and Properties