Our current working model of the atom is based on quantum mechanics – which incorporates the idea of quantized energy levels, the wave properties of electrons, and the uncertainties associated with electron location and momentum. If we know their energies, which we do, then the best we can do is to calculate a probability distribution that describes the likelihood of where a specific electron might be found (if we were to look for it, of course). If we were to find it, we would know next to nothing about its energy, which implies we would not know where it would be in the next moment. We refer to these probability distributions by the anachronistic, misleading, and Bohrian term “orbitals”. Why misleading? Because (to a normal person) the term orbital implies that the electron actually has a defined and observable orbit, something that is simply impossible to know (can you explain why?)

2.1 Electrons & Orbitals
2.2 Quanta
2.3 Spectroscopy
2.4 Beyond Bohr
2.5 Periodic table
2.6 Orbitals plus
2.7 Quantum numbers

Another common and often useful way to describe where the electron is in an atom is to talk about the electron probability density, or electron density for short. In this terminology, electron density represents the probability of an electron being within a particular volume of space; the higher the probability, the more likely it is in a particular region at a particular moment. Of course, you can't really tell if the electron is in that region at any particular moment, because if you did you would have no idea of where the electron would be in the next moment.

A mathematical description of the behavior of electrons in atoms was developed by Erwin Schrödinger (1887–1961), and extended by Max Born (1882–1970). Schrödinger used the idea of electrons as waves and described each atom in an element by a mathematical wave function using the famous Schrödinger equation (HΨ = EΨ) – we assume that you have absolutely no idea what either HΨ or EΨ are, but not to worry, you don’t really need to. The solutions to the Schrödinger equation are a set of equations (wave functions) that describe the energies and probabilities of finding electrons in a region of space. They can be described in terms of a set of “quantum numbers” (recall that Bohr’s model also invoked the idea of quantum numbers). One way to think about this is that almost every aspect of an electron (within an atom or a molecule) is quantized – that is, only defined values are allowed for its energy, probability distribution, orientation, and “spin”. It is far beyond the scope of this book to present the mathematical and physical basis for these calculations, so we won’t pretend to try. However we can use the results of these calculations to provide a model for the arrangements of electrons in an atom - that is we can introduce the idea of orbitals - which are mathematical descriptions of the probability of finding electrons in space, and determining their energies. Let us take a look at some orbitals, their quantum numbers, energies, shapes and how they can be used to explain atomic behavior.

Examining atomic structure using light: on the road to quantum numbers:

J.J. Thompson’s studies (remember them), suggested that all atoms contained electrons. We can use the same basic strategy, but in a more sophisticated way, to begin to explore the organization of electrons in particular atoms. This approach involves measuring the amount of energy it takes to remove electrons from atoms. This is known as the element’s ionization energy, which in turn relates directly back to the photoelectric effect.

All atoms are by definition electrically neutral - they contain equal numbers of positive and negatively charged particles, protons and electrons. We cannot remove a proton from an atom without changing the identity of the element, since the number of protons is how we define elements, but it is possible to add or remove an electron, leaving the atom’s nucleus unchanged. When an electron is removed or added to an atom it results in the atom having a net charge - atoms (or molecules) with a net charge are known as ions; and this process (atom/molecule to ion) is called ionization. A positively charged ion (called a cation) results when we remove an electron, a negatively charged ion (called an anion) results when we add an electron. Remember this added or removed electron becomes part of, or is removed from, the atom’s electron cloud.

Now, consider the amount of energy required to remove an electron. Clearly energy is required since we are moving the electron away from the nucleus, which attracts it. We are perturbing a stable system that exists at a potential energy minimum. We might naively predict that the energy required to move an electron away from an atom will be the same for each element. We can test this assumption experimentally. We measure what is called the ionization potential. In such an experiment we determine the amount of energy (in kilojoules per mole of molecules) required to remove an electron from an atom. Let us consider the situation for hydrogen (H): we can write the ionization reaction 63 as:  H (gas) + energy → H+ (gas) + e–.

What we discover is that it takes 1312 kJ to remove a mole of electrons from a mole of hydrogen atoms. As we move to the next element, helium (He) – which has two electrons we find that the energy required to remove an electron from helium is 2373 kJ/mol - almost twice that required to remove an electron from hydrogen!

Let us return to our model of the atom. Each electron in an atom is attracted by all the protons, located in essentially the same place (the nucleus), and at the same time the electrons repel each other. Coulomb’s law (?) tells us that the potential energy of the system is proportional to the product of the charges divided by the square of the distance between them. Therefore the energy to remove an electron from an atom should depend on the net positive charge acting upon the electron and the electron’s average distance from the nucleus. Since it is more difficult to remove an electron from a He atom than from an H atom, our tentative conclusion is that the electrons in helium must be attracted more strongly to the nucleus. In fact this makes sense: the He nucleus contains two protons, and each electron is attracted by both protons - making them more difficult to remove. They are not attracted exactly twice as strongly, since there are also some repulsive forces between the two electrons.

The size of an atom depends on the size of its electron cloud, which depends on the balance between the attractions between the protons and electrons (making it smaller), and the repulsions between electrons (which would tend to make the electron cloud expand). The system is most stable when the repulsions are balanced by the attractions – giving the lowest potential energy. If the electrons in helium are attracted more strongly to the nucleus, we might predict that the size of the helium atom would be smaller than that of hydrogen. There are several different ways to measure the size of an atom and they do indeed indicate that He is a smaller than H. Here we have yet another very counter-intuitive idea - apparently as atoms get heavier (more protons and neutrons), their volume gets smaller!

We might guess that it is smaller than Lithium and has a larger ionization energy because the electrons are attracted more strongly by the 4 positive charges in the nucleus. Again, this is the case, the ionization energy of Be is 899 kJ/mol, larger than Li, but much smaller than that of either hydrogen or helium. Following this trend, the atomic radius of Be is smaller than Li but larger than H or He. We could continue this way, empirically measuring ionization energies for each element, but how do we make sense of the patterns observed?

2.1 Electrons & Orbitals
2.2 Quanta
2.3 Spectroscopy
2.4 Beyond Bohr
2.5 Periodic table
2.6 Orbitals plus
2.7 Quantum numbers

Question to answer:

• Why are Helium atoms smaller than Hydrogen atoms?
• What factors govern the size of an atom? list all that you can. Which factors are the most important?
  

Questions to ponder:

• What would a graph of the potential energy of a hydrogen atom look like as a function of distance of the electron from the proton?
• What would a graph of the kinetic energy of a hydrogen atom look like as a function of distance of the electron from the proton?
• What would a graph of the total energy of a hydrogen atom look like as a function of distance of the electron from the proton?