Quantum (Haehner) Flashcards
equation for frequency?
v = dE/h
what energies can v possess (looking at particle character of EM radiation)
0, hv, 2hv…
what is p in the de broglie equation?
linear momentum
what is the de broglie equation?
[lambda] = h/p
how do diffraction patters arise?
from many waves interfering constructively and destructively
what is h bar?
h/2pi - planck’s constant which takes angular momentum into consideration
general schrodinger equation?
HY=EY
what is the schrodinger equation for a particle in a 1D box?
-(hbar^2/2m)*(d^2[psi]/dx^2) + V(x)[psi] = E[psi]
give some examples of observables
momentum, energy, position
what is the position operator?
x[bar]=x*…
what is the momentum operator?
p[bar]x = h(bar)/i*d/dx…
What is the Born interpretation?
gives the probability of finding a particle between x and xdx
what is the equation for probability density?
|[psi]x|^2
what is the expectation value?
the average value of a large number of observations
3 steps to forming a schrodinger equation?
1) write down the total energy in terms of kinetic and potential
2) replace momentum and position in the energy equation with appropriate operators
3) then solve the SE to obtain eigenstates and eigenvalues.
What is a polar coordinate?
A 2D system of coordinates where all points are referenced off a central point by distance and angle
What symmetry does potential energy in the schrodinger equation have?
centrosymmetric
What can polar coordinates do to the schrodinger equation?
split it into radial and angular components
What are spherical harmonics?
the normalised wave functions of the angle dependent part of the schrodinger equation.
What is a hydrogen atom?
an atom or ion with only 1 electron
What is the equation for the Coulomb interaction of an electron with a nucleus?
V(r) = - Ze^2 / 4[pi][eo]r
What is the equation for the eigenvalues for the hydrogen atom?
En = -Z^2[mu]e^4 / 32[pi]^2[eo]^2h(bar)^2n^2
What is a boundary surface?
the area in which you are most likely to find an electron in an orbital - the shape of the orbital
What is the Born-Oppenheimer approximation?
oscillations from e- arrangement to equilibrium position moves faster than nuclei –> so the wavefunction can be separated into smaller components
How does the potential energy curve differ upon observation?
observed has a deeper well
What symmetry does a diatomic wave function need to have?
antisymmetric
in an antisymmetric function, when the sign of psi changes, what happens to the sign of fermions and bosons?
fermions change sign, bosons retain sign
What does valence band theory not account for?
lone pairs (when calculating angles) or hybridisation
What is the schrodinger for the hydrogen molecule ion H2+?
-h(bar)/2me (triangle)^2 + V
If you solve the schrodinger for a molecule what can you make?
attain 1 e- wave functions called MOs
How do you normalize a wavefunction?
Put N, a normalisation factor in front, which gives the probability of finding the particle in space equal to one
What is it called when an e- can be found in an atomic orbital belonging to atoms A and B in a wavefunction?
an LCAO
What does the potential energy curve describe?
the accumulation of e- density in and out of the internuclear region.
What do two orbitals need to have in common to form an orbital
symmetry
In which diatomic elements does sp mixing occur?
How do you measure bond order?
1/2(number of e- in bonding orbitals - number of e- in anti bonding orbitals)
Why is oxygen paramagnetic?
its a triplet state
What is a coefficient? (c)
it tells you how much each atomic orbital contributes to the overall MO
What is the variation principle?
If an arbitrary wavefunction is used to calculate the energy, the value calculated is never less than the true value.
It allows you to find optimum coefficients and energies.
How would you build an optimum molecular orbital?
start with a trial wavefunction then vary the coefficients until the lowest energy is achieved.
What is [alpha]? in the energy expression
the coulomb integral
what is [beta]? in the energy expression
the resonance integral
what is S? in the energy expression
the overlap integral
What is the coulomb integral?
the sum of the potential energy an electron feels from interacting with the charge distribution of a different electron
How do you find the minimum value for energy?
set dE/dc=0 (then find roots to obtain values for coefficients)
Once you know the minimum value for energy, what other things can you find out?
coefficients and total wavefunction (or MO)
In a heteronuclear diatomic, when the energy difference is large, how do the energies of the MOs differ from AOs?
only differ slightly - implies that bonding and anti bonding effects are small.
What is the Hückel approximation?
that sigma and pi bonds should be treated differently - sigma provides shape of the molecule.
How does a computational equation work?
Looking for a minimum energy value
find derivative of energy; use variation principle; find non trivial solution determinant=/=0; polynomial then use E values for find coefficients.
What is the equation for moments of inertia?
I = sum of mr^2
What are the four types of rigid rotors? (and what are their moments?)
linear (1 moment), spherical (3 equal moments), symmetric (two equal moments), asymmetric (3 different moments)
What is the equation for the classical energy of a body rotating about a?
Ea = 1/2 Ia*wa^2
what is the equation for angular momentum around axis a?
Ja = Ia*wa
What is the energy of a rotational energy level (including the angular mometum)?
E = Ja^2/2Ia + Jb^2/2Ib + Jc^2/2Ic
What is the quantum expression for energy of a spherical rotor?
Ej = J(J+1) * h(bar)^2/2I
What is B?
rotational constant = h(bar)/4pic*I
What is the rotational wavenumber
F(J) = BJ(J+1)
what is the separation between energy levels in a spherical or linear rotor?
2B –> 4B –> 6B etc…(increases with J)
what is quantum number Mj?
the angular momentum fo a molecule on an externally fixed axis.
What values can Mj have?
quantised to real numbers : +/- 1, +/-2, …, +/-J
What is centrifugal distortion?
the effect of rotation on a molecule. It distorts the molecule, opening out bond angles and stretching bonds slightly.
What is the fixed equation for F(J) including centrifugal distortion?
F(J) = BJ(J+1) - DjJ^2(J+1)^2
What conditions would you need to be able to observe a pure rotation spectrum?
a permanent electric dipole moment
What are the rotational selection rules?
dJ = +/-1 and dMj= 0, +/-1
If a photon is absorbed by a molecule, what happens to the angular momentum of the combined system?
stays the same
If a molecule absorbs a photon and they are rotating in the same direction, what happens to J?
it increases by 1 (hv) –> the energy is absorbed
When selection rules are applied for a rigid or symmetric rotor, what happens to the allowed wavenumber for J+1
v(bar) = 2B(J+1)
What does the boltzmann distribution say about the population of the rotational levels?
decays exponentially with J
What is the equation for the boltzmann distribution?
Nj = (2J+1) exp(-hcBJ(J+1)/kT)
if degeneracy of level J is 2J+1 for linear rotor
What happens to a molecule if an electric field is applied?
distorted - the distorted molecule acquires a contribution to its dipole moment.
what is anisotropic polarisability?
when polarisability differs depending on when the field is applied wither parallel or perpendicular to the molecular axis
What are the selection rules for rotational raman?
dJ = +/-2 (0 for linear rotors)
Describe and explain raman spectra
shows energy lines for what is allowed from the selection rules. Rayleigh line is when dJ = 0, stokes (to the left) are for when dJ = +2 and antistokes (to the right) are for when dJ = -2.
Under what conditions does a particle undergo harmonic motion?
F=-kx
How is force related to potential energy?
F=-dV/dx
What is the parabolic potential energy of an harmonic oscillator?
V = 1/2 k x^2
What is the schrodinger for an harmonic oscillator?
-h(bar)^2/2[mu] *d2[psi]/dx2 + 1/2 kx^2 [psi] = E[psi]
what is [mu]?
reduced mass: m1*m2/m1+m2
How does quantisation arise?
from the boundary conditions imposed by the potential: [psi]=0 when x=+/- infinity
What is the energy derived from the schrodinger for a harmonic oscillator?
En = (n+1/2) h(bar)w (where w=sqrt(k/[mu])
What is the separation between energy levels in a harmonic oscillator?
h(bar)w
what is the zero-point energy for a harmonic oscillator?
E0=1/2 h(bar)w
what is a translational degree of freedom?
molecule as a whole moves place to place
what is a rotational degree of freedom?
the molecule rotates about a centre of mass
what is a vibrational degree of freedom? and how many vibrational modes does a molecule have?
Molecules bonds stretch and contract and bend so the structure oscillates around the most stable configuration (sometimes called internal motion)
linear: 3N-5
non-linear: 3N-6
What is the gross selection rule for the absorption of radiation by a molecular vibration?
the electric dipole moment of the molecule must change when the atoms are displaced relative to one another.
What is the specific vibrational selection rule?
dn = +/- 1 (where + = absorption and - = emission)
what are the corresponding frequencies to the vibrational selection rule?
v = dE/h
how do transitions between vibrational energy levels occur?
Stimulated by or emit infrared radiation
what is anharmonicity? What is the name of the curve that applies to this?
V=(1-exp(-a(R-Re))^2
the Morse potential energy curve. - just shows that V reaches a maximum - so the parabola trails off at Vmax
Describe a vibration-rotation spectrum
selection rule dJ = +/- 1 (also = 0 if the molecule possesses angular momentum around its principle axis)
P branch when dJ = +1
what is the selection rule for vibrational Raman
dn = +/-1 (polarisability should change as the molecule vibrates)
describe vibration-rotation raman
similar to rotational raman - selection rules dJ = 0,+/-2
O branch = -2
Q branch = 0
S branch = +2
branched structure combined simultaneous rotational transitions and vibrational excitations
what is the equation for an electric dipole moment?
[mu] = q R ( where q is the molecule, and R is the distance between moments)
what is the difference between [alpha] and [alpha]’
[alpha] is the polarizability and [alpha]’ is the polarizability volume
What is the interaction energy of charge and dipole?
V = -[mu1]q2 / 4[pi][eo]r^2 ([mu] is dipole q is charge)
potential energy between two charges in a vacuum?
V = q1q2/4[pi][eo]r
What are the combinations to the diminishing field of an electric dipole distance?
charge - potentials of the charges decrease and their charges appear to merge.
what is the interaction energy of two dipoles?
the sum of the repulsions of like charges and the attractions of opposite charges
what is the equation for collinear arrangement of dipoles?
V = - [mu]1[mu]2 / 2[pi][eo]r^3
what is the equation for parallel arrangement of electric dipoles?
V = [mu]1[mu]2 / 4[pi][eo]r^3
what are the two equations for the electric field generated by a dipole?
E(monopole) = q/4[pi][eo]r^2 E(dipole) = [mu]/2[pi][eo]r^3
What is the potential energy of interaction between two polar molecules when their dipoles are parallel?
V = [mu]1[mu]2 1-3cos^2(theta) / 4[pi][eo]r^3
in a fluid of freely rotating molecules what is the interaction between the dipoles?
averages to zero
What is the equation for the marginally favoured non-zero interaction
= -C/r^6
Which of the attractive interactions of a/b/c are temperature dependent? look at ionization energies? look at polarizablility volume?:
a) dipole-dipole, b) dipole-induced-dipole, c) induced-dipole-induced-dipole
a
c
b
How does a hydrogen bond form?
the attractive interaction between two species from a link of the for AH—B where A and B are highly negative and B possesses a lone pair of electrons
How is hydrogen bond stability explained by MO diagrams?
Three MOS can be formed by the A, H and B orbitals.
Only the two lower energy orbitals are occupied –> so there may be a net lowering of energy compared to the separate species
What is the total attractive interaction?
the sum of the three van der waals interactions –> V = C^6/r^6
What are the dependencies and limitations of C6
it depends on the identities of the molecules
limitations: only dipolar interactions accounted for; assumed molecules can rotate reasonably freely; equation relates to the interaction of pairs of molecules.
What is the name (and equation) of the potential energy graph that takes into account the repulsive and total interactions?
Lennard Jones Potential.
V = 43{(ro/r)^12 - (ro/r06}
Where does the minimum of the lennard Jones potential energy well occur?
2^(1/6) ro
When you add a non-polar molecule into a polar solvent, is energy required or released?
required - as dG>0
When you add a non-polar molecule into a polar solvent, is heat produced or absorbed?
produced
When you add a non-polar molecule into a polar solvent, what happens to entropy?
decreases (to account for positive G)
what is it called when a substance has a positive dG transfer from non-polar to polar solvent?
hydrophobic
What happens to dH and dS when the chain length of a hydrocarbon in a polar solvent is increased?
dH more +
dS more -
(so molecules become more hydrophobic)
what is the solvent cage that water makes called?
clathrate cage
What happens to the entropy when water solvent cages are formed?
-dS
How does a spontaneous association of hydrophobic molecules in polar solvents occur?
many solute molecules cluster together which means more solvent molecules are free to move. The net effect of this is an increase in entropy of the system. This increase in entropy is sufficient for the spontaneous association. –> this is the origin of hydrophobic interaction - stabilises large groups of hydrophobic molecules.
at what concentration are micelles made?
above the critical micelle concentration
what forces influence protein structure?
hydrophobicity/hydrophilicity; H bonding; Van der Waals/dispersion forces; electrostatics
how does temperature affect electrical conductivity of a metallic conductor?
T up conductivity down
how does temperature affect electrical conductivity of a semiconductor?
T up conductivity up
what is a superconductor?
it displays no electrical resistance
What are the two models of the distribution of electron which determine electrical properties
nearly-free electron approximation
tight-binding approximation
describe the nearly free-electron model
valence electrons are assumed to be trapped in a box with periodic potential
describe the tight bonding approximation
valence electrons are assumed to occupy MOs delocalised throughout the solid
How is valence band theory described by the tight-binding approximation?
N MOs are overlapped to fill in a finite range of energies.
what is the solution of the determinant to describe band width?
Ek = [alpha] + 2[beta]cos(kpi/N+1)
what is an s band // a p band
an s band is the overlap of infinite s orbitals to form a valence band (p is overlap of p orbitals)
How does a band gap occur?
if the p orbitals are sufficiently high in energy to not overlap with the s band. (if not the bands can be contiguous or overlap)
what is the band width En-E1 –> ?? as N —> infinity?
-4beta
what are the bonding and anti bonding properties of the orbitals?
k=1 is fully bonding, k=N is fully antibonding
What is the fermi level?
a thermodynamic equilibrium - when the probability of this band being occupied is 50%
what is the fermi dirac distribution?
it gives the population of the levels at temperature T:
P = 1/exp(E-[mu]/kt)+1
what two principles does the fermi-dirac distribution take into consideration?
boltzmann distribution taking into account the effect of the pauli principle
why does conductivity of a metallic solid decrease with increasing temperature?
despite the extra excitations, the increase in temperature causes more vigorous thermal motion which causes collisions which scatter the electrons so they are less efficient at transporting charge.
how do you distinguish between insulators and semiconductors?
at T=0 when all atoms provide 2 e-, 2N e- are present. when T increases e- can occupy the upper conduction bad and the solid becomes a semiconductor
what is the difference between intrinsic and extrinsic semi-conductors?
intrinsic is a pure material or a combination of different elements - the electrons must jump between the two standard conduction bands
extrinsic is when charge carriers are present as a result of the replacement of some atoms by dopant molecules
what is the difference between p and n type semi-conductors?
they are both extrinsic.
p type has a dopant with fewer e- than the host - so a small valence band aids conduction
n type has a dopant with more e- than the host - so can supply extra e- to the conduction band.
what is a p-n junction
a battery powered by p and n type dopants - where a forward bias fills holes with extra electrons to conduct electricity and a reverse bias lowers current.
