Atoms and Quantum Numbers GLY 4200 – Lecture 5 –Fall, 2018 The basic building blocks of all matter, including minerals, are atoms. The chemical and nuclear properties of atoms determine the properties of minerals, and often record information about the formation and subsequent history of the minerals. Careful study of a mineral can help unravel some of this history.

Atom Grossly oversimplified view of atom, but it does have a nucleus and electron cloud An atom is a small particle composed of two parts, the nucleus at the center, and the surrounding electron cloud. Atoms have dimensions on the order of Angstroms, or tenths of a nanometer. The nucleus is approximately 100,000 times smaller than the atom. The nucleus is composed of two types of particles:

Nucleons Protons – charge of +1 Neutrons – neutral, with a mass slightly greater than the proton Atomic number = number of protons, often denoted z Atomic number is designated as a left subscript to the chemical symbol - 1H, 14Si The number of protons determines the major properties of the atom, including what element it is. This number is called the atomic number. When atoms have different atomic numbers, they are different elements. Example: Si has 14 protons, Fe has 26 protons, O has 8, and Pb has 82. The atomic number is designated as a left subscript to the chemical symbol - 1H, 14Si, etc.

Atomic Weight The atomic weight equals the number of protons plus neutrons Atomic weight is shown as a left superscript to the chemical symbol, such as 16O The number of neutrons has only subtle effects on the chemistry of the atom. The number of protons and neutrons determines the atomic weight, shown as a left superscript to the chemical symbol, such as 16O. One element may have one or more numbers of neutrons:

Isotopes One element may have one or more numbers of neutrons: 11H is normal hydrogen, with one proton and no neutrons 12H is deuterium, or heavy hydrogen, with one proton and one neutron 13H is tritium, a radioactive form of hydrogen, with one proton and two neutrons Collectively, the different types of the atoms of one element are called isotopes Chemically, the isotopes of one element behave very similarly, with only small differences due to differing masses. Deuterium was discovered in 1931 by Harold Urey, and was the first known example of an isotope. It is present in seawater at a level of 156 ppm, or 1 atom for every 6420 atoms of hydrogen. Urey won the Nobel Prize in Chemistry in 1934 for his discovery. When formed in stars, the ratio is about 26 atoms of deuterium per million atoms of hydrogen. This is the ratio seen in gas giant planets like Jupiter. Mass fractionation effects associated with heating of ices in comets is thought to have changed this ratio. Limited data from comets suggest the deuterium to hydrogen ratio in comets is about 156 per million, close to earth’s oceans, and may suggest a cometary origin for at least some water on earth. By definition, atoms are electrically neutral Therefore, the number of electrons in the electron cloud around the nucleus must equal the atomic number. If the number of electrons does not equal the number of protons, then the particle becomes an ion, rather than an atom. There are two types of ions:

Ions Cation - the atom has lost electrons, so there is a net positive charge Anion - the atom has gained electrons, so there is a net negative charge Cations are found toward the left side and center of the periodic table, while anions are found on the right side

Planetary Model of the Atom Early models of the atom showed the electrons circling the nucleus like the planets orbit the sun Figure shows nitrogen represented this way Source: http://education.jlab.org/qa/atom_model.html It was soon realized that this model was incorrect because it gave an inaccurate description of atomic structure. If electrons orbited the nucleus in a manner which resembled the planets orbiting the sun, classical physics predicted that the electrons would spiral in and crash into the nucleus within a fraction of a second. Scientists learned that electrons could only occupy certain orbits (usually referred to as energy levels or shells) around the nucleus. They also discovered that only a certain number of electrons could fit in each energy level.

Johann Balmer Discovered a formula for the spectral lines of hydrogen in 1885 Images: http://www.neatherd.org/astronomy/Inside%20Atoms.htm Swiss mathematician who discovered a formula basic to the development of atomic theory and the field of atomic spectroscopy. This turned out to be a special case of the Rydberg formula, discovered by Johannes Rydberg, in 1888, which is used to predict the wavelengths of photons (of light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. The explanation for why matter behaved this way had to wait for Neils Bohr.

Neils Bohr Danish physicist 1885-1962 Formulated the next model of the atom, now called the Bohr model Won the 1922 Nobel prize for physics Image: http://www.neatherd.org/astronomy/Inside%20Atoms.htm Neils Bohr, Danish physicist, refined the description of atoms. He placed electrons circling the nucleus in concentric circles around the nucleus.

Bohr Model of Nitrogen Electrons are in discrete orbitals Image: http://education.jlab.org/qa/atom_model.html Neils Bohr, Danish physicist, refined the description of atoms. He placed electrons circling the nucleus in concentric circles around the nucleus. Electrons are in discrete orbitals Two of the electrons are in the first energy level The other five are in the second energy level

Spectral lines of Elements Source: http://www.hi.is/~hj/QuantumMechanics/quantum.html This model failed to correctly predict many properties of matter, especially the spectra of elements heavier than hydrogen, so it was discarded in favor of the quantum mechanical description of atoms. A very simplified view of the quantum mechanical description says that electrons are only allowed to occupy certain fixed energy states. The quantum mechanical model cannot tell exactly where an electron is at a given moment or where it is going. It allows the calculation of the probability that an electron will be found in a given volume of space, but that isn't the same as knowing where that electron is.

De Broglie Waves An electron standing wave vibrating in an orbit around a nucleus of an atom Only integral numbers of wavelengths are allowed Source: http://www.clickandlearn.org/chemistry/DeBroglie.htm In 1924, a French physicist named Louis de Broglie suggested that, like light, electrons could act as both particles and waves. De Broglie's hypothesis was soon confirmed in experiments that showed electron beams could be diffracted or bent as they passed through a slit much like light could. So, the waves produced by an electron confined in its orbit about the nucleus sets up a standing wave of specific wavelength, energy and frequency (i.e., Bohr's energy levels) much like a guitar string sets up a standing wave when plucked.

Standing Wave, One Loop One String Vibrating, 1/2 wavelength Two nodes Source: http://www.clickandlearn.org/chemistry/DeBroglie.htm If a string is fixed on both ends, then the only waves that can occur are those with zero amplitude at those fixed ends; such points of zero amplitude are called nodes. The allowed vibrations are called standing waves. One String Vibrating, 1/2 wavelength Two nodes Quantum number n = 1

Standing Wave, Two Loops One String Vibrating, 1 wavelength 3 nodes Quantum number n = 2

Standing Wave, Four Loops As long as the number of wavelengths per orbit is integral, the waves reinforce. If the number is not integral, the waves interfere destructively, and this corresponds to non-allowed states. One String Vibrating, 2 wavelengths 5 nodes Quantum number n = 4

Schrődinger’s Equation Source: http://www.hi.is/~hj/QuantumMechanics/quantum.html Erwin Rudolf Josef Alexander Schrödinger (12 August 1887 – 4 January 1961) was an Austrian physicist who developed a number of fundamental results in the field of quantum theory, which formed the basis of wave mechanics: he formulated the wave equation (stationary and time-dependent Schrödinger equation) Quantum theory predicts the allowed orbitals as solutions of the Schrödinger equation. This is a complex equation, and solutions can only be approximated. Quantum theory describes the allowed orbits an electron may have using four quantum numbers:

Principal Quantum Number n has values 1,2,3..... The Principal Quantum Number refers to the angular momentum of the electron n determines which shell of electrons is being discussed May be referred to as n =1, n = 2, etc. In X-ray terminology, shells are assigned letters, K,L,M,N,O,P,Q for n = 1 to 7

Azimuthal Quantum Number ℓ may take values 0, 1, 2, 3, ... (n-1) The Azimuthal Quantum Number Indicates the symmetry of the orbital ℓ determines the subshell to which an electron belongs s, p, d, f are used to designate ℓ = 0 to 3

Magnetic Quantum Number m has allowed values are 0, ±1, ±2, ±3, ..... ±ℓ The Magnetic Quantum Number refers to the orientation of the orbital in space

Spin Quantum Number s has allowed values are ±½ The Spin Quantum Number refers to the direction of rotation of the electron itself The subshell designation describes the symmetry of the orbital.

s Orbitals s orbitals are spherically symmetric The radius increases as n increases ℓ = 0 One orbital, two electrons

p Orbitals p orbitals are aligned along the three principal axes ℓ = 1 Three orbitals, six electrons

d Orbitals Four lobes, oriented as shown ℓ = 2 Five orbitals, ten electrons

f Orbitals Complex shapes ℓ = 3 Seven orbitals, fourteen electrons

Wolfgang Pauli Austrian physicist, 1900-1958 Formulated the Exclusion Principle, which today bears his name Won the Noble Prize in Physics, 1945 Image: http://quasar.physik.unibas.ch/~aste/pauli.html Wolfgang Pauli, Austrian physicist, used quantum numbers to formulate the “Exclusion Principle”, which now bears his name.

Pauli Exclusion Principle Proposed in 1925 The Pauli Exclusion Principle States that no two electrons in an atom can have the same four quantum numbers Since any given orbital has n, ℓ, and m the same, there can only be two electrons per orbital, with s = ±½