Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.
"Physicists Put Atom in Two Places at Once." This was the headline in the science section of the New York Times on May 28, 1996. “Impossible!” you say. “How could they do that?” you wonder. This event is impossible at the macroscopic level at which classical mechanics governs the world. But it is entirely possible according to quantum mechanics, which governs the particulate world of atoms, protons, and electrons. The physicists who caused a single beryllium atom to exist in two states at the same time, separated by a distance of 83 nanometers, made use of a quantum mechanical trait called spin.
The spin attribute is one of the four quantum mechanical characteristics needed to describe an electron completely. Each of these characteristics is described by what is known as a quantum number. If we consider quantum numbers to be the “address” of each electron within an atom, each address has four parts, and no two electrons have the exact same address.
The Bohr model of the hydrogen atom gave us our first understanding that electrons were governed by non-classical mechanics, and this model worked well for explaining the properties of the electron in the hydrogen atom. However, it failed for all other atoms. In 1926, Erwin Schrödinger devised a new model of the atom which is now known as the quantum mechanical model.
The Quantum Mechanical Model of the Atom Schrödinger’s atomic model is framed mathematically in terms of what is known as a wave equation. Solutions of wave equations are called wave functions. The solutions to a wave equation define the volume in space where an electron with a particular energy is likely to be found. This volume in space is called an orbital. Each orbital is characterized by three quantum numbers.
The Pauli Exclusion Principle
This principle states that no two electrons in an atom can have the same four quantum numbers. If two electrons occupy the same orbital, they must have different spins.
The four quantum numbers (for electronic wavefunctions) are:
1. The principal quantum number, n.
The allowed values of the principal quantum number are n = 1, 2, 3, ..., 7. Electrons with the same value of n are said to have the same principal energy level.
2. The angular momentum quantum number, l.
Angular momentum quantum numbers depend on principal quantum numbers. For n = 1, l = 0. For n = 2, l = 0 or 1. For n = 3, l = 0, 1, or 2. For n = 4 (and higher), l = 0, 1, 2, or 3. This pattern can be summarized as l = 0, 1, ..., n – 1. Notice that all principal energy levels are divided into one or more sublevels.
Angular momentum quantum numbers are often referred to by using letter designations which correspond with the numerical values. l = 0 is also called the s sublevel, l = 1 is p, l = 2 is d, and l = 3 is f. Electron energies are described by the principal energy level and the sublevel. Thus an electron with n = 3 and l = 1 is referred to as a 3p electron.
3. The magnetic quantum number, ml.
Magnetic quantum numbers depend on angular momentum quantum numbers. The pattern is ml = –l, ..., 0, ..., +l. Thus for l = 0, the only allowed value of ml is 0. When l = 1, ml can be –1, 0, or +1. For l = 2, ml = –2, –1, 0, +1, and +2. When this pattern is followed for l = 3, there are seven possible ml values (can you write them?).
4. The electron spin quantum number, ms.
The values of ms are +½ and –½ . Electrons can be thought of as spinning on an axis, where one ms value corresponds to a clockwise rotation and the other value corresponds to a counterclockwise rotation.
The Periodic Table
The periodic table serves as a guide to both order of increasing electron energies and the order in which electrons fill orbitals. Electrons occupy the lowest energy orbitals available, and as the numbers of electrons in an atom increases, the outermost electrons occupy higher and higher energy levels. The periodic table below illustrates the correspondence of electron energy levels and position on the periodic table.
s orbitals are being filled in Groups 1A–2A, p orbitals are being filled in groups 3A–8A, d orbitals fill in the B Groups, and f orbitals fill in the lanthanide and actinide series.
For s and p orbitals, the period number corresponds with the principal energy level. For d orbitals, the fourth period corresponds to n = 3, the fifth period to n = 4, and so on. The lanthanides correspond to n = 4, and the actinides have n = 5.