1	AHL Atomic Structure II: Modern Theory
2	Learning objectives: Concepts Concepts: Aufbau principle, Pauli exclusion principle, Hund’s rule, orbital, s orbital, p orbital, d orbital, f orbital, lobe, Skills: Give a sequence of orbitals from lowest energy to highest Give a sequence of orbitals from closest to the nucleus to furthest out Draw and name s and p orbitals Give a simple description of an orbital in terms of probabilities State the rules governing the order in which orbitals are filled
3	Preamble: Perspective Look closely at the sketch. Do you see a woman? How old do you think she is? What does she look like? What is she wearing? How do your impressions compare with those of your friends?
4	Quantum Mechanical Model of Atoms Schrodinger’s wave equation describing electrons in an atom is like an algebraic equation with variables just like a mechanical wave equation which also is a mathematical equation with variables describing it’s amplitude, wavelength etc. When the equation is solved, each set of values of the variables which constitutes one solution to the equation, describes the characteristics and position of a single electron in the atom. “Position” is not to be understood as a description of a particular electron’s location in a single point in space but within a volume of space referred to as an orbital. The volume of space, within which there is approximately 97% chance of the electron being found, is called an orbital (not to be confused with orbit which has a fixed radius).
5	Orbital How is this like a mechanical wave?
6	Mechanical standing wave The standing waves caused by the vibration of a guitar string fastened at both ends. Each dot represents a node (a point of zero displacement).
7	Electron as a Standing wave The hydrogen electron visualized as a standing wave around the nucleus: This would result if the two ends of the vibrating string were brought together to form a circular vibrating string. Nodes in this instance indicate points where the probability density is zero. Remember that this model has been built up to rationalize experimental data and is purely mathematical!
8	Solution to the wave equation On solving the wave equation, in general, it is found that: Electrons are found in what are called energy levels (or shells which are completely different from the shells in atoms according to Bohr’s model of an atom). Analogy: An atom can be thought of as a filing cabinet with several drawers one on top of the other. Each drawer in a filing cabinet can then be likened to an energy level in an atom. Each energy level (shell) is made up of fixed number of subshells. If each drawer is likened to an energy level in an atom, the individual folders within a drawer can then be likened to subshells within a shell of an atom. 3. Each subshell is made up of clusters but fixed numbers of what are called orbitals. The sheets of paper containing information filed away in each folder can be likened to orbitals.
9	Solution to the wave equation These energy levels are denoted by the letter n and numbered from inside out. The energy level closest to the nucleus would be the 1^st shell (n=1), the next shell n=2 etc. Continuing with the analogy, the bottom drawer can be likened to the first shell (n=1), the next one up the second (n=2) etc. The subshells within these shells are distinguished by different letters s, p, d, f , g etc. [Incidentally, the letters s, p, d and f stand for sharp, principal, defuse and fine, terms that described the appearance of spectra of elements.] The number of subshell within any shell is equal to where the shell is in the sequence. The 1^st shell (n=1) contains 1 subshell (subshell s); the 2^nd shell (n=2) contains 2 subshells (subshells s and p); he 3^rd shell (n=3) contains 3 subshell (subshells s, p and d) etc. It’s as if like the bottom drawer had only 1 folder, the next one up 2 folders, the next one up 3 folders etc.
10	Solution to the wave equation Now, the different folders in our filing cabinet of an atom are called orbitals and there are a fixed number of each of them within any subshell. s contains one orbital (also called s orbital), while p contains 3 (each one of them called p orbital), d contains 5 (each one of them called f orbital), and f contains 7 (each one called g orbital) etc. 4. An orbital can contain a maximum of 2 electrons with opposed spins (Pauli’s exclusion principle). That means, s subshell with it’s single orbital (s) in it, can accommodate a maximum of 2 electrons. In other words, the first energy level, shell # 1 (n=1), which contains only 1 subshell (s), which in turn contains only one orbital, referred to as an s orbital (named after the subshell it is found in), can accommodate a maximum of only 2 electrons. P subshell, on the other hand, with it’s 3 (p) orbitals, can accommodate a maximum of 6 electrons. That is to say then, the second energy level, shell # 2 (n=2), which contains 2 subshells, s and p, can accommodate a maximum of (2+6) 8 electrons etc.
11	Solution to the wave equation: Shells, Subshells, Orbitals and Electrons
12	Solution to the wave equation: Shape of Orbital within a shell So far so good! What should have been obvious is that you have a situation where, going back to the analogy, a drawer can have more than one folder (all shells except n=1 contain more than one subshell), and also that a folder can contain different number of information (subshells such as p, d and f can contain more than one orbital). In other words, electrons in atoms are found in orbitals. Three questions arise then: Firstly, how are the orbitals in subshells within the same shell (s, p and d in n=3 for example) distinguished from one another? Secondly how are orbitals within the same subshell (the three p orbitals in p subshell in n=3) differentiated? And lastly, how are orbitals in different shells (for instance the s orbitals in n=1 and n=2) are different from one another?
13	Solution to the wave equation: Shape of Orbital within a shell 5. Orbitals in different subshells but within the same shell are distinguishable by their shape.
14	Solution to the wave equation: Shape of Orbital and Relative Energy Whereas in the Bohr’s models, all electrons within a shell were assumed to be at the same distance from the nucleus and therefore at the same energy level, here, within a shell, the different subshells are said to be at different relative energy levels themselves. And that is indicated by the orbitals found within them being of different shapes. Relative energies of the sub-shells within an energy level (within a shell) is s < p < d < f etc. in general. There are some exceptions, of course. Consistent with Bohr’s model, it is presumed that p orbitals within a shell are at a higher energy level than s orbitals within the same shell because the electrons in p orbitals, on average, are farther away from the nucleus than s orbitals electrons. That conclusion is based on experimental data of course, and the data that supports this is no other than ionization energies of elements within the same period which we will return to later (see AHL Periodicity: Ionization Energy). Now, what about orbitals within the same subshell? How are for example, the three p orbitals within a p subshell distinguished from one another?
15	Solution to the wave equation: Orientation of Orbital 6. Orbitals within a subshell are distinguishable by their orientation. They are all at have the same energy level and are said to be degenerate. Shown below are the three different p orbitals.
16	Solution to the wave equation: Orientation of Orbital Orientation of the five d orbitals.
17	Solution to the wave equation: Size of Orbital And finally how are orbitals in different shells distinguished from one another? 7. Orbitals in different shells (energy levels) are distinguishable by their size (the average distance from the nucleus, also indicated by n) Orbitals in higher energy levels are bigger.
18	Solution to the wave equation: Size of Orbital Similarly, 3p is bigger than 2p, and 4p is bigger still. Here again, consistent with Bohr’s model of the atom, on average, 3p electrons would be found farther away from the nucleus than 2p electrons, and therefore they would be said to be at a higher energy level.
19	Relative energy of different orbitals Energy levels for a single-electron chemical species, such as hydrogen, He⁺ and Li²⁺ etc.
20	Relative energy of different orbitals This represents the energy levels for multi-electron chemical species. Notice that within an energy level, s is at a lower energy level than p, and p is at a lower energy than d etc. (In fact 4s is actually at a lower energy level than 3d which you do not have to concern yourselves with.) Within one-electron species such as hydrogen, the different subshells within a shell are found at the same energy level (see next slide). All electrons in 2p, for example, are said to be degenerate. All electrons in 3d can also be said to be degenerate.
21	Particle-wave duality Now to return to the figure of the woman you saw in the beginning of this set of slides. By now you should have noticed that there is a woman as well as a young lady in the picture. At any time, how YOU look at it will determine whether you see one or the other but never both at the same time! Therefore, the picture at any time it is viewed will either APPEAR to be that of the woman or the young girl. Similarly, what you should have noticed is that electrons have the dual characteristics of a particle and a wave. Which one it appears to be depends how you “look” at it, the perspective you take—which in this case is the experiment you perform to characterize it. Electrons will never be both at the same time, just as you weren’t able see both the women in the picture at the same time. That is, we have no single experiment which gives us evidence of both a particle and a wave just as there are no single pair of eye that can see at the same time both the girl and the woman.
22	Practice Questions 1. For an electron in a 1s orbital, where is the electron density greatest? What does this mean in terms of the location of the electron? 2. What is the energy level for which there are three and only three different types of orbitals? 3. Give an example of a d-orbital that is of lower energy than a p-orbital.
23	Practice Questions 4. Indicate the atoms which have the following electron arrangements: (a) 3d electrons, but no electrons of higher energy (b) 3p electrons, but no electrons of higher energy (c) 6s electrons, but no electrons of higher energy
24	Practice Questions: Structured Questions 1. M99/6. Use modern theory of the atom to answer each of the following. (a) List the d, f, p and s orbitals in the order of increasing relative energy. [2] (b) Give the number of each type of orbital, d, f, p and s at each energy level. [2] 2. Explain the following observations. (a) S orbital in s subshell in two different shells have the same shape. [1] (b) Orbitals in p subshell in two different shells have different sizes. [1]
25	Practice Questions: Structured Questions (c) P orbitals within the same shell have the shape and size. [1] (d) P orbitals within the same shell are orientated in different directions. [1] 3. Explain why orbitals within the same shell may be different in appearance and orientation. [2]
26	Practice React Questions 1. Explain how you can use the periodic table to determine the order in which orbitals fill in polyelectronic atoms (so that you do not have to memorize it).