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General Chemistry Ch. 7 - Quantum Theory and Atomic Structure

Electromagnetic radiation Energy propagated by means of electric and magnetic fields that alternately increase and decrease in intensity as they move through space.
Two independent variables that describe the wave properties of electromagnetic radiation Frequency and wavelength
Frequency Symbol nu, is the number of cycles the wave undergoes per second and is expressed in units of 1/second, or S^-1, AKA hertz (Hz).
Wavelength Symbol lambda, is the distance between any point on a wave and the corresponding point on the next crest (or trough) of the wave, that is, the distance the wave travels during one cycle. Expressed in meters or angstroms.
The speed of a wave ALL types of electromagnetic radiation travel at the same speed: 299,792,458 m/s, the speed of light (c).
Speed of light calculation c = nu * lambda. I.e. frequency * wavelength.
Amplitude of a wave The height of the crest (or depth of the trough) of each wave. The amplitude of an electromagnetic wave is a measure of the strength of its electric and magnetic fields. Thus amplitude is related to the intensity of the radiation.
The dimness/brightness of light is related to its Amplitude
Electromagnetic spectrum The continuum of wavelengths/frequencies of radiant energy
List electromagnetic spectrum regions by lowest frequency (highest wavelength) to highest frequency (lowest wavelength) Radio waves, microwaves, infrared, visible(red -> violet), ultraviolet, x-ray, gamma rays.
How should you quantitatively refer to wavelength regions? Lambda = x. E.g. for the visible region of color red lambda = 750.
What phenomenon describes why electromagnetic radiation travels at different velocities through different media (e.g. vacuum vs. air vs. water) Refraction
Refraction A phenomenon in which a wave changes its speed and therefore its direction as it passes through a phase boundary.
Why does light split into different colors as it passes through a prism (this is called dispersion of light)? The new angle of refraction as light passes through a substance depends on the materials on either side of the boundary and the wavelength of light, so each incoming wave in a prism is refracted at a slightly different angle.
Why does a rainbow form the color band? Light refracted through the droplets hit your eye at different angles depending on the distance the droplet it from your eye. The droplets closest to you (closest to the ground) are being refracted at the most acute angles.
How does refraction of energy differ from the behavior of particles If you throw a pebble into water it wont undergo refraction. That is, rather than immediately changing direction, the trajectory will gradually curve as it slows down.
Diffraction When a wave strikes the edge of an object, it bends around it in a phenomenon called diffraction.
How does diffraction of energy differ from particle behavior? When energy passes through a slit, it bends around both edges of the slit and forms a semicircular wave on the other side of the opening.
Interference If waves of light pass through two adjacent slits the emerging circular waves interact with each other through the process of interference.
Interference pattern results from… Waves that interfere constructively (in phase) their amplitudes are summed and waves that interfere destructively (out of phase), cancel each other out. This creates the diffraction (interference) pattern.
How does the interference pattern differ from particle movement Particles, when moving through such a slit, will continue to move in a straight line.
Three phenomena involving matter and light that confounded physicists at the turn of the 20th century (1) Blackbody radiation, (2) the photoelectric effect, and (3) atomic spectra
Blackbody A blackbody is an idealized object that absorbs all the radiation incident on it.
Blackbody radiation Light given off by a hot blackbody
What did Max Planck propose about hot blackbodies? He proposed that the hot blackbody could emit (or absorb) only certain quantities of energy: E = n*h*nu, where n = quantum number, h = Planck’s constant, and nu = frequency
If an atom itself can have only certain quantities of energy, then… The energy of an atom is quantized: it exists only in certain fixed quantities, rather than being continuous.
Quantum Plural: quanta, are “packets”, or definite amounts of, energy.
An atom changes its energy state by… Emitting (or absorbing) one or more quanta.
The energy of the emitted (or absorbed) radiations is equal to… The difference in the atom’s energy states: deltaE_atom = delta(n*h*nu)
Equation describing when an atom in a given energy state changes to a single adjacent state deltaE = h*nu (why? Because n = 1 because it’s the smallest measurement, a single quantum)
The photoelectric effect The flow of current when monochromatic light of sufficient frequency shines on a metal plate. The current arises because energy transferred to the metal by the light causes electrons to break free.
Confusing features about the photoelectric effect that challenged the wave model of energy (1) The presence of a threshold frequency and (2) the absence of a time lag
Photoelectric effect: The presence of a threshold frequency Light shining on a metal must have a minimum frequency or no current flows. This causes confusion because the wave theory associates energy with amplitude (intensity), not with frequency (color).
Photoelectric effect: Absence of a time lag Current flows from the moment light of this minimum frequency shines on the metal, regardless of the light’s intensity. The wave theory predicts there would be a time lag before the current so the electron could break free
Photoelectric effect: where did Einstein step in? Einstein proposed that light itself is particulate, that is, quantized into small bundles of electromagnetic energy, later called photons.
Energy of a photon E_photon = h*nu = deltaE_atom
Einstein’s photon theory explanation for the presence of a threshold frequency A beam of light consists of a large number of photons. Brightness is related to the number of photons striking a surface per unit time. A photon of certain minimum energy must be absorbed for an electron to be freed.
Einstein’s photon theory explanation for the absence of a time lag An electron cannot “save up” energy from several photons below the minimum energy until it has enough to break free, rather one electron breaks free the moment it absorbs one photon of enough energy (hence: quantized).
Current is weaker in dim light than bright light because… Fewer photons of enough energy are present resulting in fewer electrons being broken free per unit time.
Planck’s quantum theory and Einstein’s photon theory assigned properties to energy that always had been reserved for matter: Fixed quantity and discrete particles
How does the photon model of energy fit the facts of diffraction and refraction? The photon model doesn’t replace the wave model; we must accept both as reality.
Line spectrum A series of fine lines of individual colors separated by colorless (black) spaces. This band represents the light emitted from excited gas.
Three postulates Bohr made in his model of the hydrogen atom 1. The H atom has only certain allowable energy levels, 2. The atom does not radiate energy while in one of its stationary states, and 3. The atom changes to another stationary state only be absorbing or emitting a photon…
Stationary states Energy levels whereby each of these states is associated with a fixed circular orbit of the electron around the nucleus.
How does an atom change to another stationary state? It must absorb or emit a photon whose energy equals the difference in energy between the two states: E_stateA – E_stateB where E_stateA is higher than that of E_stateB
Lower energy level of an atom means that its electron… Orbits at a smaller radius
When the electron is in the first (lowest) orbit it is said to be in the… Ground state
When an electron is in the second or any higher orbit, the atom is said to be in an… Excited state
Limitations of the Bohr model (1) The illustrated model of the H atom is incorrect, (2) the math only predicts the bands for one-electron species because electron repulsions and attractions are present in those with more than one electron.
Equation for calculating the energy levels of an atom E = -2.18^-18 * (Z^2/n^2) where Z is the charge of the nucleus, and n is the energy level.
Equation to find the energy difference between two energy levels h*nu = -2.18^-18*(1/n^2_final – 1/n^2_initial), and since h*nu = hc/lambda, thus: 1/lambda = ((-2.18^-18)/hc)*(1/n^2_final – 1/n^2_initial)
Calculating Ionization energy of an atom Quantity of energy needed to remove an electron from an atom, e.g. H(g) -> H+(g) + e-: -2.18^-18*(1/infinity – 1/1^2) = -2.18^-18. Then for 1 mole of atoms: -2.18^-18*(6.022*10^23) = 1.31*10^6 J
Definition of ionization of an atom The quantity of energy required to form 1 mol of gaseous H+ ions from 1 mol of gaseous H atoms
Louis de Broglie’s proposal about matter in the early 1920s If energy is particle-like, perhaps matter is wave-like
de Broglie wavelength Louis de Broglie combined Einstein’s mass equivalence (E=mc^2) with Planck’s photon energy (E=hc/lambda) to obtain lambda = (h/mu) where h = constant, m = mass and u = speed.
Conceptually what does the de Broglie wavelength equation state? That matter travels in waves, and that the wavelength is inversely proportional to the object’s mass and speed.
So if electrons travel in waves, then they should display… Diffraction, and therefore an interference pattern in the double-slit experiment.
If electrons have properties of energy, do photons have properties of matter? How does the de Broglie equation approach this? We can calculate momentum (p), the product of mass and speed, for a photon of a given wavelength. Substitute c for you and re-write de Broglie’s equation: lambda = h/p, solve for p: p = h/lambda
Conceptually describe the implications of showing that photons display properties of matter with regard to p = h/lambda The inverse relationship between p and lambda means that shorter wavelengths should have greater momentum. Thus, a decrease in a photon’s momentum should appear as an increase in its wavelength.
Experimentally, has the photon’s matter-like behavior been observed? In 1923, Arthur Compton directed a beam of x-ray photons at a sample of graphite and observed that the wavelength of the reflected photons increased. This means the photons transferred some of their momentum.
The fact that matter can be energy-like and energy can be matter-like ultimately implies what? That both matter and energy show both behaviors; each possesses both “faces”.
The dual character of matter and energy is known as The wave-particle duality
Uncertainty principle States that it is impossible to know the exact position and momentum (mu) of a particle simultaneously. Uncertainty principle equation: deltaX*m*deltaU >= h/(4*pi) where deltaX is the uncertainty in position and deltaU in speed.
Mathematically the uncertainty principle states that the more accurately we know the position of the particle (smaller deltaX), then…, thus… Then the less accurately we know its speed (larger deltaU). Thus, unlike classical physics, we can’t calculate the exact trajectory of an electron.
What does the uncertainty principle say about Bohr’s model? It means that we cannot assign fixed paths for electrons, such as the circular orbits of Bohr’s model. All we can ever hope to know is the probability of finding an electron in a given region of space.
Quantum mechanics The branch of physics that examines the wave motion of objects on the atomic scale
What model did Erwin Schrodinger develop? He derived a model that describes an atom that has certain allowed quantities of energy due to the allowed frequencies of an electron whose behavior is wavelike and whose exact location is impossible to know.
Schrodinger equation (Hamiltonian operator)*(psi) = (E)*(psi)
In the Schrodinger equation what does psi and the Hamiltonian operator represent? Psi = wave function, a mathematical description of the electron’s matter-wave in terms of position in three dimensions. H.O. = a set of mathematical operations that, with psi, yields an allowable energy level.
Each solution to the Schrodinger equation (i.e. the energy state of the atom) represents… …the given atomic orbital.
Atomic orbital “orbital” in the quantum-mechanical model bears no resemblance to an “orbit” in the Bohr model: an orbit was an electron’s path around the nucleus, whereas, an orbital is a mathematical function with no direct physical meaning.
The Schrodinger equation allows us to… Describe where an electron “probably” is, even though it is impossible to know precisely where it is most likely to be found.
Psi^2 The probability density; a measure of the probability that the electron can be found within a particular tiny volume of the atom
Electron density diagram For a given energy level, we can depict the probability of where an electron can be found on a dotted diagram called the electron density diagram. Also called, an electron cloud.
Summary of the result of an electron density diagram Higher probability of finding an electron in the second layer than the first, but then drops off and probability quickly diminishes with greater distance.
Radial probability distribution plot A chart that shows the layers around an atom and the dots where the electrons might be found according to their probabilities
Probability contour A shaped that defines the volume around an atomic nucleus within which an electron spends a given percentage of its time. We don’t know the exact volume of an atom due to the electron’s orbital, but we can be ~90% sure.
As a result of Schrodinger’s research we can conclude that each atomic orbital has… A distinctive radial probability distribution and 90% probability contour.
An atomic orbital is specified by… Three quantum numbers: (1) the principle quantum number n, (2) the angular momentum quantum number l, and (3) the magnetic quantum number m_l.
Principal quantum number (n) It is a positive integer. It indicates the size of the orbital thus the distance from the nucleus of the peak in the radial probability distribution plot. The higher the n value, the higher the energy level. n = 1 = ground state
Angular momentum quantum number (l) Is an integer from 0 to n - 1. It is related to the shape of the orbital and is sometimes called the orbital shape (or azimuthal) quantum number. n limits l, e.g. for an orbital of n = 2, l can have only a value of 0 or 1.
Magnetic quantum number (m_l) An integer from -l (the letter) through 0 to +l. It describes the orientation of the orbital in the space around the nucleus and is sometimes called the orbital-orientation quantum number. Thus if l is 2, m_l = -2 through +2
If n = 3, how many orbitals are there That means l = 0 & m_l = 0; l = 1 & m_l = -1, 0, +1; l = 2 & m_l = -2, -1, 0, +1, +2. That’s a total of nine different m_l’s, meaning when n = 3 there are nine orbitals.
Energy levels (or shells) are given by The n value: the smaller the n value, the lower the energy level and the greater the probability of the electron being closer to the nucleus.
Energy sublevels (or subshells) Designates the orbital shape. l = 0 = s sublevel; l = 1 = p sublevel; l = 2 = d sublevel; l = 3 = f sublevel
Sublevels are named by… Joining the n value and the letter designation, e.g. the sublevel (or subshell) with n = 2 and l = 0 is called the 2s sublevel.
Orbital Each allowed combination of n, l and m_l values specifies one of the atom’s orbitals. Thus, the three quantum numbers that describe an orbital express its size (energy), shape, and spatial orientation.
s orbital: description and shape 1s orbital: spherical, highest probability density at nucleus, 2s orbital larger sphere, highest probability farther away from nucleus, 3s orbital: even larger sphere, highest probability density farther from nucleus.
s orbital node Between the nucleus region and the next region where there is a probability density, there is a area called the node where the probability density = 0. More nodes appear as the s orbital grows to larger energy states.
p orbital: description and shape Has two regions (lobes) of high probability, one on either side of the nucleus. The nucleus lies at the nodal plane of the dumbbell-shaped orbital. Only levels with n = 2 or higher can have a p orbital. An electron spends equal time in each lobe.
The lowest energy p orbital 2p.
How does the magnetic quantum number (m_l) of the p orbital effect its spatial orientation? Unlike an s orbital, each p orbital does have a specific orientation in space. Each m_l value refers to three mutually perpendicular p orbitals. They correspond with the x, y, and z axes: p_x, p_y, and p_z
d orbital: shape and description It can have 5 different orientations (e.g. for l = 2: -2, -1, 0, 1, 2). Study shapes on pg 294.
orbitals higher than l = 2 l = 3 are f orbitals and must have a principal quantum number of at least n = 4. There are seven f orbitals, each with a complex multilobed shape. l = 4 are g orbitals. we will not discuss those because they play no role in bonding
Special case of H atom The energy state of the H atom depends only on the principal quantum number, n. For the H atom only, all four n = 2 orbitals have the same energy, all nine n = 3 orbitals have the same energy, etc.
Created by: Intellex_