Equations Flashcards
Draw the conversion chart.
State the Born-Oppenheimer approximation.
Ŷtot = ŶelŶnuc
Etot = Eel + Enuc
Ŷ: wavefunction
Etot: total energy
el: electron
nuc: nucleus
State the Boltzmann Law.
At thermal equilibrium, the relative population of the ith energy level is given by:
ni/n0 = gi/g0 e-∆E/kBT
g: degeneracy
∆E: energy gap
n/n: ratio of the population at ni compared to n0
State the equation for the energies of hydrogenic orbitals.
E = - (Z2RX)/n2
Z: nuclear charge
n: principal quantum number
RX: Rydberg constant for tht atom or ion
State the Rydberg constant.
RX/cm-1 = (µe4)/8h3ce02
*Value depends on the mass of the nucleus.
µ: reduced mass
State the equation for calculating ionisation energy.
∆E = Z2RX (1/n12 - 1/n22)
State the generic atomic term symbol + annotate.
2S+1LJ
2S + 1: Spin multiplicity, where S is the total spin quantum number.
L: total orbital angular momentum quantum number.
J: total angular momentum quantum number (how L and S are coupled).
What is the atomic term symbol for closed-shell atoms and ions?
1S0
e.g. He, Xe, Li+, Mg2+
What is the term symbol for alkali metal atoms?
2S1/2
i.e. outer shell is Xs1
What is the atomic term symbol for excited alkali metal atoms?
2P3/2 and 2P1/2
State the selection rules for what we see in a spectrum.
- ∆J = 0, +/- 1 (not 0 ⇔ 0).
- ∆S = 0
State the Pauli exclusion principle.
No two electrons in an atom/ion can have all quantum numbers the same.
What are Hund’s Rules for energy ordering?
When can we apply them?
- The term with largest S is lowest in energy.
- For a given S, the term with the largest L is lowest in energy.
- For a term with several levels:
- If the sub-shell is less than half full, the lowest J level is lowest in energy.
- If the sub-shell is more than half full, the highest J level is lowest in energy.
⇒Only applies to ground state atoms.
What is the atomic term symbol for a group state N atom?
4S3/2
What is the atomic term symbol for a ground state O atom?
3P0
State the equation for the moment of inertia (I).
I = Σimiri2
mi: mass of the ith particle
ri: its perpendicular distance from the axis
State the equation for the moment of interia for diatomic molecules.
I = µR2
µ: m1m2 ÷ m1 x m2
I: rotational equivalent of mass
State the eigenvalues for rotational energy levels.
F (J) (=EJ) = BJ (J+1)
J: rotational quantum number
B: rotational constant
State the equation for rotational constant B in Hz and cm-1.
- B (Hz) = h ÷ 8π2I
- B (cm-1) = h ÷ 8π2Ic
⇒ B in cm-1 is most common, with c being in units of cm-1
What are the rotational selection rules? (3)
- Heterodiatomics must have a permanent dipole moment to exhibit a pure rotational spectrum.
- Transitions occur for ∆J = +/- 1
- Transitions observed at 2B(J+1).
State the equation of B used to extract bond lengths.
B = 1 ÷ µR2
What are the other points important to rotational spectroscopy? (3)
- R is isotope independent, is determined by electronic structure.
- B is isotope dependent - spectra will have different spacings.
- In the absense of an applied magnetic field, each K levels exhibits (2J+1)-fold degeneracy arising from projection quantum number MJ, levels are split when a magnetic field is applied.
What is the equation for the most populated rotational level?
Jmax = (-/kBT ÷ 2B) - 1/2
-/: square root
State the equation for rotational terms + why the correction term is required.
F(J) = BJ(J+1) - D[J(J+1)]2
D: centrifugal distortion constant (cm-1)
The correction term is required because molecules are not rigid rotors, bonds stretch slightly during rotation.
State the equation for bond strength from vibrational frequency with units.
we2 = 4B3 ÷ D
in (cm-1)2
State the B values for H2, HCl and CO with the standard equation for B.
B = h2 ÷ 2I
where h = h ÷ 2π
- H2: 60.85 cm-1
- HCl: 10.59 cm-1
- CO: 1.93 cm-1
h in equation is the h with the cross stick.
State Hooke’s Law.
F = -kFx
- kF: force constant (Nm-1)
- x: displacement from equilibrium
Give the equation for classical harmonic frequency of oscillation.
vvib = (1 ÷ 2π) x (-/kF ÷ µ)
where µ = (m1m2) ÷ (m1 + m2)
Give the 2 equations for harmonic vibrational energy levels.
- we = (vvib ÷ c) = (1 ÷ 2πc) x (-/kF ÷ µ)
- G(v) = (v + 1/2) x we
v: vibrational quantum number (0, 1, 2…)
we: vibrational constant (cm-1)
kF: force constant (Nm-1)
µ: reduced mass
c: speed of light in cm-1
State the generic molecular term symbol.
What is the molecular term symbol for N2 ground state?
1Σg+
What is the molecular term symbol for NO ground state?
2Π1/2
What is the molecular term symbol for O2 ground state?
3Σg-
State the general principles for electronic state labels (3).
- Ground electronic state - label X.
- Higher electronic states with the same spin multiplicity as the ground state are labelled A, B, C…
- Higher electronic states with a different spin multiplicity are labelled a, b, c…
What is the assumption of the Frank-Codon principle?
Electronic transitions take place on such a short timescale that the nuclei remain frozen (R unchanged) during the transition.
What does the Frank-Codon principle say about vibrational structure?
The overlap of the vibrational wavefunctions is governed by the nature of the electronic states involved.
What is the equation for the dissociation limit in electronic spectroscopy?
G (vmax) = De = (we2) ÷ (4wexe)
What is the equation for the lower state dissociation energy?
D0” = v0 + D0’ - ∆vatomic
v: v with squiggle over the top
