# Tanner's General Chemistry

## Atomic Orbitals

Electrons exist in different energy levels. Electrons have wave properties. The kinetic energy of an electron corresponds to a wavelength and thus a frequency. The position of an electron is described in terms of the probability of finding it in a certain position. The probability corresponds to the amplitude of the associated wave. The Schrodinger equation relates the energy of a system to its wave properties. The equation can be solved exactly for the hydrogen atom with one proton and one electron. This yields a description of quantized energy levels and orbital configurations. This scheme is used for larger atoms with certain adjustments. Electrons are added to the various orbitals, filling the higher energy levels as the size of the atom increases.

The are three orbital quantum numbers that describe an orbital. They are called n, l and ml. The principal quantum number n determines the size of the atom. The second orbital quantum number l determines the shape of the orbital in a general way. The third quantum number ml determines the orientation, for example, with respect to an applied magnetic field.

The value of n can be any positive integer (1, 2, 3, 4, etc.) The value of l is from zero to n-1. Thus for n = 1, the value of l is 0. For n = 2, l can be 0 or 1. For n = 3, l can be 0, 1 or 2. For n = 4, l can have values of 0, 1, 2 or 3. The various shapes that correspond to the values of l are referred to as s, p, d and f orbitals.

 l 0 1 2 3 orbital s p d f

A given orbital can be described by combining the first and second quantum numbers n and l.

1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, etc.

The third quantum number has the range -l to l. Thus for l = 0 there is only one value for ml (0). For l = 1 there can be three values for ml (-1, 0 or 1). For l = 2, ml can have five values (-2, -1, 0, 1 or 2). For l = 3, ml can have seven values (-3, -2, -1, 0, 1, 2 or 3). Thus for the s orbital there is only one value for ml. For a p orbital there are 3 orientations. For a d orbital there are five orientations. For an f orbital there are 7 orientations.

 orbital s p d f orientations 1 3 5 7

There is a fourth quantum number ms. This is called the spin quantum number. It can have two values, 1/2 and -1/2. This is because an electron can orient in two ways in an applied magnetic field. You might imagine an electromagnet with the wire coiled clockwise or counter clockwise. Thus an orbital specified by the first three quantum numbers can be occupied by only two electrons.

The shapes or orbitals are described by surfaces through regions of equal probability. The shapes most commonly used are those that include 99% probability. An s orbital is spherical. An s orbital is shown below.

A p orbital has two parts separated by a nodal plane where the probability is zero. There are three orientations available for a p orbital. They are named pz, py and px. These are shown below.

A d orbital has four lobes. The probability is zero between the lobes. There are five possible orientations. These are shown below.

The first quantum number determines the size, i.e., a 1s orbital is smaller than a 2s orbital which is smaller than a 3s orbital, etc. The energy of an orbital is determined mostly by the first two quantum numbers. The order in increasing energy is given below.

1s, 2s, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f @ 5d, 6p, 7s, 5f @ 6d

When building the electronic configuration of many electron atoms we start by placing electrons in the lowest energy orbitals and work up from there. For hydrogen the single electron goes into the 1s orbital. This is the ground state, the state of lowest energy for the hydrogen atom. The electron may be elevated into higher energy orbitals by absorbing the energy of a photon of the right frequency. As the electron falls back to the ground state energy is released. The energy of the photon emitted depends on the energy difference between the orbits involved in the transition.

For helium the two electrons, one of spin 1/2 and the other of spin -1/2, are placed in the 1s orbital. The 1s orbital is filled with two electrons. This is represented as 1s2. The next electron (lithium) goes into the orbital with the next highest energy, the 2s orbital. This is written 1s22s1. The 2s orbital is filled at beryllium with two electrons. The electron configuration of the beryllium atom in the ground state is represented in the form 1s22s2. With boron the next electron is placed in one of the 2p orbitals. They are all equivalent in energy. This is shown as 1s22s2 2p1.

The carbon atom has six electrons. The lowest energy is acieved by placing the next electron in one of the unoccupied 2p orbitals. Furthermore the lowest energy is achieved if both the 2p electrons have the same spin (parallel spin). Such electrons are called unpaired. The lowest energy carbon atom electron configuration can be represented as 1s22s2 2p2 or 1s22s2 2px1 2py1.

For a list of electronic configurations for all the elements go to Electronic Configurations.