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193 Cards in this Set
- Front
- Back
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Light |
c = hv |
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heat capacity at constant volume |
3R |
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Borh Frequency |
E = hv |
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short wavelength |
high momentum |
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long wavelength |
low momentum |
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Normalizing the wavefunction |
integral of square is equal to 1 |
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probability of finding particle |
abs of wavefunction squared |
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node |
where a wavefunction passes through zero |
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operator |
function that acts upon the wavefunction |
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wavefunction of particle in a box |
sine function |
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2D particle |
L1 = rectangle length, L2 = rectangle width |
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3D particle |
L1 = square length, L2 = square width, L3 = square height |
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degeneracy |
for two dimensional and higher systems different quantum numbers can lead to the same energy levels |
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quantum tunneling |
penetration into or through classically forbidden areas. If walls are thin (so v falls to zero again after a finite distance) then the wave function will vary smoothly within the wall before oscillating on the other side |
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conditions for tunneling |
slopes must be continuous at edges of barrier (ikA - ikB) = (kC - kD) |
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all wavefunctions approach zero |
because gaussian functions go quickly to zero as displacement increases |
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because y^2 is proportional to x^2 * (mkf)^1/2 |
wavefunctions decay more rapidly for large masses and large force constants |
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as v increases |
hermite polynomials grow larger so wavefunction spreads over wider range as v increases |
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rotational motion |
angular momentum Jz is perpendicular to xy plane |
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positive angular momentum |
clockwise rotation |
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negative angular momentum |
counterclockwise rotation |
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orbital angular momentum quantum number |
l = 0, 1, 2, ... |
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magnetic quantum number |
ml = l, l-1, ... -l |
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operators do not commute |
therefore cannot specify more than one component of momentum (square of magnitudes does commute) |
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n |
principle quantum number |
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ms |
electron can be either +1/2 or -1/2 |
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shells and subshells (n) |
n = 1(K), 2 (L), 3(M), 4 (N) |
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shells and subshells (l) |
l = 0(s), 1(p), 2(d), 3(f), 4(g), 5(h), 6(i) |
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electron closer to the nucleus |
low average potential energy |
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electron further away from the nucleus |
higher average potential energy |
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Radial distribution function |
probability electron will be found between inner and outer surfaces of shell P(r) |
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p orbitals |
px = xf(r), py = yf(r), pz = zf(r) |
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d orbitals |
dxy = xyf(r), dyz = yzf(r), dxz = xzf(r), dx2y2 = 1/2(x2-y2)f(r), dz2 = (1/2sqrt3)(3z2-r2)f(r) |
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electron undregoing transition |
ejects ecess energy as a photon of electromagnetic radiation with frequency v |
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conservation of total angular momentum |
change in angular momentum of the electron must compensate for the angular momentum of the photon |
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selection rules for angular momentum |
(delta)l = +/- 1, (delta)ml = 0, +/- 1 |
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orbital approximation |
approximate one electron to occupy just one orbital so combination is the product of all wavefunctions |
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pauli exclusion principle |
no more than 2 electrons may occupy any given orbital, and if two do occupy the same orbital their spins must be paired (antiparallel) |
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penetration |
likelyhood that an electron will be found closer to the nucleus (will feel nuclear charge more strongly) |
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less penetration |
more sheilding |
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more sheilding |
less penetration |
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positive electron affinity |
energy is released when the electron attaches to the atom
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negative electron affinity |
energy is absorbed when the electron attaches to the atom |
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doppler broadening |
spectrum is sharpest when sample is colder because increasing temperature increases speed of atom which distorts line width |
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lifetime broadening |
energy levels blur over time which leads to natural line width of transition |
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quantum defects |
accounts for differences in binding energies for the outermost electrons |
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singlet |
paired spin arrangement (antiparallel) that results in momentum cancelation |
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triplet |
unpaired spin arrangement (parallel) that results in no momentum cancellation |
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spin orbit coupling |
interaction between magnetic moment from electrons spin and magnetic moment from orbital momentum |
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high total momentum |
when spin and orbit are parallel |
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low total momentum |
when spin and orbit are antiparallel |
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fine structure |
splitting of spectral lines based on the two j values (arising from spin orbit coupling) |
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total orbital angular momentum (L) |
L = 0(S), 1(P), 2(D), 3(F), 4(G), 5(H), 6(I) L = l1 + l2, l1+l2-1, ... abs(l1-l2) |
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when coupling more than 2 |
couple the 1st two then couple the next and so on, keeping record of how many terms you get |
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multiplicity (S) |
S = s1 + s2, s1 + s2 - 1,... abs(s1 - s2) multiplicity is then 2S + 1 |
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Total angular momentum (J) |
J = L + S, L + S - 1,... abs(L - S) |
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notation of orbital descriptions |
left: multiplicity (S) center: total orbital angular momentum (L) right: total angular momentum (J) |
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selection rules for orbital transitions |
(delta)S: 0 (delta)L: 0, +/- 1 (delta)l: +/- 1 (delta)J: 0, +/- 1 |
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born oppenheimer approximation |
the nuclei is so much heavier than the electron that it may be treated as stationary and only the electron moves |
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valence bond theory |
a bond forms when an electron in an atomic orbital on one atom pairs its spin with that of an electron in an atomic orbital on another atom |
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sigma bond |
electron pairing between two head-on facing orbitals |
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pi bond |
electron pairing between electrons in two side-by-side orbitals |
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promotion |
excitation of an electron to an orbital of higher energy (imaginary concept but useful) |
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sp3 hybrid orbitals |
h1 = s + px + py + pz; h2 = s - px - py + pz h3 = s - px + py - pz; h4 = s + px - py - pz |
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sp2 hybrid orbitals |
h1 = s + 2^1/2 py h2 = s + (3/2)^1/2px - (1/2)^1/2py h3 = s - (3/2)^1/2px - (1/2)^1/2py |
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sp hybrid orbitals |
h1 = s + pz; h2 = s - pz |
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distribution of an electron in a molecule |
psi squared |
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bonding molecular orbital |
wavefuncitons constructively interfere |
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antibonding molecular orbital |
wavefunctions destructively interfere |
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orbital filling rules |
1. electrons are added to the lowest energy orbitals 2 at a time 2. if several degenerate orbitals are available, the electrons are added singly before double occupancy 3. electrons in multiple degenerate orbitals must have parallel spins |
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zero net overlap |
bonding and antibonding orbitals cancel each other out |
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going left to right across a period |
orbital energies get lower |
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N2 and left on the periodic table orbitals |
double < single < double* < single* |
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O2 and right on the periodic table orbitals |
single < double < double* < single* |
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bond order |
1/2(bonding orbital e-'s - antibonding orbital e-'s) |
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greater bond order |
shorter bond, greater strength, higher dissociation energy |
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lesser bond order |
longer bond, lesser strength, lower dissociation energy |
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polar bond |
atomic orbital with the lowest energy makes the greater contribution to the molecular orbital |
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variation principle |
if an arbitrary wavefunction is used to calculate energy, the value calculated is never less than the true energy |
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secular equations |
have a solution if the determinant of the coefficients is zero a - E B - ES B - ES a - E |
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strongest bonding and antibonding effects |
obtained when the two contributing orbitals have similar energies |
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Huckel Approximation |
ignores overlap and interactions between atoms that are not neighbors - all overlap integrals set equal to zero - all resonance integrals between non-neighbors = 0 - all remaining resonance integrals = B - solved by finding a translation that makes H diagonal |
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Hartee Fock Equations |
Write down many electron wavefunctions as a product of one electron wavefucntions |
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hartee fock (f) expresses |
- kinetic energy of electron
- potential energy of interaction between other electrons - repulsive interactions betwen electrons - effects of spin correlations between electrons |
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self consistent field |
guessing an initial wavefunction then solving, plugging in new wavefunction, and solving again, repeating until energies and wavefunctions no longer change |
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semi empirical method |
integrals estimated by examining spectroscopic data or physical properties and setting certain integrals = 0 |
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ab initio method |
attempt is made to calculate all of the integrals in the Fock and overlap matrecies |
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# of integrals |
4th power of number of atomic orbitals |
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gaussian type orbitals |
use e^-&r2 to approximate atomic orbitals because the product of two gaussians is another gaussian |
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density funcitonal theory |
energy of a molecule is a function of electron density E[P] and density is a funciton of position P(r) and is solved self consistently |
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isodensity surface |
constant total electron density |
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solvent accesible surface |
sphere of solvent rolls accross molecule to map surface |
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electrostatic potential surface |
subtract charge due to electron density from charge due to nuclei (red negative blue positive) -can be used to identify electron poor areas subject to nucleophilic attack |
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decreased LUMO energy |
increases ability of molecule to accept an electron into the LUMO therefor increasing the standard potential |
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smaller HOMO LUMO gap |
wavelength of transition increases |
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emission spectroscopy |
transition from high energy E1 to lower energy E2 emits the excess energy as a photon |
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raman radiation (stokes) |
incident photons that collide and lose energy |
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raman radiation (anitstokes) |
incident photons that collide and gain energy |
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raman radiation (rayleigh) |
incident photons that collide with no change in energy |
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vibrational transition spectrum |
infrared radiation |
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rotational transition spectrum |
microwave radiation |
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rotational (microwave) transitions |
lower frequency and smaller linewidth than vibrational |
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vibrational (infrared) transitions |
higher frequency and larger linewidth than rotational |
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absorption and emission |
in order for a molecule to absorb/eject a photon of frequency v the molecule must have a dipole oscillating at the same frequency, v |
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dipolar |
charge distribution changes so a change in polarity |
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permanent electric dipole |
gives rise to a rotational spectrum |
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rigid rotor |
body that does not distort under the stress of rotation (treat molecules as rigid rotors unless otherwise specified) |
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moment of inertia |
mass of each atom multiplied by the square of its distance from the rotational axis (each axis has its own moment of inertia |
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linear rotors |
- 2 equal moments of inertia and a 3rd = 0 - degeneracy (2J + 1) - F(J) = BJ(J + 1) - only occurs around an axis perpendicular to the line of the atoms and has zero angular momentum |
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spherical rotors |
- 3 equal moments of inertia - degeneracy (2J + 1)^2 - F(J) = BJ(J+1) - large molecules have closely spaced rotational energy levels because rotational constant is inversely proportional to momentum |
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symmetrical rotors |
- 2 equal moments of inertia and a 3rd not = 0 - oblate: parallel I > perpendicular I - prolate: parallel I < perpendicular I - degeneracy 2(2J + 1) for k not = 0 and (2J + 1) for k = 0 - F(J,K) = BJ(J + 1) + (A - B)K^2 |
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K = 0 |
no component of angular momentum around the principle axis so energy depends on perpendicular momentum |
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K = +J/-J |
angular momentum arises from rotation about principle axis so energy depends on parallel momentum |
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starck effect |
splitting of states due to an electric field (only effective for volatiles/ vaporized substances) |
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centrifugal distortion |
stretching of bond lengths and increases moment of inertia, bringing energy levels closer together |
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purely rotational spectrums |
polar molecules |
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Because B values for small molecules are small |
rotational transitions occur in the microwave region |
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rotationally inactive |
linear molecules and homonuclear diatomics |
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rotation selection rules |
(delta)J = +/- 1 (delta)Mj = 0, +/- 1 J + 1 is an absorption, J - 1 is an emission |
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strongly polar molecules |
intense rotational lines |
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spacing of rotational spectrum |
series of lines seperated by 2B |
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isotropically polaraizeable |
same distortion is induced no matter the direction of the applied field (atoms, spherical rotors) |
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anisotropically polarizeable |
different distortion is induced in different electric field directions (nonspherical rotors) - raman transitions occur only for anisotropically polarizeable molecules |
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rotational raman transition selection rules |
linear rotors: (delta)J = 0, +/- 2 symmetric rotors: (delta)J = 0, +/- 1, +/- 2 (delta)K = 0 |
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J increases |
wavenumber ,v, decreases |
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stokes lines (J + 2) |
lower frequency (lose energy) and have displacements 6B, 10B, 14B from vi when J = 0, 1, 2 |
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antistokes lines (J - 2) |
higher frequency (gain energy) and have displacements 6B, 10B, 14B from vi when J = 2, 3, 4 |
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when two identical bosons exchange |
wavefunction must not change |
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orthohydrogen |
parallel nuclear spins (J not = 0) |
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parahydrogen |
paired (antiparallel) nuclear spins (J = 0) |
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greater force constant |
steeper the walls of the potential well and the stiffer the bond |
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graphical representation of molecular vibrational potential energy |
parabola which gets steeper as the force constant increases |
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effective mass |
(product of masses)/(sum of masses) - measures mass that is moved during vibration |
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vibrational gross selection rules |
- the electric dipole moment of a molecule must change when the atoms are displaced relative to one another - does not need a permanent electric dipole |
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infrared inactive |
stretching of homonuclear diatomics |
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vibrational specific selection rule |
(delta)v = +/- 1 - v + 1 is absorption, v - 1 is emission |
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fundemental transition |
transition 1 <-- 0 is dominant spectral transition |
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anharmocity |
at high vibrational excitations the swing of the molecule breaks from the parabolic approximation - all values of v are technically allowed but transitions of (delta)v > 1 are weak |
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convergence of energy levels |
energy levels converge at high quantum numbers due to the anharmocity Xe |
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birge sponer plot |
graphical technique to determine the dissociation energy (area under curve) of a bond - linear extrapolation so overestimate) |
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vibration - rotation spectra |
vibrational transitions either accelerate or retard molecular rotation so each vibrational transition is accompanied by a rotational transition |
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spectral branches |
rotational quantum number j changes by +/- 1 during a vibrational transition of a diatomic molecule |
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P branch |
all transitions where v --> v + 1 and (delta)J = -1 Vp = V - 2BJ |
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Q branch |
all transitions where v --> v + 1 and (delta)J = 0 Vq = V |
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R branch |
all transitions where v --> v + 1 and (delta)J = +1 Vr = V + 2B(J + 1) |
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Relative Frequencies |
P < Q < R |
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combination differences |
rotational constant of the vibrationally excited state, B1, is different from that of the ground vibrational state, B0 |
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vibrational raman spectra gross selection rule |
polarizibility should change as the molecule vibrates (both homonuclear and heteronuclear diatomics are raman active) |
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O branch |
all transitions where v --> v +1 and (delta)J = -2 Vo = Vi - V - 2B + 4BJ |
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Q branch |
all transitions where v --> v + 1 and (delta)J = 0 Vq = Vi - V |
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S branch |
all transitions where v --> v + 1 and (delta)J = +2 Vs = Vi - V - 6B - 4BJ |
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relative raman frequencies |
S < Q < O |
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Normal Mode |
an independent, synchronous motion of atoms or groups of atoms that can be excited without leading to the excitation of any other normal mode and without involving translation or rotation of the molecule |
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linear normal modes |
3N - 5 (N = number of atoms) |
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nonlinear normal modes |
3N - 6 (N = number of atoms) |
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bending vs. stretching frequencies |
bending frequency < stretching frequency |
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asymmetric stretch |
infrared active with parallel band (has no Q branch) |
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bending modes |
infrared active with perpendicular band (has Q branch) |
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force field |
set of all force constants corresponding to all displacements of the atoms |
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tumbling |
random changing of rotational state frequency (liquids) |
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beer - lambert law |
transmitted intensity varies with length L of the sample and the molar concentration [J] of the absorbing species |
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molar absorbancy coefficient (extinction) |
greater when absorption is more intense |
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transmittance (T) |
T = I/Io |
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absorbance (A) |
A = log(Io/I) |
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integrated absorption coefficient |
sum of absorption coefficients over the entire band - for lines of similar widths, the integrated absorption coefficients are proportional to the height of the lines |
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total orbital angular momentum of all electrons (lambda) |
lambda = 0(sigma), 1(pi), 2(delta) |
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sigma orbitals (electronic) |
lambda = 0 (sigma) |
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pi orbitals (electronic) |
lambda = 0 (sigma) for paired spins, +/- 2 (delta) for parallel spins |
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multiplicity (electronic) |
2S + 1 (H2+ is 2, H2 is 1, and O2 is 3) |
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parity (electronic) |
measurement of symmetry in molecule g = +1, u = -1 g x g = g; u x u = g; g x u = u |
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parity for closed shell homonuclear |
g (must also look at inversions between planes to determine + and - sign) |
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parity for heteronuclear |
no parity |
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total angular momentum (omega, electronic) |
omega = lambda + sigma where sigma = S, S-1, S-2,... -S |
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notation of electronic state |
left: multiplicity (2S + 1) center: lambda right: omega |
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electronic selection rules |
(delta)lambda = 0, +/- 1 (delta)S = 0 (delta)sigma = 0 (delta)omega = 0, +/- 1 |
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franck condon principle |
because the nuclei are so much more massive than the electrons, an electronic transition takes place very much faster than the nuclei can respond |
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dynamic state |
nuclear wavefunction does not change during electronic transition |
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band head |
when the lines on either side (P or R) begin to converge |
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longer bond than in the ground state |
R branch begins to converge to a band head |
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shorter bond than in the ground state |
P branch begins to converge to a band head |
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ligand-field splitting parameter (deltao) |
difference in energy between (dxy, dyz, dxz) and (dx2-y2 and dz2) states - not large so in visible spectrum, responsible for many colors of d-metal complexes |
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t2g |
dxy, dxz, dyz |
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eg |
dx2-y2 and dz2 |
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charge-transfer transitions |
transfer of electrons from metal to ligand or vice versa |
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pi to pi star transition |
C=C double bond is excited to promote pi electron to pi star orbital (ultraviolet emission) |
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n to pi start transition |
C=O double bond is excited to promote oxygen electron into C=O pi start orbital |
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circular dichroism |
difference between left-circular and right-circular polarized light by a chiral molecule can be used to determine the R or S state of a molecule |
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flourescence |
the spontaneous emission of radiation occuring within a few nanoseconds after the exciting radiation is extinguished (occurs at lower intensities because of vibrational energy lost) |
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phosphoresence |
spontaneous emission persists over longer periods because of intersystem crossing which allows a transition from the singlet to triplet state |
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stimulated absorption |
transition from low to high energy driven by electromagnetic field oscillating at transition frequency - stimulated absorption and emission rates are the same for a given intensity of radiation |
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spontaneous emission |
excited state can emit photon to return to lower state spontaneously - spontaneous emission increases as the separation of upper and lower states increases |
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dissociation |
at certain energies the molecule's bonds break and the unquantized fragments appear as a continuous band above this point |
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interconversion |
radiationless conversion between multiplicities (singlet 1 to singlet 0) resulting in a blurred region between two well defined regions |
