• Choose a category
• All
• Classic-Fiction

By Root 399 0

constants. In contrast to Bohr’s theory, the Schrödinger Equation has been successfully applied to a tremendous number of other problems including atoms other than hydrogen and small and large molecules. As mentioned above, for systems larger than the hydrogen atom, that is, atoms and molecules involving more than two particles, the Schrödinger Equation cannot be solved exactly. However, many powerful approximation techniques have been developed that enable accurate solutions to the Schrödinger Equation for atoms, molecules, and other types of quantum mechanical problems. With the advent of computers, and the current tremendous power of computers, it is possible to solve the Schrödinger Equation for very large and complex molecules. As discussed in subsequent chapters, molecules have shapes. The solutions to the Schrödinger Equation for a molecule give its energy levels and its wavefunctions. The wavefunctions provide the necessary information to describe the shapes of molecules.

THERE ARE FOUR QUANTUM NUMBERS

The energies of the different states of the hydrogen atom only depend on a single quantum number, n. However, there are actually four quantum numbers associated with electrons in atoms. These come out of solving the hydrogen atom with quantum theory. One of these only comes into play for atoms and molecules that have more than one electron. In that sense, the hydrogen atom is a special case because it has only one electron. In the hydrogen atom, in addition to the principal quantum number n, the two other quantum numbers are l and m. l is called the orbital angular momentum quantum number and m is called the magnetic quantum number. These two quantum numbers, when combined with n, determine how many different states are associated with a particular energy, and they determine the shapes of the wavefunctions. The fourth quantum number is s. It is called the spin quantum number. When Bohr solved the hydrogen atom problem with old quantum theory, the electron moved in orbits that had different energies and shapes. Schrödinger’s correct quantum solution to the hydrogen atom gave the energies and the wavefunctions, which, in correspondence to Bohr’s orbits, are called “orbitals.” In discussing atoms and molecules, we often use the term wavefunction and orbital interchangeably. The orbitals are probability amplitude waves that obey Heisenberg’s Uncertainty Principle, in contrast to Bohr’s orbits.

As stated above, the principal quantum number, n, can take on values, n ≥ 1, that is, 1, 2, 3, 4, etc. l can have values from 0 to n-1 in integer steps. m can take on values from l to -l in integer steps. s can only take on two values, +½ or -½. These values are summarized in Table 10.1.

For historical reasons, the states with different quantum numbers l are given different names. An s orbital has l = 0. A p orbital has l = 1. A d orbital has l = 2. An f orbital has l = 3. For our discussions of all atoms, we will only need to go to f orbitals, that is, l = 3. As shown below, these different orbital types have different shapes.

Because the energies of the states (orbitals) of the hydrogen atom only depend on the quantum number n, there will be more than one state with the same energy for n > 1. For n = 1, l = 0, and m = 0 (see the table). Therefore, there is a single orbital with n = 1. It has l = 0, so it is referred to as the 1s orbital. The 1 is the n value, and s means that l = 0. For n = 2, l can equal 0, giving rise to the 2s orbital. However, for n = 2, l can also equal 1. For l = 1, m can equal 1, 0, or -1 (see the table). l = 1 is a p orbital, and there are three different p orbitals that can be called 2p1, 2p0, and 2p-1. The 2 is the principal quantum number n. The p means l = 1, and the three subscripts are the three possible m values. So for n = 2, there are four different states.

TABLE 10.1. Quantum Numbers.

If n = 3, then l can equal 0, to give the 3s orbital. l can also equal 1 with m = 1, 0, and -1, to give 3p1, 3p0, and 3p-1. In addition, l can equal 2. For l = 2, m can equal 2, 1, 0, -1, and -2.