X-Rays from Atoms The highest photon energy available in a hydrogen atom is in the ultraviolet part of the electromagnetic spectrum Other atoms can emit much more energetic photons – larger Z, more electric potential energy is available May applications use X-ray photons obtained from an electron transition from E 2 to E 1 where n=2  n=1 in heavier atoms These are called K α X-rays See table 29.1 for the energy of K α X-rays produced by some elements Section 29.3

Continuous Spectrum If an absorbed photon has more energy than is needed to ionize an atom, the extra energy goes into the kinetic energy of the ejected electron This final energy can have a range of values and so the absorbed photon can have a range of values This produces a continuous absorption spectrum Section 29.3

Bohr and de Broglie The allowed electron orbits in the Bohr model correspond to standing waves that fit into the orbital circumference Since the circumference has to be an integer number of wavelengths, 2 π r = n λ This leads to Bohr’s condition for angular momentum Section 29.3

Problems with Bohr’s Model The Bohr model was successful for atoms with one electron H, He +, etc. The model does not correctly explain the properties of atoms or ions, that contain two or more electrons Physicists concluded that the Bohr model is not the correct quantum theory It was a “transition theory” that help pave the way from Newton’s mechanics to modern quantum mechanics Section 29.3

Modern Quantum Mechanics Modern quantum mechanics depends on wave functions and probability densities instead of mechanical ideas of position and motion To solve a QM problem, you use Schrödinger’s equation The solution gives the wave function, including its dependence on position and time Four quantum numbers are required for a full description of the electron in an atom Bohr’s model used only one Section 29.4

Quantum Numbers, Summary Section 29.4 Electrons with a particular value of n live in what we call a “shell”

Orbital Quantum Number ℓ is the orbital quantum number – an integer LESS THAN n Allowed values are ℓ = 0, 1, 2, … n – 1 The orbital angular momentum L of an electron is L = ℓ h/2π Section 29.4

Orbital Magnetic Quantum Number m is the orbital magnetic quantum number Allowed values of m = - ℓ, -ℓ + 1, …, -1, 0, 1 …, ℓ,. The “multiplicity” is {2 ℓ + 1} different possibilities You can think of m as giving the “direction” of the angular momentum, ℓ, of the electron relative to some chosen axis (which picks out a special direction). m is often called the “projection” of l onto this axis. In particular, m = +- ℓ means ℓ is exactly along (or opposite to) the axis. m does, for orbital angular momentum, what the +-1/2 projections do for SPIN Section 29.4

Spin Quantum Number s is the spin quantum number. s = + ½ or – ½ These are often referred to as “spin up” and “spin down” This gives the direction of the electron’s spin angular momentum Since only spin ½ electrons are involved, we need only to specify the projection of the spin, not its total value which is always ½. Spin ½ means angular momentum = ½ unit S = ½ h/2π Section 29.4

Electron Shells and Probabilities A particular quantized electron state is specified by all four of the quantum number n, ℓ, m and s The solution of Schrödinger’s equation also gives the wave function of each quantum state From the wave function, you can calculate the probability for finding the electron at different locations around the nucleus Plots of probability distributions for an electron are often called “electron clouds” Section 29.4

Electron Clouds Section 29.4

Electron Cloud Example Ground state of hydrogen n = 1 The only allowed state for ℓ is ℓ = 0 This is an “s state” The only allowed state for m is m = 0 The allowed states for s are s = ± ½ The probability of finding an electron at a particular location does not depend on s, so both of these states have the same probability The electron probability distribution forms a spherically symmetrical “cloud” around the nucleus See fig A, or previous slide Section 29.4

Multielectron Atoms The general electron energy levels and probability distributions of multielectron atoms follow a similar pattern as hydrogen Use the same quantum numbers There are two main differences between hydrogen and multielectron atoms The values of the electron energies are different for different atoms The spatial extent of the electron probability clouds varies from element to element Section 29.5

Pauli Exclusion Principle Each quantum state in the atom can be occupied by only one electron This is an example of the Pauli exclusion principle, which says that two identical [matter] particles cannot occupy the same quantum state. Each electron in an atom is described by a unique, different set of quantum numbers. If at least one of the four quantum numbers is different, that distinguishes such an electron from all the other electrons in the atom This includes spin up vs spin down – so, for example, the two electrons in He can both be in the ground state n=0, one with s = +½, the other, -½ Section 29.5

Electron Configuration Shorthand notation for electron configurations: 1s 1 1 : n =1 s : ℓ = 0 Superscript 1 : 1 electron No direct information about electron spin, but 2 would mean two electrons with spin up and down, (which is max allowed) 1s 2 2s 2 2p 2 2 electrons in n = 1 with ℓ = 0 2 electrons in n = 2 with ℓ = 0 2 electrons in n = 2 with ℓ = 1 Section 29.5

Filling Energy Levels The energy of each level depends mainly on the value of n -- but not entirely In multielectron atoms, the order of energy levels is more complicated For shells higher than n = 2, the energies of subshells from different shells begin to overlap In general, with increasing Z the energy levels fill with electrons in the following order: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f Note how 4s sneaks in ahead of 3d. Due to its lower energy level. etc. etc. Section 29.5

Order of Energy Levels Section 29.5

Chemical Properties of Elements The periodic table was first assembled by Dmitry Mendeleyev in the late 1860’s – a case of “numerology” or inspired “pattern mongering” Only much later did the QM patterns explain the periodic table Mendeleyev and other chemists had noticed that many elements could be grouped according to their chemical properties Mendeleyev organized his table by grouping chemically-related elements in the same column His table had a number of “holes” because many elements had not yet been discovered: prediction Section 29.6

Chemical Properties, cont. Mendeleyev could not explain the regularities in the periodic table. Now we know! The electron energy levels and configurations in the atom cause its chemical properties When an atom reacts chemically, some of its electrons may be shared with (or else transferred to) other atoms (making chemical bonds) The bonding electrons are those occupying the highest energy levels – called valence electrons Section 29.6

Electrons and Shells Mendeleyev grouped elements into columns according to their common bonding properties and chemical reactions, which rely on the valence electrons When a shell has all possible states filled it forms a closed shell, (the “noble gases”) Elements in the same column in the periodic table have the same number of valence electrons Section 29.6

Electrons and Shells The last column in the periodic table contains elements with completely filled shells These elements are largely inert (the “noble gases”) They almost never participate in chemical reactions Elements in the first column in the periodic table have one valence electron, which they easily donate (highly reactive) to other atoms, leaving a closed shell Elements in the next to last column in the periodic table have one electron “missing” from a closed shell, and are eager to accept one more electron. In both cases, these elements are unusually reactive, chemically. Section 29.6

Structure of the Periodic Table The rows correspond to different values of the principal quantum number, n Since the n = 1 shell can hold only two electrons, the row contains only two elements For n = 2, ℓ = 0 (2l+1) = 1 * (2 electrons [spin!]). ℓ = 1 with m = 1, 0, -1 (2l+1 = 3) * (2 e) n = 2, there are 8 possible states = a closed shell n=3 allows ℓ = 0,1,2 multiplicities 1,3,5 = 9 times (2 e) n=3, closed shell has 18 states, n=3 states become 18 long, 10 longer than n=2 row To keep “closed-shell +- one electron” columns lined up, need to INSERT the extra 10, spreading the table apart Section 29.6

Periodic Table Section 29.6

Structure of the Periodic Table But n=4, ℓ =0 gets ahead of n=3, ℓ =2, these ten states drop down to the next line of the periodic table. Notice the chemical similarity of these ten elements – such as iron, copper, manganese, zinc, chromium, cobalt, nickel, etc n=5, ℓ=4 (multiplicity 2*4+1) would cause the table to spread 14 columns wider. Instead, for tidiness, the fourteen extra states (plus the element that was in the slot) is shown as rows of 15 states below the main table These are the lanthanide series (rare earths) and similarly the actinide series: very heavy, mostly radioactive, for nuclear reasons (as opposed to atomic reasons.) Section 29.6

Example Electron Configurations Section 29.6

Atomic Clocks Atomic clocks are used as global time standards The clocks are based on the accurate measurements of certain spectral line frequencies Cesium atoms are popular One second is now defined as exactly the time it takes a cesium clock to complete 9,192,631,770 ticks Section 29.7

Incandescent Light Bulbs The incandescent bulb contains a thin wire filament that carries a large electric current Type developed by Edison The electrical energy dissipated in the filament heats it to a high temperature The filament then acts as a blackbody and emits radiation Section 29.7

Fluorescent Bulbs This type of bulb uses gas of atoms in a glass container An electric current is passed through the gas This produces ions and high-energy electrons The electrons, ions, and neutral atoms undergo many collisions, causing many of the atoms to be in an excited state These atoms decay back to their ground state and emit light Section 29.7

Neon and Fluorescent Bulbs A neon bulb contains a gas of Ne atoms Fluorescent bulbs often contain mercury atoms Mercury emits strongly in the ultraviolet The glass is coated with a fluorescent material The photons emitted by the Hg atoms are absorbed by the fluorescent coating The coating atoms are excited to higher energy levels When the coating atoms undergo transitions to lower energy states, they emit new photons The coating is designed to emit light throughout the visible spectrum, producing “white” light Section 29.7

Lasers Lasers depend on the coherent emission of light by many atoms, all at the same frequency In spontaneous emission, each atom emits photons independently of the other atoms It is impossible to predict when it will emit a photon The photons are radiated randomly in all directions In a laser, an atom undergoes a transition and emits a photon in the presence of many other photons that have energies equal to the atom’s transition energy A process known as stimulated emission causes the light emitted by this atom to propagate in the same direction and with the same phase as surrounding light waves Section 29.7

Lasers, cont. Laser is an acronym for light amplification by stimulated emission of radiation The light from a laser is thus a coherent source Mirrors are located at the ends of the bulb (laser tube) One of the mirrors lets a small amount of the light pass through and leave the laser Section 29.7

Lasers, final Laser can be made with a variety of different atoms One design uses a mixture of Ne and He gas and is called a helium-neon laser The photons emitted by the He-Ne laser have a wavelength of about 633 nm Another common type of laser is based on light produced by light-emitting diodes (LEDs) These photons have a wavelength around 650 nm These are used in optical barcode scanners Section 29.7

Force Between Atoms Consider two hypothetical atoms and assume they are bound together to form a molecule The binding energy of a molecule is the energy required to break the chemical bond between the two atoms A typical bond energy is 10 eV Section 29.7

Force Between Atoms, cont. Assume the atom is pulled apart by separating the atoms a distance Δx The magnitude of the force between the atoms is A Δx of 1 nm should be enough to break the chemical bond This gives a force of ~1.6 x N Section 29.7