Course Outline

General Chemistry--Unit 1

04/25/09

�Before You Begin:

To master this material, you need to be familiar with the parts of the atom and periodic law. It also helps if you delay worrying about the incongruities until after you get familiar with the usefulness of quantum mechanics.

"Not only is the universe stranger than we imagine, it is stranger than we can imagine."

Sir Arthur Eddington

Introduction

Some people like history. The history of the development of quantum theory can give an open minded student a lot of insight into the theory itself. Another section of these lecture notes describes the history of the development of atomic theory and gives the details of the experimental evidence and the theoretical limitations that made earlier atomic theories inadequate. But, you don’t need to know its history to use quantum mechanics. In a nutshell, the basic problem with other atomic theories is that they treat electrons as though they were bits of classical matter with definite, knowable positions, momentum, and energies. Unfortunately, all matter has wavelike properties. These wavelike properties are not significant for a cannon ball or an automobile, so classical mechanics works just fine. But, for very small bits of matter, like electrons, the wave properties are important.

We don’t need to know the wave properties of an electron for a lot of useful chemical applications. We can think in terms of indivisible Dalton atoms to balance reaction equations and work stoichiometry problems. However, if we really need details about the electrons, we need quantum mechanics. The really bad news is that the mathematics needed for quantum mechanics requires advanced calculus. The students who cannot work these problems for themselves will have to trust that someone out there knew how to do it, trust that someone understands this stuff.

So, a vital question is “When do you have to use QM?” The typical freshman chemistry student needs quantum mechanics to understand molecular bonding. That is about ten per cent of all of the material covered in a year. Not only do you seldom need quantum mechanics, but a trusting soul can skip straight to the conclusions and never worry about the puzzling philosophical issues such as Schrödinger’s cat. But there are always atypical freshman chemistry students. They want to know why. They ask “How come the d orbitals are shaped so funny?” The answer, of course, is “Rocks are hard, water is wet, and quantum mechanics says so.” For those students who find that answer unsatisfactory, we have included another section with answers to some very pesky questions. But first, let’s cover the basics.

Schrödinger’s wave equation:

The heart of quantum mechanics is the Schrödinger’s wave equation, a description of the wave properties of an electron. In Cartesian coordinates, one form of the wave equation looks like this:

Where ψ is the wave function, ħ = h/2π and h is Planck’s constant, m is mass of the electron, Ze² is the nuclear charge, and E is energy.

The wave equation has a set of solutions called wave functions, ψ. For the hydrogen atom, these solutions are similar to the energies and orbits of the Bohr model. The function cannot be solved for an atom with many electrons or a molecule.  We can find approximate solutions by setting boundary conditions on the function and by making simplifications, such as treating a bond as though it were a spring.

Probability Density

 Although a set of wave functions describe the energy of an electron, they don’t really correspond to anything we can visualize. If we square the wave functions, however, we get a new set of functions that we can almost understand. The square of a wave function, ψ², is known as the probability density. The probability density is a map of the volume around the nucleus likely to contain an electron, and, just as important, it gives us an idea of where the electron is most likely not going to be.

Orbitals

An orbital is a possible energy state of an electron. Most people find it easier to think visually, so they link the idea of an orbital with its probability density. In that respect, an orbital is a volume likely to contain an electron with a specific energy. For the simplest case of a single electron in a hydrogen atom at its lowest energy, the wave function and probability density match the Bohr model, except that we are stuck with the notion of an electron probably somewhere in a spherical volume instead of traveling in a circular orbit. At higher energies, the probability density volumes have more complicated shapes, and it becomes more difficult to distinguish among them.

Quantum Numbers

The wave functions for an atom with a single electron have three index numbers called quantum numbers. These indices allow a set of solutions to share the same form and yet allow each solution to be unique.  A set of quantum numbers is sort of like a finger print: even though you have the same basic form as other human beings, you are unique and you have a unique finger print. Your finger print isn’t what makes you an individual, but it is a way to differentiate you from others. A set of quantum numbers doesn’t make an orbital, but it allows us to distinguish among them. But a set of quantum numbers is more than just an identifier. Their values give us information about the shape and orientation of the probability density.

1. The principle quantum number, n, indicates the energy of the electron in that orbital. The higher the value of n is, the greater the energy the electron has. The principle quantum number has values that range from 1 to infinity, theoretically, but the principle quantum number also indicates the distance to the nucleus. High energy electrons far from the nucleus tend to get lost. Which brings us to an important digression: if an atom has got only one electron, doesn’t it only have one orbital? NO! Every atom has an infinite number of orbitals because every wave equation has an infinite number of solutions. At any given instant an electron is occupying only one of these orbitals, but all the other possible energy states still exist.
2. The azimuthal quantum number, l, indicates the shape of the probability density that the electron occupies (or the shape of the orbital, if you prefer). The possible values for l depend on the value for n of that orbital, and l has values from zero up to n – 1. For example, if n = 3, l = 0, 1, 2. Many students balk at this point and demand to know which value is the right one. They are all equally right. Remember that we are building a large set of orbitals.
3. The magnetic quantum number, m_l, indicates the orientation of the probability density, or the direction the orbital is pointing. The value the magnetic quantum number can have depends on the value of the azimuthal quantum number with values ranging from negative l up to positive l. For example, if l = 2, m_l = -2, -1, 0, 1, 2. 

Shells, Subshells, and Orbitals in the Many Electron Atom

For atoms with more than one electron, we can use the hydrogen atom orbitals as a starting point. For an atom with more than one electron, the value of n is only a rough guideline to the electron’s energy. The value for the l quantum number gives orbitals that vary in energy, even when they have the same n value. In general, for equal values of n, the lower the value of l is, the lower the energy is.  

The set of orbitals with the same n value is called a shell. The set of orbitals with the same values for both n and l are called a subshell. The energy of an electron in any of these orbitals is exactly the same as long as the value for both n and l are the same. We use the word degenerate to signify that different orbitals have the same energy. The subshell orbitals have names, though they don’t seem much like names. They are labeled with a number representing the shell (or n value) and a letter that was originally used to describe experimental results in spectroscopic experiments.

The orbitals also have characteristic shapes depending on their l values. The shapes of these orbitals are important because they are used in various bond theories to predict and explain molecular shapes.  

If an atom has more than one electron, we need an additional index that the hydrogen atom wave functions don’t need. The spin quantum number, m_s, describes the magnetic direction of the electron in that energy state. No matter what the values of the other quantum numbers might be, the two possible values for the spin quantum number are positive and negative one half. We can think of the spin quantum number as designating whether an electron is a magnetic north or a magnetic south. Diagrams of electrons use arrows pointing either up or down to illustrate this property.

In any given atom, each electron has a unique energy state at any given instant. The first three quantum numbers describe the total energy, probable distance, the most likely volume of space, and the spin quantum number designates magnetic direction. That means that a unique set of four quantum numbers is all we need to identify a unique energy state. Pauli Exclusion Principle states that, in any given atom, none of the electrons can have the same values for all four quantum numbers. Because three of the numbers identify the orbital and each electron can have a spin of +/- ½, each orbital can hold at most two electrons.

4Concept Check: Why isn't there a 2d orbital set?

Answer: The number 2 refers to the principle quantum number, or n value. If n =2, the possible values for l are 1 and 0, because allowed values of l depend on the value of n. When l = 1, the name of the orbital set is p (for reasons we will cover in another section). When l = 0, the name of the orbital set is s (we'll explain the names later, I promise!). A d orbital has an l value of 2; but, if n = 2, l ≠ 2. There isn't a 2d orbital set because it goes against the quantum mechanics index numbering rules.

For most people, this is not a satisfying answer. A more complicated answer is that the shells are subdivided into smaller volumes called orbitals. Each orbital has to be unique. The second shell is too small, too simple, and has too little energy to be subdivided into s and p AND have any space left over to call d. The s and p subdivisions account for all of the possibilities.

Of course, the real problem is that it is an unfortunate question, rather like asking "how many 'E's are in the word balloon?"

 Shapes of Some of the Orbitals

A 1s Orbital:

The Set of 2p Orbitals:

The Set of 3d Orbitals:

One of the Set of 3f Orbitals:

Electron Configuration

An atom has an infinite number of orbitals, though many of them are so high in energy and so far from the nucleus that an electron would get lost out there. When all of the electrons are in the lowest energy orbitals the atom is said to be in its ground state. The most valuable quantum mechanics skill for a freshman chemistry student is being able to determine how many electrons occupy which orbitals in any given atom’s ground state. When an atom gains energy one of its electrons moves to a higher energy state. The energy is given off when the electron moves to a lower energy state. The amount of energy needed to move electrons to higher and lower energy levels is a unique characteristic for each element and each molecule. Many important instruments use this aspect of quantum mechanics to identify elements and compounds in a field known as spectroscopy.

In an atom with more than one electron, the orbitals within a subshell have the same energy, but the orbitals in different subshells have slightly different energies, even if they are in the same shell. The s orbitals are lower in energy than the p which are lower than the d and so on. Life would be nice if the energy ranking was the same as the order in the table above, but it isn’t. The d and f orbitals are so much higher in energy that they are higher than the s and p orbitals of the next higher shell. The Pauli Aufbau filling order is a ranking of the orbitals from lowest energy to higher energies:

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p7s 5f 6d 7p

We can use the filling order to determine how many electrons are in which orbitals for the ground state of an atom. Count the number of electrons and distribute them to the lowest energy subshells remembering the maximum number of electrons that each subshell can hold. The electron configuration is a list of the subshells that are occupied in the ground state with a superscript by each subshell name to indicate the number of electrons in that subshell. For example, phosphorus has 15 electrons. An s subshell can hold two electrons, and a p subshell can hold six. Starting at the lowest energy, we distribute electrons until we have a total of 15:

P: 1s² 2s² 2p⁶ 3s² 3p³

As atoms get larger in atomic number, the electron configuration becomes more and more tedious and cumbersome. A useful abbreviation is to use the symbol for a noble gas to signify the core electrons. For example, lead has 82 electrons. Its electron configuration is

Pb: 1s² 2s² 2p⁶ 3s² 3p³ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶ 6s² 4f¹⁴ 5d¹⁰ 6p²

Xenon has the same electron configuration for the first 54 of these electrons. This inner core of electrons is not very likely to take part in a chemical change, so we can write the electron configuration as

Pb: [Xe] 6s² 4f¹⁴ 5d¹⁰ 6p²

which is a lot shorter.

There are a variety of diagrams and pneumonic devices for memorizing the Pauli Aufbau filling order, but the easiest for us is the periodic table. See the next section for this useful device.  

4Concept Check: What is the electron configuration of fluorine?

Answer: F: 1s² 2s² 2p⁵  

In the next section, we will learn a way of remembering the filling order using the periodic table.

Electrons are negatively charged and like charges repel one another. It is slightly lower in energy if a partly filled subshell has its electrons even distributed thou space. Hund’s Rule states that when determining the ground state of an atom, each orbital in a subshell gets one electron before any gets a second electron. In the phosphorus example above, it is lower in energy if the three electrons in the 3p subshell are each in a different orbital. An orbital diagram used arrows pointing up and down to distinguish spin quantum states. All the orbitals have two electrons except those in the last subshell.

The orbital diagram of phosphorus shows Hund's rule in action. The last three electrons each occupy a separate one of the 3 p orbitals. Yes, the electrons can pair up in this subshell, but that would take energy, so it wouldn't be the ground state.

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