The Quantum Mechanical Picture of the Atom

What is Quantum Mechanics? Although Bohr’s planetary model worked well in describing the electron behavior in an atom of hydrogen, it did not work for any other atom. Through the work of de Broglie and others, it was determined that electrons in atoms can be treated as waves more effectively than as small compact particles traveling in circular or elliptical orbits. As a result, a new approach, called Quantum Mechanics, was taken to describe the behavior of very small particles such as electrons.

Heisenberg Uncertainty Principle One of the underlying principles of quantum mechanics is that it is impossible to precisely determine the paths that electrons follow as they move about the atom’s nucleus. Heisenberg’s Uncertainty Principle: It is impossible to determine accurately both the momentum and the position of an electron (or any other very small particle) simultaneously. Consequently, a statistical approach is taken in which the probability of finding an electron within a specified region of space is determined.

Foundations of Quantum Mechanics Atoms can exist only in certain energy states. In each energy state, the atom has a definite energy. When an atom changes its energy state, it must emit or absorb just enough energy to bring it to the new energy state (quantum condition). The energy lost (or gained) by an atom as it goes from higher to lower (or vice versa) energy states is equal to the energy of the photon emitted (or absorbed) during the transition. The allowed energy states of atoms can be described by sets of numbers called quantum numbers.

Contributions to Energy States The energy states of the electrons result from contributions from three different sources 1. Kinetic energy of the electrons as they move around the nucleus. 2. Potential energy caused by the attraction between the nucleus and the electrons. 3. Potential energy caused by the repulsion between electrons.

Quantum Mechanics Quantum Mechanical treatment of electrons is highly mathematical. In 1926, Schrodinger modified an existing equation that described the electron by a kind of standing-wave mathematics. In this approach, the electron is described by a three-dimensional wave function, Ψ. In a given space around the nucleus, only certain “waves” can exist.

Quantum Mechanics (continued) Each allowed wave corresponds to a stable energy state for the electron and is described by a set of four quantum numbers. The set of quantum numbers result from the solving of the Schrodinger wave equation and identifies the area where the electron is most likely to be found. The set of quantum numbers give information about the shapes and orientations of the regions in space in which the probability of finding an electron is high. These regions are called atomic orbitals. Electron density is directly proportional to the probability of finding an electron at that point.

Quantum Numbers Quantum numbers are used to designate the electronic arrangement in all atoms. Electronic arrangements are also referred to as electron configurations. Quantum numbers also play important roles in describing the energy levels of electrons and the shapes of the orbitals that describe the distributions of electrons in space.

Principal Quantum Number The principal quantum number, n, describes the main energy level, or shell, an electron occupies. It also indicates the size of the orbital. Principal quantum numbers may be any positive integer: n = 1, 2, 3, 4, … As “n” increases, the orbital becomes larger and the electron spends more time farther from the nucleus. As “n” increases, the energy of the electron becomes higher since they are less tightly bound to the nucleus.

Principal Quantum Number (continued) There is a maximum number of electrons that may be found in each energy level according to the equation: 2n2. energy level electron capacity n=1 2 n=2 8 n=3 18 n=4 32

Angular Momentum (Subsidiary) Quantum Number The angular momentum quantum number, l, designates the shape of the region in space that an electron occupies. This shape is also sometimes referred to as the sublevel that an electron may occupy. The angular momentum quantum number may take integral values from 0 up to and including (n-1): l = 0, 1, 2, 3, … (n-1)

Angular Momentum Quantum Number (continued) Each value of l is given a letter notation. Each letter corresponds to a different sublevel. value of l sublevel shape electron capacity 0 s spherical 2 1 p hourglass (2 lobes) 6 2 d 4 lobes 10 3 f 8 lobes 14 4 g unknown 18

Orbital Shapes s orbital p orbital f orbital d orbital

Magnetic Quantum Number The magnetic quantum number, ml , designates the specific orbitals within a sublevel. Orbitals within a given sublevel differ in their orientations in space, but not in their energies. Two or more orbitals that have the same energy are said to be degenerate. Each atomic orbital can only accommodate two electrons. Within each sublevel, ml may take any integral values from -l through zero and including +l.

Magnetic Quantum Number (continued) The maximum value of ml depends on the value of l. value of l sublevel value of ml # of orbitals electron capacity 0 s 0 1 2 1 p -1, 0, 1 3 6 2 d -2, -1, 0, 1, 2 5 10 3 f -3,-2, -1, 0, 1, 2, 3 7 14

Spin Quantum Number The spin quantum number, ms, refers to the spin of the electron and the orientation of the magnetic field produced by this spin. The values of ms are either +1/2 or -1/2. If two electrons occupy the same orbital, they will have opposite spins. Therefore, one will have a value of ms = +1/2 and the other will have a value of ms = -1/2. Substances that contain unpaired electrons are weakly attracted to magnetic fields and are said to be paramagnetic. Substances in which all electrons are paired are very weakly repelled by magnetic fields are called diamagnetic.

Pauli Exclusion Principle The electronic structures of atoms are governed by the Pauli Exclusion Principle. The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of four quantum numbers.

Electron Capacity of sublevel Electron capacity of Energy Level Summary n l sublevel ml # of orbitals Electron Capacity of sublevel ms Electron capacity of Energy Level 1 s 2 +1/2 - 1/2 8 p -1, 0, +1 3 6 18 d -2, -1, 0, +1, +2 5 10 4 32 P f -3, -2, -1, -0, +1, +2,+3 7 14