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12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 Synopsis Spin The Helium Atom Spin Orbitals and the Pauli Principle Other Atoms and the Aufbau Principle Perturbation Theory Variation Theory Linear Variation Theory Comparison of Variation and Perturbation Theories Simple Molecules and the Born-Oppenheimer Approximation Introduction to LCAO-MO Theory Properties of Molecular Orbitals Molecular Orbitals of Other Diatomic Molecules Summary Atoms and Molecules W E HAVE SEEN HOW QUANTUM MECHANICS provides tools for understanding some simple systems, up to and including the hydrogen atom itself An understanding of the H atom is a crucial point because it is real, not a model system Quantum mechanics showed that it can describe the hydrogen atom like Bohr’s theory did It also describes other model systems that have applications in the real world (Recall that all of the model systems—particle-in-a-box, 2-D and 3-D rigid rotors, harmonic oscillators— could be applied to real systems even if the real systems themselves weren’t exactly ideal.) As such, quantum mechanics is more applicable than Bohr’s theory and can be considered “better.” We will conclude our development of quantum mechanics by seeing how it applies to more complicated systems: other atoms and even molecules What we will find is that explicit, analytic solutions to these systems are not possible, but quantum mechanics does supply the tools for understanding these systems nonetheless 12.1 Synopsis In this chapter, we will consider one more property of the electron, which is called spin Spin has dramatic consequences for the structure of matter, consequences that could not have been considered by the standards of classical mechanics We will see that an exact, analytic solution for an atom as simple as helium is not possible, and so the Schrödinger equation cannot be solved analytically for larger atoms or molecules But there are two tools for studying larger systems to any degree of accuracy: perturbation theory and variational theory Each tool has its advantages, and both of them are used today to study atoms and molecules and their reactions Finally, we will consider in a simple way how quantum mechanics considers a molecular system Molecules can get very complicated However, we can apply quantum mechanics to molecules We will finish this chapter with an introduction to molecular orbitals and how they are defined for a very simple molecule, H2ϩ Simple as this system is, it paves the way for other molecules 370 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 12.2 Spin 371 12.2 Spin Not long before quantum mechanics was developed, an important experimental observation was made In 1922, Otto Stern and W Gerlach attempted to measure the magnetic moment of the silver atom They passed vaporized silver atoms through a magnetic field and recorded the pattern that the beam of atoms made after it passed through the magnetic field Surprisingly, the beam split into two parts The experiment is illustrated in Figure 12.1 Attempts to explain this in terms of the Bohr theory and quantized angular momentum of electrons in their orbits failed Finally, in 1925, George Uhlenbeck and Samuel Goudsmit proposed that this result could be explained if it was assumed that the electron had its own angular momentum This angular momentum was an intrinsic property of the electron itself and not a consequence of any motion of the electron In order to explain the experimental results, Uhlenbeck and Goudsmit proposed that components of the intrinsic angular momentum, called spin angular momentum, had quantized values of either ϩᎏ12ᎏប or Ϫᎏ12ᎏប (Recall that h has units of angular momentum.) Since that proposal, it has become understood that all electrons have an intrinsic angular momentum called spin Although commonly compared to the spinning of a top, the spin angular momentum of an electron is not due to any rotation about the axis of the particle Indeed, it would be impossible for us to determine that an electron is actually spinning Spin is a property of a particle’s very existence This property behaves as if it were an angular momentum, so for all intents and purposes it is considered an angular momentum Like the angular momentum of an electron in its orbit, there are two measurables for spin that can be observed simultaneously: the square of the total spin and the z component of the spin Because spin is an angular momentum, there are eigenvalue equations for the spin observables that are the same as for ˆ L and ˆ Lz , except we use the operators ˆ S and ˆ Sz to indicate the spin observables We also introduce the quantum numbers s and ms to represent the quantized values of the spin of the particles (Do not confuse s, the symbol for the Glass plate Beam split into two N Magnet S Oven Beam of Ag atoms Figure 12.1 A diagram of the Stern-Gerlach experiment A beam of silver atoms passed through a magnetic field splits into two separate beams This finding was used to propose the existence of spin on the electron Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 372 C H A P T E R 12 Atoms and Molecules spin angular momentum, with s, an orbital that has ᐉ ϭ 0.) The eigenvalue equations are therefore ˆ S 2⌿ϭ s(s ϩ 1)ប2⌿ (12.1) Sz⌿ϭ ms ប⌿ (12.2) The values of the allowed quantum numbers s and ms are more restricted than for ᐉ and mᐉ All electrons have a value of s ϭ ᎏ12ᎏ The value of s, it turns out, is a characteristic of a type of subatomic particle, and all electrons have the same value for their s quantum number For the possible values of the z component of the spin, there is a similar relationship to the possible values of mᐉ and ᐉ: ms goes from Ϫs to ϩs in integral steps, so ms can equal Ϫᎏ12ᎏ or ϩᎏ12ᎏ Thus, there is only one possible value of s for electrons, and two possible values for ms Spin also has no classical counterpart Nothing in classical mechanics predicts or explains the existence of a property we call spin Even quantum mechanics, at first, did not provide any justification for spin It wasn’t until 1928 when Paul A M Dirac incorporated relativity theory into the Schrödinger equation that spin appeared as a natural theoretical prediction of quantum mechanics The incorporation of relativity into quantum mechanics was one of the final major advances in the development of the theory of quantum mechanics Among other things, it led to the prediction of antimatter, whose existence was verified experimentally by Carl Anderson (with the discovery of the positron) in 1932 Example 12.1 What is the value, in Jиs, of the spin of an electron? Compare this to the value of the angular momentum for an electron in s and p orbitals of an H-like atom Solution The value of the spin angular momentum of an electron is determined by using equation 12.1 We must recognize that the operator is the square of the total spin, and to find the value for spin we will have to take a square root We get spin ϭ ͙s(s ϩ 1ෆ2 ϭ ෆ)ប Ί 1 6.626 ϫ 10Ϫ34Jиs ᎏᎏ ᎏᎏ ϩ ᎏᎏ 2 2 ϭ 9.133 ϫ 10Ϫ35 Jиs The angular momentum of an electron in an s orbital is zero, since ᐉ ϭ for an electron in an s orbital In a p orbital, ᐉ ϭ 1, so the angular momentum is 6.626 ϫ 10Ϫ34 Jиs ͙ᐉ(ᐉ+ 1ෆប ϭ ͙1ෆ ෆ) и ᎏᎏ ϭ 1.491 ϫ 10Ϫ34 J и s 2 which is almost twice as great as the spin The magnitude of the spin angular momentum is not much smaller than the angular momentum of an electron in its orbit Its effects, therefore, cannot be ignored The existence of an intrinsic angular momentum requires some additional specificity when referring to angular momenta of electrons One must now Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 12.2 Spin 373 differentiate between orbital angular momentum and spin angular momentum Both observables are angular momenta, but they arise from different properties of the electron: one from its motion about a nucleus, the other from its very existence The spin angular momentum of an electron can have only certain specific values Spin is quantized Like the z component of orbital angular momentum, ms has 2s ϩ possible values In the case of the electron, s ϭ ᎏ12ᎏ, so the only possible values of ms are Ϫᎏ12ᎏ and ϩᎏ12ᎏ The specification of an electron’s spin therefore represents two other quantum numbers that can be used to label the state of that electron In practice, however, it is convenient to not specify s, since it is always ᎏ12ᎏ for electrons This gives us a total of four quantum numbers: the principal quantum number n, the orbital angular momentum quantum number ᐉ, the orbital angular momentum z component mᐉ, and the spin angular momentum (z component) ms These are the only four quantum numbers needed to specify the complete state of an electron Example 12.2 List all possible combinations of all four quantum numbers for an electron in the 2p orbital of a hydrogen atom Solution In tabular form, the possible combinations are Symbol n ᐉ mᐉ ms Possible values Ϫ1 ϩᎏ12ᎏ or Ϫᎏ12ᎏ ϩᎏ12ᎏ or Ϫᎏ12ᎏ ϩᎏ12ᎏ or Ϫᎏ12ᎏ There are a total of six possible combinations of the four quantum numbers for this case 2468 cmϪ1 H 400Å 500Å 600Å 700Å 21 cmϪ1 Figure 12.2 A very high resolution spectrum of the hydrogen atom shows a tiny splitting due to the spin on the electron This splitting is caused by the electron spin interacting with the nuclear spin of the hydrogen atom’s nucleus (a proton) Although not considered until now, the ms of the electron in a hydrogen atom is either ϩᎏ12ᎏ or Ϫᎏ12ᎏ A fascinating astronomical consequence of spin is the fact that an electron in hydrogen has a slightly different energy depending on the relative spin orientations of the electron and the proton in the nucleus (A proton also has a characteristic spin quantum number of ᎏ12ᎏ.) If an electron in a hydrogen atom changes its spin, there is a concurrent energy change that is equivalent to light having a frequency of 1420.40575 MHz, or a wavenumber of about 21 cmϪ1, as shown in Figure 12.2 Because of the pervasiveness of hydrogen in space, this “21-cmϪ1 radiation” is important for radio astronomers who are studying the structure of the universe Finally, since spin is part of the properties of an electron, its observable values should be determined from the electron’s wavefunction That is, there should be a spin wavefunction part of the overall ⌿ A discussion of the exact form of the spin part of a wavefunction is beyond our scope here However, since there is only one possible observable value of the total spin (s ϭ ᎏ12ᎏ) and only two possible values of the z component of the spin (ms ϭ ϩᎏ21ᎏ or Ϫᎏ21ᎏ), it is typical to represent the spin part of the wavefunction by the Greek letters ␣ and , depending on whether the quantum number ms is ϩᎏ21ᎏ or Ϫᎏ21ᎏ, respectively Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 374 C H A P T E R 12 Atoms and Molecules Spin is unaffected by any other property or observable of the electron, and the spin component of a one-electron wavefunction is separable from the spatial part of the wavefunction Like the three parts of the hydrogen atom’s electronic wavefunction, the spin function multiplies the rest of ⌿ So for example, the complete wavefunctions for an electron in a hydrogen atom are ⌿ ϭ Rn,ᐉ и ⍜ᐉ,mᐉ и mᐉ и ␣ for an electron having ms of ϩᎏ12ᎏ A similar wavefunction, in terms of , can be written for an electron having ms ϭ Ϫᎏ12ᎏ 12.3 The Helium Atom e1 r12 e2 r1 2+ r2 Figure 12.3 Definitions of the radial coordi- nates for the helium atom In the previous chapter, it was shown how quantum mechanics provides an exact, analytic solution to the Schrödinger equation when applied to the hydrogen atom Even the existence of spin, discussed in the last section, does not alter this solution (it only adds a little more complexity to the solution, a complexity we will not consider further here) The next largest atom is the helium atom, He It has a nuclear charge of 2ϩ, and it has two electrons about the nucleus The helium atom is illustrated in Figure 12.3, along with some of the coordinates used to describe the positions of the subatomic particles Implicit in the following discussion is the idea that both electrons of helium will occupy the lowest possible energy state In order to properly write the complete form of the Schrödinger equation for helium, it is important to understand the sources of the kinetic and potential energy in the atom Assuming only electronic motion with respect to a motionless nucleus, kinetic energy comes from the motion of the two electrons It is assumed that the kinetic energy part of the Hamiltonian operator is the same for the two electrons and that the total kinetic energy is the sum of the two individual parts To simplify the Hamiltonian, we will use the symbol ᭞2, called del-squared, to indicate the three-dimensional second derivative operator: Ѩ2 Ѩ2 Ѩ2 ᭞2 ϵ ᎏᎏ2 ϩ ᎏᎏ2 ϩ ᎏᎏ2 Ѩx Ѩy Ѩz (12.3) This definition makes the Schrödinger equation look less complicated ᭞2 is also called the Laplacian operator It is important to remember, however, that del-squared represents a sum of three separate derivatives The kinetic energy part of the Hamiltonian can be written as ប2 ប2 Ϫᎏᎏ᭞21 Ϫ ᎏᎏ᭞22 2 2 where ᭞21 is the three-dimensional second derivative for electron 1, and ᭞22 is the three-dimensional second derivative for electron The potential energy of the helium atom has three parts, all coulombic in nature: there is an attraction between electron and the nucleus, an attraction between electron and the nucleus, and a repulsion between electron and electron (since they are both negatively charged) Each part depends on the distance between the particles involved; the distances are labeled r1, r2, and r12 as illustrated in Figure 12.3 Respectively, the potential energy part of the Hamiltonian is thus 2e 2e e2 ˆ ϭ Ϫᎏ V ᎏ Ϫ ᎏᎏ ϩ ᎏᎏ 4⑀0r1 4⑀0r2 4⑀0r12 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 12.3 The Helium Atom 375 where the other variables have been defined in the previous chapter The in the numerator of each of the first two terms is due to the 2ϩ charge on the helium nucleus The first two terms are negative, indicating an attraction, and the final term is positive, indicating a repulsion The complete Hamiltonian operator for the helium atom is 2 2e 2e e2 ˆ ϭ Ϫᎏបᎏ᭞21 Ϫ ᎏបᎏ᭞22 Ϫ ᎏ H ᎏ Ϫ ᎏᎏ ϩ ᎏᎏ 2 2 4⑀0r1 4⑀0r2 4⑀0r12 (12.4) This means that for the helium atom, the Schrödinger equation to be solved is ប2 ប2 2e 2e e2 Ϫᎏᎏ᭞21 Ϫ ᎏᎏ᭞22 Ϫ ᎏᎏ Ϫ ᎏᎏ ϩ ᎏᎏ ⌿ ϭ Etot⌿ (12.5) 2 2 4⑀0r1 4⑀0r2 4⑀0r12 where Etot represents the total electronic energy of a helium atom The Hamiltonian (and thus the Schrödinger equation) can be rearranged by grouping together the two terms (one kinetic, one potential) that deal with electron only and also grouping together the two terms that deal with electron only: ប2 2e 2e e2 ˆ ϭ Ϫᎏបᎏ᭞21 Ϫ ᎏ H ᎏ ϩ Ϫᎏᎏ᭞22 Ϫ ᎏᎏ ϩ ᎏᎏ (12.6) 2 4⑀0r1 2 4⑀0r2 4⑀0r12 This way, the Hamiltonian resembles two separate one-electron Hamiltonians added together This suggests that perhaps the helium atom wavefunction is simply a combination of two hydrogen-like wavefunctions Perhaps a sort of “separation of electrons” approach will allow us to solve the Schrödinger equation for helium The problem is with the last term: e 2/4⑀0r12 It contains a term, r12, that depends on the positions of both of the electrons It does not belong only with the terms for just electron 1, nor does it belong only with the terms for just electron Because this last term cannot be separated into parts involving only one electron at a time, the complete Hamiltonian operator is not separable and it cannot be solved by separation into smaller, one-electron pieces In order for the Schrödinger equation for the helium atom to be solved analytically, it either must be solved completely or not at all To date, there is no known analytic solution to the second-order differential Schrödinger equation for the helium atom This does not mean that there is no solution, or that wavefunctions not exist It simply means that we know of no mathematical function that satisfies the differential equation In fact, for atoms and molecules that have more than one electron, the lack of separability leads directly to the fact that there are no known analytical solutions to any atom larger than hydrogen Again, this does not mean that the wavefunctions not exist It simply means that we must use other methods to understand the behavior of the electrons in such systems (It has been proven mathematically that there is no analytic solution to the so-called three-body problem, as the He atom can be described Therefore, we must approach multielectron systems differently.) Nor should this lack be taken as a failure of quantum mechanics In this text, we can only scratch the surface of the tools that quantum mechanics provides Quantum mechanics does provide tools to understand such systems Atoms and molecules having more than one electron can be studied and understood by applying such tools to more and more exacting detail The level of detail depends on the time, resources, and patience of the person applying the tools In theory, one can determine energies and momenta and other observables to the Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 376 C H A P T E R 12 Atoms and Molecules same level that one can know such observables for the hydrogen atom—if one has the tools Example 12.3 Assume that the helium wavefunction is a product of two hydrogen-like wavefunctions (that is, neglect the term for the repulsion between the electrons) in the n ϭ principal quantum shell Determine the electronic energy of the helium atom and compare it to the experimentally determined energy of Ϫ1.265 ϫ 10Ϫ17 J (Total energies are determined experimentally by measuring how much energy it takes to remove all of the electrons from an atom.) Solution Using equation 12.6 and neglecting the electron-repulsion term by assuming that the wavefunction is the product of two hydrogen-like wavefunctions: ⌿He ϭ ⌿H,1 ϫ ⌿H,2 the Schrödinger equation for the helium atom can be approximated as ΄ ប2 ប2 2e 2e Ϫᎏᎏ᭞21 Ϫ ᎏᎏ ϩ Ϫᎏᎏ᭞22 Ϫ ᎏᎏ ⌿H,1⌿H,2 Ϸ EHe⌿H,1⌿H,2 2 4⑀0r1 2 4⑀0r2 ΅ where EHe is the energy of the helium atom Because the first term in brackets is a function of only electron and the second term in the brackets is a function of only electron 2, this Schrödinger equation can be separated just like a two-dimensional particle-in-a-box can be separated Understanding this, we can separate the Schrödinger equation above into two parts: ប2 2e Ϫᎏᎏ᭞21 Ϫ ᎏᎏ ⌿H,1 ϭ E1⌿H,1 2 4⑀0r1 ប2 2e Ϫᎏᎏ᭞22 Ϫ ᎏᎏ ⌿H,2 ϭ E2⌿H,2 2 4⑀0r2 where EHe ϭ E1 ϩ E2 These expressions are simply the one-electron Schrödinger equations for a hydrogen-like atom where the nuclear charge equals An expression for the energy eigenvalue for such a system is known From the previous chapter, it is Z 2e4 E ϭ Ϫᎏ ᎏ 8⑀20h2n2 for each hydrogen-like energy For this approximation, we are assuming that helium is the sum of two hydrogen-like energies Therefore, EHe ϭ EH,1 ϩ EH,2 e 4 22e4 22e4 ϭ Ϫᎏ ᎏ 2ᎏ 2 Ϫ ᎏ 2ᎏ 2 ϭ Ϫᎏ 8⑀0h n 8⑀0h n ⑀0h2n2 where we get the final term by combining the two terms to the left Keep in mind that is the reduced mass for an electron about a helium nucleus, and that the principal quantum number is for both terms Substituting the values of the various constants, along with the value for the reduced mass of the electron-helium nucleus system (9.108 ϫ 10Ϫ31 kg), we get EHe ϭ Ϫ1.743 ϫ 10Ϫ17 J Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 12.4 Spin Orbitals and the Pauli Principle 377 which is low by ϳ37.8% compared to experiment Ignoring the repulsion between the electrons leads to a significant error in the total energy of the system, so a good model of the He atom should not ignore electron-electron repulsion The example above shows that assuming that the electrons in helium—and any other multielectron atom—are simple combinations of hydrogen-like electrons is naive assumption, and predicts quantized energies that are far from the experimentally measured values We need other ways to better estimate the energies of such systems 12.4 Spin Orbitals and the Pauli Principle Example 12.3 for the helium atom assumed that both electrons have a principal quantum number of If the hydrogen-like wavefunction analogy were taken further, we might say that both electrons are in the s subshell of the first shell—that they are in 1s orbitals Indeed, there is experimental evidence (mostly spectra) for this assumption What about the next element, Li? It has a third electron Would this third electron also go into an approximate 1s hydrogen-like orbital? Experimental evidence (spectra) shows that it doesn’t Instead, it occupies what is approximately the s subshell of the second principal quantum shell: it is considered a 2s electron Why doesn’t it occupy the 1s shell? We begin with the assumption that the electrons in a multielectron atom can in fact be assigned to approximate hydrogen-like orbitals, and that the wavefunction of the complete atom is the product of the wavefunctions of each occupied orbital These orbitals can be labeled with the nᐉ quantum number labels: 1s, 2s, 2p, 3s, 3p, and so on Each s, p, d, f, subshell can also be labeled by an mᐉ quantum number, where mᐉ ranges from Ϫᐉ to ᐉ (2ᐉ ϩ possible values) But it can also be labeled with a spin quantum number ms, either ϩ12 or Ϫ12 The spin part of the wavefunction is labeled with either ␣ or , depending on the value of ms for each electron Therefore, there are several simple possibilities for the approximate wavefunction for, say, the lowest-energy state (the ground state) of the helium atom: ⌿He ϭ (1s1␣)(1s2␣) ⌿He ϭ (1s1␣)(1s2) ⌿He ϭ (1s1)(1s2␣) ⌿He ϭ (1s1)(1s2) where the subscript on 1s refers to the individual electron We will assume that each individual ⌿He is normalized Because each ⌿He is a combination of a spin wavefunction and an orbital wavefunction, ⌿He’s are more properly called spin orbitals Because spin is a vector and because vectors can add and subtract from each other, one can easily determine a total spin for each possible helium spin orbital (It is actually a total z component of the spin.) For the first spin orbital equation above, both spins are ␣, so the total spin is (ϩᎏ12ᎏ) ϩ (ϩᎏ12ᎏ) ϭ Similarly, for the last spin orbital, the total spin is (Ϫᎏ12ᎏ) ϩ (Ϫᎏ12ᎏ) ϭ Ϫ1 For the middle two spin orbitals, the total (z-component) spin is exactly zero To summarize: Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 378 C H A P T E R 12 Atoms and Molecules Approximate wavefunction Total z-component spin ⌿He ϭ (1s1␣)(1s2␣) ⌿He ϭ (1s1␣)(1s2) ⌿He ϭ (1s1)(1s2␣) ⌿He ϭ (1s1)(1s2) ϩ1 0 Ϫ1 At this point, experimental evidence can be introduced (The necessity of comparing the predictions of theory with experiment should not be forgotten.) Angular momenta of charged particles can be differentiated by magnetic fields, so there is a way to experimentally determine whether or not atoms have an overall angular momentum Since spin is a form of angular momentum, it should not be surprising that magnetic fields can be used to determine the overall spin in an atom Experiments show that ground-state helium atoms have zero z-component spin This means that of the four approximate wavefunctions listed above, the first and last are not acceptable because they not agree with experimentally determined facts Only the middle two, (1s1␣)(1s2) and (1s1)(1s2␣), can be considered for helium Which wavefunction of the two is acceptable, or are they both? One can suggest that both wavefunctions are acceptable and that the helium atom is doubly degenerate This turns out to be an unacceptable statement because, in part, it implies that an experimenter can determine without doubt that electron has a certain spin wavefunction and that electron has the other spin wavefunction Unfortunately, we cannot tell one electron from another They are indistinguishable This indistinguishability suggests that the best way to describe the electronic wavefunction of helium is not by each wavefunction individually, but by a combination of the possible wavefunctions Such combinations are usually considered as sums and/or differences Given n wavefunctions, one can mathematically determine n different combinations that are linearly independent So, for the two “acceptable” wavefunctions of He, two possible combinations can be constructed to account for the fact that electrons are indistinguishable These two combinations are the sum and the difference of the two individual spin orbitals: ⌿He,1 ϭ ᎏᎏ[(1s1␣)(1s2) ϩ (1s1)(1s2␣)] ͙2ෆ ⌿He,2 ϭ ᎏᎏ[(1s1␣)(1s2) Ϫ (1s1)(1s2␣)] ͙2ෆ The term 1/͙2ෆ is a renormalization factor, taking into account the combination of two normalized wavefunctions These combinations have the proper form for possible wavefunctions of the helium atom Are both acceptable, or only one of the two? At this point we rely on a postulate proposed by Wolfgang Pauli in 1925, which was based on the study of atomic spectra and the increasing understanding of the necessity of quantum numbers Since electrons are indistinguishable, one particular electron in helium can be either electron or We can’t say for certain which But because the electron has a spin of ᎏ21ᎏ, it has certain properties that affect its wavefunction (the details of which cannot be considered here) If electron were exchanged with electron 2, Pauli postulated, the complete wavefunction must change sign Mathematically, this is written as ⌿(1, 2) ϭ Ϫ⌿(2, 1) Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 12.4 Spin Orbitals and the Pauli Principle 379 The switch in order of writing the labels and implies that the two electrons are exchanged Electron now has the coordinates of electron 2, and vice versa A wavefunction having this property is called antisymmetric (By contrast, if ⌿(1, 2) ϭ ⌿(2, 1), the wavefunction is labeled symmetric.) Particles having half-integer spin (ᎏ21ᎏ, ᎏ23ᎏ, ᎏ25ᎏ, ) are collectively called fermions The Pauli principle states that fermions must have antisymmetric wavefunctions with respect to exchange of particles Particles having integer spins, called bosons, are restricted to having symmetric wavefunctions with respect to exchange Electrons are fermions (having spin ϭ ᎏ12ᎏ) and so according to the Pauli principle must have antisymmetric wavefunctions Consider, then, the two possible approximate wavefunctions for helium They are ⌿He,1 ϭ ᎏᎏ[(1s1␣)(1s2) ϩ (1s1)(1s2␣)] ͙2ෆ (12.7) ⌿He,2 ϭ ᎏᎏ[(1s1␣)(1s2) Ϫ (1s1)(1s2␣)] ͙2ෆ (12.8) Are either of these antisymmetric? We can check by interchanging electrons and in the first wavefunction, equation 12.7, and get ⌿(2, 1) ϭ ᎏᎏ[(1s2␣)(1s1) ϩ (1s2)(1s1␣)] ͙2ෆ (Note the change in the subscripts and 2.) This should be recognized as the original wavefunction ⌿(1, 2), only algebraically rearranged (Show this.) However, upon electron exchange, the second wavefunction, equation 12.8, becomes ⌿(2, 1) ϭ ᎏᎏ[(1s2␣)(1s1) Ϫ (1s2)(1s1␣)] ͙2ෆ (12.9) which can be shown algebraically to be Ϫ⌿(1, 2) (Show this, also.) Therefore, this wavefunction is antisymmetric with respect to exchange of electrons and, by the Pauli principle, is a proper wavefunction for the spin orbitals of the helium atom Equation 12.8, but not equation 12.7, represents the correct form for a spin-orbital wavefunction of the ground state of He The rigorous statement of the Pauli principle is that wavefunctions of electrons must be antisymmetric with respect to exchange of electrons There is a simpler statement of the Pauli principle It comes from the recognition that equation 12.8, the only acceptable wavefunction for helium, can be written in terms of a matrix determinant Recall that the determinant of a ϫ matrix written as ͉ ͉ a d c b is simply (a ϫ b) Ϫ (c ϫ d ), which is remembered mnemonically as ϩ a d→ Ϫc ϫ d → aϫb Ϫ c b The proper antisymmetric wavefunction, equation 12.8, for the helium atom can also be written in terms of a ϫ determinant: ͉ ͉ ⌿He ϭ ᎏᎏ ͙2ෆ ͉ ͉ 1s1␣ 1s1 1s2␣ 1s2 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part (12.10) INDEX group frequency regions in vibrational spectroscopy, 504–506 threshold frequency, 253 Freundlich isotherm, 786 frictional proportionality constant, in ionic solutions, 235–236 fugacity description, 110–114 in liquid/liquid systems, 170 fundamental equation of chemical thermodynamics, 114 fusion description, 51, 143 heat of fusion, 51, 146 G galvanic cell, 215 gamma function, 673 gamma rays, 464 Gamow, George, 298 gases catalysis, 783–788 heat of adsorption, 787 liquid/gas systems, 183–184, 194 phase diagrams, 154–159, 174–175 vaporization, see vaporization vapor pressure, see vapor pressure gas laws Boyle’s gas law, 6, 15, 50 Charles’s gas law, collisions, 666–671 description, 1, 21 diffusion, 671–677 effusion, 671–677 first law of thermodynamics, 26–28 ideal gas constant, 7, 9, 553 ideal gas law, 7, 609, 668 kinetic theory of gases, 47, 651–679 monatomic gases, 604–608 nonideal gases, 10–17 partial derivatives, 8–10, 18–21, 96–99 partition function, 604–608 postulates, 652–656 pressure, 652–656 second law of thermodynamics, 77–78 velocity distributions, 656–666 zeroth law of thermodynamics, 1–23 gauss, 561 Gauss, Karl F., 319 Gaussian-type function, 319 Germer, Lester H., 268 Gibbs, J Willard, 159, 586 Gibbs, Josiah W., 93 Gibbs free energy for chemical equilibrium, 123–128 825 chemical potential relationship, 108–110, 114, 118, 121, 604 derivation from partition functions, 638–639 description, 89, 92–96, 114 in electrochemical reactions, 210–213, 216–217, 221 Eyring equation, 722–723 for ionic solutions, 228–229 isothermic processes, 95, 147 in single-component systems, 159–160 spontaneity determination, 92–93, 108 statistical thermodynamics, 601–602, 610–611 surface tension relationship, 768–771, 779 variation with temperature, 105–108 Gibbs phase rule for multiple-component systems, 166–169, 189 for single-component systems, 154–159 for solid/solid solutions, 189 Gibbs surface energy, 768–771, 779 glass annealing, 780 properties, 732 glass pH electrode, 223 Graham, Thomas, 677 Graham’s law, 677 grating, 740 great orthogonality theorem, in group theory, 438–441, 537 gross selection rule, 472, 513, 520 Grotrian, Walter, 533 Grotrian diagrams, 533–534 ground state configuration of elements, 387 for electron orbitals, 382, 539 in harmonic oscillators, 323 partition functions, 618–619 group frequency regions, in vibrational spectroscopy, 504–506 group theory great orthogonality theorem, 438–441, 537 in vibrational spectroscopy, 498 H half-cells, 215 half-life, rate laws, 688–690, 701–702 half-reactions, redox reactions, 214–216 Hamilton, William R., 244 Hamiltonian function description, 244–248, 285–286, 300 for helium atom, 374–375 kinetic energy relationship, 245, 286–287 in three-dimensional rotations, 341–342 in variation theory, 395–396 hard-sphere model, gas particle collisions, 666 harmonic oscillator classical harmonic oscillator, 316–318, 624 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 826 INDEX harmonic oscillator (continued) description, 315–329 hydrogen atom, 332–333 ideal harmonic potential, 491 motion equations, 245–246 quantum-mechanical harmonic oscillator, 318–324, 484–487 vibration treatment, 484–485, 490–491, 624 wavefunctions, 321–329 heat adiabatic systems, 33, 41, 48–49 Carnot cycle, 68–73, 94 changing temperatures, 29, 58–60 description, 4, 24–32 heat of adsorption, 787 heat of formation, 55–57 heat of fusion, 51, 146 heat of sublimation, 146 heat of vaporization, 51–53, 146 mechanical equivalent, 30 specific heat, 31, 40 heat capacity constant volume heat capacity, 39 of crystals, 644–645 derivation from partition functions, 638–640 entropy change, 75–76 first law of thermodynamics, 31, 39–42, 46–50 Heisenberg, Werner, 269, 279–280 Heisenberg’s uncertainty principle, 279–281 Heitler, W., 446 helium charge, 374–375, 396–397 Grotrian diagram, 533 lasers, 556 partition function, 604–608 Schrödinger equation, 374–376, 413 wavefunctions, 376–378, 396 Helmholtz energy description, 89, 92–96, 114 isothermic processes, 92–94 statistical thermodynamics, 601–602, 604, 610–611 Henry, William, 184 Henry’s law, in liquid/gas systems, 183–184 Hermite, Charles, 279, 326 Hermite polynomials, 326–327 Hermitian operators, 279 Hertz, Heinrich, 253 Hess, Germain H., 54 Hess’s law description, 54, 56, 61 entropy changes, 82 in redox reactions, 216–217 Hinshelwood, Cyril N., 786 Hooke’s-law harmonic oscillator, see Harmonic oscillator Hook’s law motion equations, 245–246, 316 vibration treatment, 484 Hückel, Erich, 230, 543 Hückel approximations, in electronic spectroscopy, 543–546 Hund’s rule, 384, 532, 538 hybrid orbitals, symmetry, 450–456 hydrochloric acid, vibrational parameters, 490–491, 507–508, 625 hydrogen atom Bohr’s theory, 262–267 central force problem, 352–353, 365 electronic spectrum, see electronic spectroscopy electron spin, 373 harmonic oscillation, 332–333 quantum mechanics, 262–267, 352–365, 373 quantum numbers, 373, 380 symmetry, 442–443, 633 wavefunctions, 355–365 hyperfine coupling, 569–571 I ideal gas constant, 7, 9, 553 ideal gases description, fugacity, 110–113 Gibbs free energy variation, 108 ideal gas law, 7, 609, 668 Joule-Thomson coefficients, 44–45, 103–104 kinetics, 651–679 real gases compared, 11 identity element, 420 immiscibility, 182 inertia, moment of inertia, 334 inertial axes, 467–468 inexact differentials, 35 infrared radiation characteristics, 464–465 fingerprint regions, 504–506 vibrational symmetry, 494–496, 499–501, 503 integrals Coulomb integrals, 449 overlap integrals, 398, 407 resonance integrals, 407 symmetry, 441–443, 449 integrated rate laws, 686–688 intensive variables, 216 interface effects, 771–777 interference, 742 internal energy chemical changes, 37, 53–58 in electrochemistry, 210 enthalpy relationship, 601 first law of thermodynamics, 32–33 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part INDEX ideal liquid solution mixing, 178–179 Joule-Thomson coefficients, 42–46, 103–104 natural variable equations, 96–99, 104 spontaneity determination, 90 state function, 33–36, 38–42 internal pressure, 28 interstitial defect, 759 inversion center of inversion, 420 inversion temperature, 45 ions Debye-Hückel Theory, 230–234, 646 electrochemistry, see electrochemistry ionic crystals, 732, 734, 752–759 ionic radius, 752–754 ionic strength, 228, 230–234 ionic transport, 234–237 ions in solution, 225–234 ion-specific electrodes, 223–224 pH, see pH redox reactions, see redox reaction ion-specific electrodes, 223–224 irreversible processes, 28, 74–75 isenthalpy, 43, 90–91 isobaric change, 42 isochoric change description, 42 Helmholtz energy, 92 isoelectric point, 136 isolated systems, 32, 75 isothermic processes description, 28–29, 41, 58 entropy, 72–73, 92 Freundlich isotherm, 786 fugacity, 111 Gibbs free energy, 95, 147 Helmholtz energy, 92–94 isothermal compressibility, 20, 94, 102 Langmuir-Hinshelwood isotherms, 786 Langmuir isotherms, 784, 786 phase transitions, 146–147 kinetic energy Hamiltonian function relationship, 245, 286–287 harmonic oscillation, see harmonic oscillator Lagrange’s equations, 246 overview, 243, 259, 652, 656 quantum mechanics, 259 two-dimensional rotations, 334 kinetics chain reactions, 714–719 collisions, 666–671 consecutive reactions, 696–702 diffusion, 671–677 effusion, 671–677 equilibrium for simple reactions, 694–696 mechanisms, 706–710 oscillating reactions, 714–719 overview, 47, 651–652, 677, 680–681, 725 parallel reactions, 696–702 postulates, 652–656 pressure, 652–656 radioactivity, 688–690, 701–702 rate laws, 681–694 steady-state approximation, 710–714 temperature dependence, 683, 702–706 thermodynamics compared, 680, 694 transition-state theory, 719–725 velocity distributions, 656–666 kinetic theory of gases collisions, 666–671 diffusion, 671–677 effusion, 671–677 overview, 47, 651–652, 677 postulates, 652–656 pressure, 652–656 velocity distributions, 656–666 Kirchhoff, Gustav R., 248–249, 257 Knudsen, Martin, 674 Knudsen cells, 674 Kohlrausch, Friedrich, 237 Kohlrausch’s law, 237 L J Jeans, James H., 256 j-j coupling scheme, 526 joule, 30 Joule, James P., 30 Joule-Thomson coefficients description, 42–46, 103–104 inversion temperature, 45 K Kamerlingh-Onnes, Heike, 46 Kelvin, Kepler, Johannes, 752 Lagrange, Joseph L., 243 Lagrange function, 244, 246–248 Laguerre polynomials, 354 Landé, Alfred, 567 Landé g factor, 566–567 Langmuir, Irving, 777 Langmuir-Blodgett film, 777 Langmuir-Hinshelwood isotherms, 786 Langmuir isotherms, 784, 786 Langrange’s method of undetermined multipliers, 595 Laplace-Young equation, 773, 776 Laplacian operator, 299, 374 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 827 828 INDEX lasers, 550–557 lattice structure, see crystals law of corresponding states, 645–646 laws, see also specific laws description, laws of motion Brownian motion, 676 classical mechanics, 242–248, 280, 316–318, 653 collisions, see collisions description, 242–248 Franck-Condon principle, 539–541 momentum, see momentum Newton’s laws of motion, 242–243, 653–654 Pauli exclusion principle, 377–382, 413, 532, 537, 630–631 quantum mechanics, see quantum mechanics rotation, see rotation spin, see spin vibration, see vibrational spectroscopy LCAO-MO theory, 405–409 Le Chantelier’s principle, 133 Lewis, Gilbert N., 228, 261 light characteristics, 464 classical properties, 253–257 photoelectric effect, 253, 259 quantum mechanics, see quantum mechanics linear combination in perturbation theory, 391 symmetry-adapted linear combinations, 443–446 linear momentum, 334 Lineweaver-Burk plot, 714 liquids boiling point elevation, 194, 196 catalysis, 783–788 Clapeyron equation, 151 description, 51–53, 143 heat of vaporization, 51–53, 146 liquid/gas systems, 183–184, 194 liquid/liquid systems, 169–179, 193, 201 multicomponent systems, 169–188 normal boiling point, 144 normal melting point, 143 oscillating reactions, 718 phase diagrams, 154–159, 174–175, 753 phase transitions, 143, 145–148 solutions, see solutions surface interface effects, 771–777 surface tension, 766–771 London, F W., 446 Lord Kelvin, Lorentz, Hendrik, 564 Lotka, Alfred, 718 M macroscopic rules, 24 Madelung constant, 757–758 magnetic resonance imaging, 560, 582 magnetic spectroscopy electric charges, 561–564 electron spin resonance, 567–571 magnetic dipoles, 561–564 magnetic inductions, 561–564 nuclear magnetic resonance, 571–582 overview, 560–561, 582–583 Zeeman spectroscopy, 560, 564–567 magnetogyric ratio, 574 magnetons Bohr magneton, 564, 568 nuclear magneton, 572 Maiman, Theodore, 554 Marsden experiment, 251–252 mass, see also momentum classical turning point, 328 harmonic oscillation, 330–333 reduced mass, 330–333, 339 three-dimensional rotations, 341–347 two-dimensional rotations, 333–341 Maxwell, James C., 101, 252, 586, 651 Maxwell-Boltzmann distribution, 593–602, 663, 666 Maxwell relationships application, 103–105 derivation from natural variable equation, 162 description, 99–103 mean free path, 667–669 mechanical equivalent of heat, 30 melting, 143 metals, see also specific metals; specific properties alloys, 188, 191 amalgams, 188 annealing, 780 bonding, 732–733 corrosion, 217–218 electroplating, 215 methane, vibrational modes, 488 methylacetylene, infrared absorption spectra, 503–504 Michaelis-Menten equation, 714 microscopic rules, 24 microsystems, statistical thermodynamics, 590–593 microwaves, 464–465, 514 Miller indices, 744–752, 778–780 Millikan, Robert, 251 Millikan oil drop experiment, 251–252 mixing enthalpy, 78–79 entropy, 78–79 internal energy of ideal liquid solutions, 178–179 molality, of solutions, 193–194, 226–227 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part INDEX molar energy, 656 molar heat capacity, 40, 47–48 molar volume, 10 molecularity, 707 molecules Born-Oppenheimer approximation, 403–405, 539 centrifugal distortions, 479–481 diatomic molecules, see diatomic molecules electronic partition functions, 621–623 fingerprint regions, 504–506 LCAO-MO theory, 405–409 orbital properties, 409–415 overview, 370, 413 polyatomic molecules, see polyatomic molecules property derivation from partition functions, 637–640 rotation, see rotation rotational-vibrational spectroscopy, 506–511 symmetry, 427–430, 482–483, 631 vibration, see vibrational spectroscopy mole fraction description, 78 vapor-phase mole fractions, 173–174 moment of inertia, 334 momentum angular momentum, see angular momentum average values, 294–295, 329 classical definition, 280 conjugate momenta, 244 de Broglie wavelength relationship, 267–269, 280 harmonic oscillation, see harmonic oscillator linear momentum, 334 three-dimensional rotations, 341–347 two-dimensional rotations, 333–341 monatomic gases kinetics, 656 partition functions, 604–608 Morse potential, 492–493 motion, see laws of motion multicomponent systems, see also single-component systems colligative properties, 193–202 description, 142 equilibria, 166–205 Gibbs phase rule, 166–169, 189 Henry’s law, 183–184 liquid/gas systems, 183–184, 194 liquid/liquid systems, 169–179, 193, 201 liquid/solid solutions, 185–188, 194 nonideal two-component liquid solutions, 179–183 overview, 166, 201–202 solid/solid solutions, 188–193 N natural variables for enthalpy, 91 equations, 96–99, 104, 144 Helmholtz energy, 92 in single-component systems, 144, 159–162 in state functions, 90 negative deviation, in vapor pressure, 179 Nernst, Walther H., 218 Nernst equation, in nonstandard conditions, 218–223 Newton, Isaac, 242 Newton’s laws of motion, 242–243, 653–654 nodes, 362 nondegenerate perturbation theory, 386–394, 402–403 nonideal gases description, 10–17 fugacity, 110–113 non-spontaneous changes, 67 normality description, 236 orthonormality, 307 normalization, 283–285, 303, 335–336, 435 nuclear decay, kinetics, 688–690, 701–702 nuclear magnetic dipole, 572 nuclear magnetic resonance, 571–582 nuclear magneton, 572 nuclear partition functions, statistical thermodynamics, 617–621, 633 O observables in quantum mechanics, 276–279, 288, 347–352 rotating systems, 347–352 occupation numbers, 588 Ohm’s law, 236 Onsäger, Lars, 237 Onsäger equation, 237 operators, 276–279, 288 Oppenheimer, J Robert, 404 orbital properties Aufbau principle, 382–386 Hückel approximations, 543–546 hybrid orbitals, 450–456 LCAO-MO theory, 405–409 molecular orbitals, 409–415 orbital angular momentum, 373, 522–525, 535 electron systems, 543–546 spin angular momentum compared, 373 spin orbitals, 377–382 symmetry, see symmetry term symbols, 526–534 order, entropy, 79–81, 602 orders, in rate laws, 683–685 orthogonality orthogonality theorem, 438–441, 537 wavefunctions, 306–307 orthonormality, 307 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 829 830 INDEX oscillating reactions, kinetics, 714–719 osmotic pressure applications, 200–201 description, 196–201 overlap integrals, 398, 407 overtone spectroscopy, 489, 503–504 oxidation-reduction reaction, see redox reaction P parallel reactions, kinetics, 696–702 partial molar quantity, chemical potential, 108–110, 114 partial pressures, in liquid/liquid systems, 171–175 particle-in-a-box solution degeneracy, 303–306, 605 description, 288–295 monatomic gases, 605–606 three-dimensional solution, 299–303 in variation theory, 395–396 partition functions description, 586, 596–600 electronic partition functions, 617–623 equilibria, 640–644 molecular partition function, 628 molecular properties derivation, 637–640 monatomic gases, 604–608 nuclear partition functions, 617–621, 633 rotational partition function, 634–636 of a system, 636–637 in transition-state theory, 721–722 Pascal, units of measure, path-dependent qualities, 34–35, 77 path-independent qualities, 34 Pauli, Wolfgang, 378 Pauli exclusion principle, 377–382, 413, 532, 537, 630–631 electron systems, Hückel approximations, 543–546 permeability membranes, 196–197, 200 of a vacuum, 561 permittivity of free space, 208 perturbation theory in quantum mechanics, 386–394, 402–403 variation theory compared, 402–403 pH glass pH electrode, 223 ion-specific electrodes, 223–224 isoelectric point, 136 measurement, 223–224 phase changes, first law of thermodynamics, 50–53 phase diagrams description, 201–202 for liquid/liquid systems, 174–177 in nonideal two-component liquid solutions, 180–183 for single-component systems, 154–159 for solid/solid systems, 190–192 phase rule, see Gibbs phase rule phase transitions enthalpy, 55–57, 146–147 entropy, 147–148, 160 Gibbs free energy, 146–147 phase diagrams, 154–159, 174–175 in single-component systems, 143, 145–148 phonon, 548 phosphorescence, 548–550 photoelectric effect classical mechanics, 253 quantum mechanics, 259 photons description, 464 fluorescence, 548–550 lasers, 550–556 quantized vibrational energy levels, 511 photosynthesis, thermodynamics, 60–61 physisorption, 787–788 Planck, Max K E L., 257 Planck’s constant, 258, 485 Planck’s radiation law, 258–259, 550 point groups, in symmetry operations, 420–435 polar coordinates description, 334 spherical polar coordinates, 341–342 polarizability, dipole moments, 513 polyatomic molecules electronic spectroscopy, 541–543 Franck-Condon principle, 541 rotational motion, 466–467, 634–636 vibrational motion, 481–484, 493–494, 500, 541, 626–627 polymorphism, 143 position, Heisenberg’s uncertainty principle, 279–281 position operators, 278, 288 positive deviation, in vapor pressure, 179 postulates kinetic theory of gases, 652–656 in quantum mechanics, 273, 309–310 potential energy Born-Oppenheimer approximation, 403–405, 539 central force problem, 353 description, 244 Hamiltonian function relationship, 245 harmonic oscillation, see harmonic oscillator Lagrange’s equations, 246 lattice energies of ionic crystals, 755–759 Morse potential, 492–493 particle-in-a-box solution, 288–292, 299–303 tunneling, 296–299 vibration treatment, 484–485, 491 power, definition, 255 power density, of light, 254, 256 power flux, 259 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part INDEX pressure chemical equilibrium relationship, 125–129 Clapeyron equation, 148–152, 155 common units, 2–3 constant-pressure heat capacity, 41–43 critical pressure, 155–156 equations of state, 5–9, 105 equilibria in single-component systems, 141–165 equilibrium constant, 643 films, 777–778 fugacity relationship, 113 gradients, 672 Henry’s law, 183–184 internal pressure, 28 isobaric change, 42 Joule-Thomson coefficients, 42–46, 103–104 kinetic theory, 655–656 in liquid/gas systems, 183–184 osmotic pressure, 196–201 partial pressures, 171–175 phase diagrams, 154–159, 174–175 SI units, 2–3 standard temperature and pressure, 7–8 surface interface effects, 771–777 vapor pressure, see vapor pressure principal inertial axes, 467–468 principal quantum number, 355–356 principle of equal a priori probabilities, 592 Prokhorov, Alexander, 554 propagating reactions, kinetics, 714–719 proportionality constant, 257 pseudo rate constant, 692–693 Q quadratic equation, 401 quantum mechanics Aufbau principle, 382–386 average values, 293–296, 329 Bohr’s theory of the hydrogen atom, 262–267 Born interpretation, 281–283 Born-Oppenheimer approximation, 403–405, 539 central force problem, 352–358, 365 classical harmonic oscillator, 316–318, 624 de Broglie equation, 267–269, 280 degeneracy, 303–306, 605, 618, 631–632 harmonic oscillator, 315–329 helium atom, 374–378, 396 historical perspectives, 257–262, 269–270 hydrogen atom, 262–267, 352–365, 373 LCAO-MO theory, 405–409 linear variation theory, 398–402 nondegenerate perturbation theory, 386–394, 402–403 normalization, 283–285, 303, 335–336 observables, 276–279, 288, 347–352 831 operators, 276–279, 288 orbital properties, 409–415 orthogonality, 306–307 overview, 273–274, 309–310, 315–316, 365–366, 370, 413 particle-in-a-box solution, 288–292, 299–303, 605–606 Pauli exclusion principle, 377–382, 413, 532, 537, 630–631 perturbation theory, 386–394, 402–403 postulates, 273, 309–310 pre-quantum mechanics, see classical mechanics probabilities, 281–283 quantum energy, 257–258, 304–305 quantum-mechanical harmonic oscillator, 318–324, 484–487 reduced mass, 330–333, 339 Schrödinger equation, see Schrödinger equation selection rules, 462–463, 471–473, 487–490 of spectroscopy, see spectroscopy spin, 371–374 spin orbitals, 377–382 symmetry, see symmetry three-dimensional rotations, 341–347 tunneling, 296–299 two-dimensional rotations, 333–341 uncertainty principle, 279–281 variation theory, 394–397, 402–403 of vibration, 484–487 wavefunctions, see wavefunctions quantum numbers angular momentum, 357, 521–525 centrifugal distortions, 479–481 description, 264, 291 hydrogen atom, 373, 380 letter designation, 358 Pauli exclusion principle, 377–382, 413, 532, 537, 630–631 principal quantum number, 355–356 rotational spectroscopy, 477–478 rotational-vibrational spectroscopy, 507–508 term symbols, 526–534 vibrational spectroscopy, 90 zero-point energy, 323 R radial nodes, 362 radiation, see specific types radiationless transitions, 548 radioactivity historical perspectives, 253 kinetics, 688–690, 701–702 radio waves, 464–465 Raman, Chandrasekhara, 511 Raman spectroscopy, 511–514 Raoult’s law in liquid/gas systems, 183 in liquid/liquid systems, 171–174, 178–179, 193 in nonideal two-component liquid solutions, 179–180 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 832 INDEX rate laws first-order reactions, 686–694 half-life, 688–690, 701–702 initial reaction rate, 681 integrated rate laws, 686–688 orders, 683–685 pseudo rate constant, 692–693 rate constant, 683, 720–722 rate-determining step, 709–712 second-order reactions, 688–690, 693–694 steady-state approximation, 710–714 temperature dependence, 683, 702–706 transition-state theory, 719–725 Rayleigh, John W S., 256 Rayleigh-Jeans law, 256–257 Rayleigh scattering, 511 reaction quotient in chemical equilibrium, 124–125, 137 in ionic solutions, 233 in nonstandard potentials, 218–223 reactions Belousov-Zhabotinsky reaction, 718–719 biochemical reactions, 60–62, 85, 218 catalysis, 713–714, 783–788 chain reactions, 714–719 competing reactions, 696–702 concurrent reactions, 696–702 consecutive reactions, 696–702 electrochemical reactions, 210–213, 216–217, 221 elementary processes, 706–710 enzyme-catalyzed reactions, 713–714 equilibrium for simple reactions, 694–696 first-order reactions, 686–694 formation reactions, 54–55 half-reactions, 214–216 initial reaction rate, 681 kinetics, see kinetics nonstandard chemical reactions, 220–221 nuclear decay, 688–690, 701–702 oscillating reactions, 714–719 parallel reactions, 696–702 propagating reactions, 714–719 rate-determining step, 709–712 rate of reaction, 681–694 reaction profile, 720 redox reactions, 211–215 second-order reactions, 688–690, 693–694 temperature coefficient of reaction, 219 thermodynamics, see thermodynamics real gases description, fugacity, 110–113 ideal gas compared, 11 redox reaction, electrochemisty, 211–215 reflection plane, 420 representation of symmetry operations, 432–440 repulsion, charged particles, 207–209, 374, 404 repulsive range parameter, 757–758 resistance, 236 resistivity, 236 resonance integrals, 407 reverse osmosis, 201 reversible processes Carnot cycle, 68–73, 94 description, 28–29, 75 entropy, 72–74, 92 Helmholtz energy, 92 right-hand rule, 561–562 Roentgen, Wilhelm, 740 root-mean-square speed, 657–658, 664–665 rotation diatomic molecules, 466, 474, 479, 628–634 hydrogen atom central force problem, 352–353, 365 molecule rotation, 466–471, 482 observables, 347–352 polyatomic molecules, 466–467, 634–636 rotational degrees of freedom, 482–483 rotational temperature, 629–630, 635 three-dimensional rotations, 341–347 two-dimensional rotations, 333–341 rotational spectroscopy, see also electronic spectroscopy; vibrational spectroscopy mechanisms, 473–479 molecule rotations, 466–471, 482 overview, 461–462, 514 rotational-vibrational spectroscopy, 506–511 selection rules, 471–473 Russell-Saunders coupling, electronic spectroscopy, 526–534 Rydberg, Johannes R., 250 Rydberg constant, 250, 262, 266, 357, 521 S Sackur-Tetrode equation, 610 salt bridge, 214 sarin nerve gas, rotational spectrum, 476 saturated calomel electrode, 216 saturated solution, 185 scanning tunneling microscopy, 298–299 Schawlow, Arthur, 554 Schottky defect, 759 Schrödinger, Erwin, 269, 285 Schrödinger equation for central force problem, 353 description, 285–289 harmonic oscillator, 318–320 for helium atom, 374–376, 413 for hydrogen-like ions, 357 in particle-in-a-box solution, 300–301 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part INDEX in three-dimensional rotations, 341, 352 time-dependent Schrödinger equation, 286, 308–309, 318 in two-dimensional rotations, 334 second law of thermodynamics Carnot cycle, 68–73, 94 disorder concept of entropy, 602–603 entropy, 72–79, 81–85, 602 overview, 66 second-order reactions, rate laws, 688–690, 693–694 secular determinant, in linear variation theory, 399–400 selection rules description, 462–463 for electronic spectroscopy, 520 gross selection rule, 472, 513, 520 for Raman spectroscopy, 513 for rotational spectroscopy, 471–473 for vibrational spectroscopy, 487–490 self-diffusion, 675 semiconductors crystal defects, 759–760 zone refining, 192–193 semipermeable membrane, 196–197, 200 shells, see also orbital properties description, 356 term symbols, 526–534 shielding, 396 shielding constant, 574 sigma orbital, 410 silicon, zone refining, 192–193 simple distillation, liquid/solid solutions, 185–186 single-component systems, see also multicomponent systems chemical potential, 159–162 Clapeyron equation, 148–152, 155 Clausius-Clapeyron equation, 152–154 equilibria, 141–165 Gibbs phase rule, 154–159 natural variables, 144, 159–162 overview, 141–145, 162 phase diagrams, 154–159 phase transitions, 143, 145–148 SI units pressure, 2–3 temperature, 3, volume, 2–3 Slater, J C., 446 Slater determinants, 380–382 Smoluchowski, Marian, 676 solids amorphous solids, 732 chemical equilibrium, 129–132, 143–144, 194 crystals, see crystals interface effects, 771–777 liquid/solid solutions, 185–188, 194 Miller indices, 744–752, 778–780 phase diagrams, 154–159 phase transitions, 143, 145–148 solidification, 143 solid/solid solutions, 188–193, 752 surfaces, 778–783 types of, 732–733 solubility, 185–188, 222 solute, 185 solutions boiling point elevation, 194, 196 chemical equilibrium, 129–132, 194 colligative properties, 193–202 Debye-Hückel Theory, 230–234, 646 diffusion, 671–677 freezing point depression, 194–195 ions in solution, 225–230, 234–237 liquid/solid solutions, 185–188, 194 molality, 193–194, 226–227 nonideal two-component liquid solutions, 179–183 osmotic pressure, 196–201 saturated solution, 185 solid/solid solutions, 188–193, 752 solubility, 185–188 solubility product constant, 222 supersaturated solution, 186 solvent, 185, 194 specific heat, 31, 40 specific heat capacity, 40 spectroscopy angular momenta electronic spectra, 521–525, 534–539 magnetic spectra, 565–566, 569 rotational spectra, 470 aromaticity, 546–548 centrifugal distortions, 479–481 classical mechanics, 248–251, 253–257 description, 463–466 diatomic molecules, 491–496, 534–539 electric charges, 561–564 electronic spectroscopy, 519–559 electron spin resonance, 567–571 fingerprint regions, 504–506 fluorescence, 548–550 Franck-Condon principle, 539–541 Hückel approximations, 543–546 hydrogen atom, 520–522 lasers, 550–556 linear molecules, 491–496 magnetic dipoles, 561–564 magnetic inductions, 561–564 magnetic spectroscopy, 560–585 molecule vibration, 481–484 multiple electrons, 526–534 nonallowed vibrational transitions, 503–504 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 833 834 INDEX spectroscopy (continued) nonlinear molecules, 498–502 normal vibration modes, 483–484 nuclear magnetic resonance, 571–582 overtone spectroscopy, 489, 503–504 overview, 461–462, 514, 519–520, 556, 560–561, 582–583 electron systems, 543–546 phosphorescence, 548–550 photoelectric effect, 253, 259 polyatomic molecules, 541–543 quantum-mechanical treatment, 484–487 Raman spectroscopy, 511–514 rotational spectroscopy, 471–479 rotational-vibrational spectroscopy, 506–511 rotations in molecules, 466–471, 482 Russell-Saunders coupling, 526–534 selection rules, 462–463, 471–473, 487–490, 513, 520 symmetry considerations, 496–498 vibrational spectroscopy, 487–504 vibrational structure, 539–541 Zeeman spectroscopy, 560, 564–567 speed, see also velocity average speed, 664–665 most probable speed, 664–665 root-mean-square speed, 657–658, 664–665 spherical harmonics, 344–345 spherical polar coordinates Cartesian coordinates compared, 341–342 description, 341 spherical top, 634 spin description, 371–374 Pauli exclusion principle, 377–382, 413, 532, 537, 630–631 spin angular momentum, 371–372, 522–525, 572 spin orbitals, 377–382 spin-orbit coupling, 523 spin-spin coupling, 577 spontaneous processes chemical potential, 108–110, 114 conditions for, 89–92, 108 description, 62, 66, 89 electromotive force relationship, 213 lasers, 550–551 prediction, 67–68 standard internationsl units pressure, 2–3 temperature, 3, volume, 2–3 standard potentials, in electrochemistry, 215–218 standard reduction potentials, 215–216 standard temperature and pressure, 7–8 Stark, Johannes, 477 Stark effect, 477–478 state, see also quantum numbers common units, equations of state, 5–9, 11, 100–101, 105 variables, 2–5, state functions change, 38–42 enthalpy, see enthalpy entropy, 72–79, 81–85 free energy, see Gibbs free energy internal energy, 33–36, 38–42 natural variable equations, 90, 96–99 in terms of partition functions, 608–613 static equilibrium, 120 statistical thermodynamics concepts, 587–590 crystals, 644–648 ensemble, 590–593 equilibria, 640–644 Maxwell-Boltzmann distribution, 593–602, 663, 666 monatomic gases, 604–608 overview, 586–587, 613, 616–617 partition functions, 586, 596–600, 604–613, 617–623, 636–637 rotations, 628–636 state functions, 608–613 thermodynamic properties, 600–604, 637–640 thermodynamic property derivation, 600–604 vibrations, 623–628 steady-state approximation, 710–714 Stefan-Boltzmann constant, 254–255, 259 steric factor, 705–706 Stern-Gerlach experiment, 371 stimulated absorption, 550 stimulated emissions, 550–552 Stirling’s approximation, 588 stoichiometric compounds, in solid/solid solutions, 191 Stokes, George G., 512 Stokes’ law, 235 Stokes’ lines, 512 sublimation Clapeyron equation, 151 description, 52, 143 heat of sublimation, 146 pressure relationship, 161–162 substitutional defect, 759 supersaturated solution, 186 surfaces catalysis, 783–788 coverage, 783–788 films, 766, 777–778 interface effects, 771–777 overview, 765–766, 788–789 solid surfaces, 778–783 surface energy, 768–771, 779 surface tension, 766–771 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part INDEX surface tension capillary action, 775–777 description, 766–771 surroundings, 2–3 symmetry character tables, 430–437 great orthogonality theorem, 438–441, 537 hybrid orbitals, 450–456 in integrals, 441–443, 449 mathematical basis, 423–427, 439 molecules, 427–430, 482–483, 631 operations, 420–423, 431–435 overview, 419–420, 456 point groups, 420–435 selection rules in spectroscopy, 462–463, 471–473, 487–490 symmetry-adapted linear combinations, 443–446 symmetry number, 634 valence bond theory, 446–450 vibrations, 494–502 wavefunctions, 429–430, 437–438, 631 systems adiabatic systems, 33, 41–49, 75, 77, 103–104 closed systems, 4, 32 description, 2–3 equilibrium, see equilibria isolated system, 32, 75 Joule-Thomson coefficients, 42–46, 103–104 multicomponent systems, see multicomponent systems observables, 276–279, 288, 347–352 partition functions, 636–637 single-component systems, see single-component systems state variables, 2–5, T Taylor-series approximation, 15, 625 temperature, see also thermodynamics boiling, see boiling Boyle temperature, 13, 15–16 Carnot cycle, 68–73, 94 change, 58–60 Clapeyron equation, 148–152, 155 common units, 3, constant temperature, 41 critical temperature, 155–156 Debye temperatures, 647 description, 3–4 efficiency relationship, 68–72 Einstein temperature, 645–646 equations of state, 5–9, 100, 105 equilibria in single-component systems, 141–165 exact differential, 100 Freundlich isotherm, 786 Gibbs free energy variation, 105–108 heat capacities, see heat capacity heat of vaporization, 51–53, 146 inversion temperature, 45 Joule-Thomson coefficients, 42–46, 103–104 Langmuir-Hinshelwood isotherms, 786 Langmuir isotherms, 784, 786 normal melting point, 143 phase diagrams, 154–159, 180–182 rate of reaction effects, 683, 702–706 rotational temperature, 629–630, 635 SI units, 3, standard temperature and pressure, 7–8 temperature coefficient of reaction, 219 vibrational temperature, 625 termination reactions, kinetics, 714–719 term symbols, quantum numbers, 526–534 Tesla, Nikolai, 561 tetrafluoroethylene, Raman spectrum, 512 theoretical plate, 176, 178 thermal de Broglie wavelength, 611–612 thermal equilibrium description, in lasers, 553 thermochemistry, 54 thermodynamics Carnot cycle, 68–73, 94 chemical changes, 53–58 concepts, 587–590 crystals, 644–648 description, 2, 24 disorder concept of entropy, 602–603 ensemble, see ensemble enthalpy, see enthalpy entropy, 72–79, 81–85, 602–604 equations of state, 5–9 equilibria, 640–644 first law of thermodynamics, 24–65 fugacity, 110–114 gas laws, 6–10 Gibbs free energy, 93 heat capacities, 31, 39–41, 46–50 internal energy, see internal energy Joule-Thomson coefficients, 42–46, 103–104 kinetics compared, 680, 694 limitations, 66–68 Maxwell-Boltzmann distribution, 593–602, 663, 666 Maxwell relationships, 99–103 monatomic gases, 604–608 natural variables, 96–99 nonideal gases, 10–17 order, 79–81, 602 overview, 1, 3–5, 21, 24, 66, 586, 616 partial derivatives, 8–10, 18–21 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 835 836 INDEX thermodynamics (continued) partition functions, 586, 596–600, 604–613, 617–623, 636–637 phase changes, 50–53 property derivation, 600–604 rotations, 628–636 second law of thermodynamics, 66–88 state, 2–3 state functions, 33–36, 38–42, 608–613 statistical thermodynamics, 586–650 surroundings, 2–3 system, 2–3 temperature change, 58–60 third law of thermodynamics, 66–88 vibrations, 623–628 work-heat relationship, 24–32 zeroth law of thermodynamics, 1–23 third law of thermodynamics Carnot cycle, 68–73, 94 entropy, 81–85, 602–604 order, 79–81, 602 overview, 66 Thompson, Benjamin, 30 Thomson, G P., 268 Thomson, Joseph J., 251, 268 Thomson, William, threshold frequency, 253 tie line description, 176 in nonideal two-component liquid solutions, 180 torr, units of measure, total power flux, 259 Townes, Charles, 554 transition moment description, 462, 489–490 for electronic transitions, 520 for magnetic transitions, 565–566 transition-state theory, 719–725 transport properties, 674 triple point, 155–156 Trouton’s rule, 148 tunneling, quantum mechanics, 296–299 tunneling microscopy, 298–299 two-component systems, see multicomponent systems U ultrahigh vacuums, 781–782 ultraviolet catastrophe, 256 ultraviolet radiation characteristics, 464–465 electronic transitions for polyatomic molecules, 542 uncertainty principle, 279–281 unexplainable phenomena, classical mechanics, 248 unit cell description, 733–738 rationalizing, 752–755 V valence bond theory, symmetry, 446–450 van der Waals, Johannes, 13 van der Waals constants, 13–14 van der Waals equation, 13–14, 16, 102 van’t Hoff, Jacobus, 198 van’t Hoff equation, 133, 198, 201, 702 vaporization Clapeyron equation, 151 description, 51–53, 143 heat of vaporization, 51–53, 146 vapor-phase mole fractions, 173–174 vapor pressure description, 153–154 in liquid/liquid systems, 169–179, 193 negative deviation, 179 in nonideal two-component liquid solutions, 179–183 phase diagrams, 154–159, 174–175 positive deviation, 179 variation theory linear variation theory, 398–402 perturbation theory compared, 402–403 in quantum mechanics, 394–397, 402–403 varying dipole moment, 488 velocity, kinetic theory of gases, 656–666 vibrational spectroscopy, see also electronic spectroscopy; lasers; rotational spectroscopy fingerprint regions, 504–506 Franck-Condon principle, 539–541 mechanisms, 487–504 molecule vibration, 481–484 nonallowed transitions, 503–504 nonfundamental transitions, 503–504 normal vibration modes, 483–484 overview, 461–462, 514 quantum-mechanical treatment, 484–487 rotational-vibrational spectroscopy, 506–511 symmetry considerations, 494–498 vibrational degrees of freedom, 482–483, 500, 541 vibrational temperature, 625–627 virial coefficients, 11–12 virial equation, 11 visible light, 464 volt, 209 Volta, Alessandro, 209 voltaic cell, 215, 220–221 volume Clausius-Clapeyron equation, 152–155 common units, equations of state, 5–9 molar volume, 10 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part INDEX natural variable equations, 96–99 SI units, 2–3 von Fraunhofer, Joseph, 525 von Helmholtz, Hermann L F., 93 von Laue, Max, 741 von Lenard, Philipp E A., 253 W water molecules crystal structure, 752–753 phase diagram, 753 reaction mechanisms, 707–708 surface tension, 767–771 vibrational parameters, 497, 509 watt, 255 wavefunctions antisymmetric wavefunctions, 379–380, 631 average values, 293–296, 329 Born interpretation, 281–283 Born-Oppenheimer approximation, 403–405, 539 degeneracy, 303–306, 605, 618, 631–632 description, 274–275 doubly-degenerate wavefunctions, 591 for harmonic oscillators, 321–329 for helium atoms, 376–378, 396 for homonuclear diatomic molecules, 536, 630 for hydrogen-like atoms, 355–365, 374 linear variation theory, 398–402 for molecular orbitals, 409–415 normalization, 283–285, 303, 335–336 orthogonality, 306–307 particle-in-a-box solution, 288–292, 299–303, 605–606 Pauli exclusion principle, 377–382, 413, 532, 537, 630–631 perturbation theory, 386–394, 402–403 Slater determinants, 380–382 spectroscopy selection rules, 462–463 symmetry, 429–430, 437–438, 631 three-dimensional rotations, 341–347, 353–354 tunneling, 296–299 two-dimensional rotations, 333–341 variation theory, 394–397 vibrational wavefunctions, 541 wavelength, de Broglie equation, 267–269, 280 wavenumber in rotational spectroscopy, 465, 469, 476 in rotational-vibrational spectroscopy, 507–509 wetting, 775–776 Wien displacement law, 255–256 work Carnot cycle, 68–73, 94 description, 24–32 electrochemistry, 210–215 energy relationship, 210–215 Gibbs free energy, see Gibbs free energy Helmholtz energy, see Helmholtz energy in surface tension, 769–771 work function, 259 X X-ray diffraction description, 741–744 Miller indices, 744–752, 778–780 X rays, 464 Y Young, Thomas, 253–254, 773, 775 Young-Dupré equation, 775 Z Zeeman, Pieter, 564 Zeeman spectroscopy, 560, 564–567 zeolites, 788 zero-point energy, 323 zeroth law of thermodynamics equations of state, 5–9 gas laws, 6–10 nonideal gases, 10–17 overview, 1, 3–5, 21 partial derivatives, 8–10, 18–21 state, 2–3 surroundings, 2–3 system, 2–3 Zhabotinsky, Anatol M., 718 zone refining, 192–193 zwitterion, 136 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 837 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 24.3050 20 19 Radium 226.0254 Francium (223) (261) Rutherfordium 104 Rf 178.49 Hafnium 72 Hf 91.224 Zirconium 40 Zr 47.88 Titanium Ti 22 4B Actinides Lanthanides 227.0278 Actinium Ac 89 Ra 88 87 Fr Lanthanum 138.9055 Barium 137.327 Cesium 132.9054 La 57 Ba 56 Cs 55 87.62 Yttrium 88.9059 Strontium Rubidium 85.4678 Y 39 Sr 38 37 Rb Scandium 44.9559 Calcium 40.078 Sc 39.0983 Ca 21 3B Potassium K Magnesium 22.9898 Sodium Mg Na 12 9.0122 11 Beryllium 6.941 Be 2A Lithium Li 1A 1.0079 Hydrogen H The Periodic Table 60 59 144.24 92 91 231.0359 Protactinium 232.0381 Thorium Pa 90 Th 238.0289 Uranium U Neodymium 140.9076 Praseodymium 140.115 Cerium Pr 237.0482 Neptunium 93 Np (145) Promethium 61 (265) Hassium 108 Hs 190.2 Osmium 76 Os 101.07 Ruthenium 44 Ru 55.847 Iron Fe 26 (244) Plutonium 94 (243) Americium 95 151.965 Europium 63 Eu (269) 110* 195.08 Platinum 78 Pt 106.42 Palladium 46 Pd 58.693 Nickel Ni 28 (247) Curium 96 157.25 Gadolinium 64 Gd (272) 111 196.9665 Gold 79 Au 107.8682 Silver 47 Ag 63.546 Copper Cu 29 1B Pu Am Cm 150.36 Samarium 62 (266) Meitnerium 109 Mt 192.22 Iridium 77 Ir 102.9055 Rhodium 45 Rh 58.9332 Cobalt Co 27 8B Nd Pm Sm (262) Bohrium 107 Bh 186.207 Rhenium 75 Re (98) Technetium 43 Tc 54.9380 Manganese Mn 25 7B (263) Seaborgium 106 Sg 183.85 Tungsten 74 W 95.94 Molybdenum 42 Mo 51.9961 Chromium Cr 24 6B Atomic weight Atomic number Symbol Ce 58 (262) Dubnium 105 Db 180.9479 Tantalum 73 Ta 92.9064 Niobium 41 Nb 50.9415 Vanadium V 23 5B 238.0289 Uranium 92 U (247) Berkelium 97 Bk 158.9253 Terbium 65 Tb (277) 112 200.59 Mercury 80 Hg 112.411 Cadmium 48 Cd 65.39 Zinc Zn 30 2B (251) Californium 98 Cf 162.50 Dysprosium 66 Dy 204.3833 Thallium 81 Ti 114.82 Indium 49 In 69.723 Gallium Ga 31 26.9815 Aluminum 13 Al (252) Einsteinium 99 Es (257) Fermium 100 (258) Mendelevium 101 168.9342 Thulium 69 Tm (209) Polonium 84 Po 127.60 Tellurium 52 Te 78.96 Selenium Se 34 32.066 Sulfur S 16 15.9994 Oxygen O 6A Fm Md 167.26 Erbium 68 Er 208.9804 Bismuth 83 Bi 121.757 Antimony 51 Sb 74.9216 Arsenic As 33 30.9738 Phosphorus 15 P 14.0067 Nitrogen N 5A (259) Nobelium 102 No 173.04 Ytterbium 70 Yb (210) Astatine 85 At 126.9045 Iodine I 53 79.904 Bromine Br 35 35.4527 Chlorine 17 Cl 18.9984 Fluorine F 7A Helium (260) Lawrencium 103 Lr 174.967 Lutetium 71 Lu (222) Radon 86 Rn 131.29 Xenon 54 Xe 83.80 Krypton 36 Kr 39.948 Argon 18 Ar 20.1797 Neon 10 Ne 4.0026 *Elements 110–112 have not yet been named 164.9303 Holmium 67 Ho 207.2 Lead 82 Pb 118.710 Tin 50 Sn 72.61 Germanium Ge 32 28.0855 Silicon 14 Si Carbon 12.011 Boron C 4A 10.811 B 3A He 8A Physical Constants Quantity Symbol Value Unit Speed of light in vacuum c 2.99792458 × 108 m/s Permittivity of free space ε0 8.854187817 × 10 –12 C2/J·m Gravitation constant G 6.673 × 10 –11 N·m2/kg2 Planck's constant h 6.62606876 × 10 –34 J·s Elementary charge e 1.602176462 × 10 –19 C Electron mass me 9.10938188 × 10 –31 kg Proton mass mp 1.67262158 × 10 –27 kg Bohr radius a0 5.291772083 × 10 –11 m Rydberg constant R 109737.31568 cm –1 Avogadro's constant NA 6.02214199 × 10 23 mol –1 96485.3415 C/mol 8.314472 J/mol·K 0.0820568 L·atm/mol·K 0.08314472 L·bar/mol·K 1.98719 cal/mol·K Faraday's constant Ideal gas constant R Boltzmann's constant k, kB 1.3806503 × 10 –23 J/K Stefan-Boltzmann constant σ 5.670400 × 10 –8 W/m2·K Bohr magneton µB 9.27400899 × 10 –24 J/T Nuclear magneton µN 5.05078317 × 10 –27 J/T Source: Excerpted from Peter J Mohr and Barry N Taylor, CODATA Recommended Values of the Fundamental Physical Constants, J Phys Chem Ref Data, vol 28, 1999 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part ... 1s2 2s2 2p6 3s1 Mg 1s2 2s2 2p6 3s2 Al 1s2 2s2 2p6 3s2 3p1 Si 1s2 2s2 2p6 3s2 3p2 P 1s2 2s2 2p6 3s2 3p3 S 1s2 2s2 2p6 3s2 3p4 Cl 1s2 2s2 2p6 3s2 3p5 Ar 1s2 2s2 2p6 3s2 3p6 K 1s2 2s2 2p6 3s2 3p6... 1s2 2s2 2p6 3s2 3p6 4s2 Sc 1s2 2s2 2p6 3s2 3p6 4s2 3d Ti 1s2 2s2 2p6 3s2 3p6 4s2 3d V 1s2 2s2 2p6 3s2 3p6 4s2 3d Cr* 1s2 2s2 2p6 3s2 3p6 4s1 3d Mn 1s2 2s2 2p6 3s2 3p6 4s2 3d Fe 1s2 2s2 2p6 3s2... 12. 1 387 Ground-state electron configurations of the elementsa H 1s He 1s2 Li 1s2 2s1 Be 1s2 2s2 B 1s2 2s2 2p1 C 1s2 2s2 2p2 N 1s2 2s2 2p3 O 1s2 2s2 2p4 F 1s2 2s2 2p5 Ne 1s2 2s2 2p6 Na 1s2 2s2