quantum mechanics part 2

what phase is rotational spectroscopy usually done in?

why?

gas phase

solids inhibit rotation, and in a liquid, too many collisions cause rapid energy transfer

what does rotational spectrum look like?

absorptions are regularly spaced

intensity of abs rises, reaches a max then falls

what is the rigid rotor model?

what does it assume?

assumes fixed bond length between two atoms in molecule

what does the moment of inertia tell us?

how easy it is to spin the diatomic molecule about its centre of mass

how does speed of rotation change for heavier masses of molecules and longer bond length?

rotor spins more slowly for a given magnitude of push

when calculating reduced mass ,what should you remember?

multiply by atomic mass unit to calculate in kg

what is J?

rotational quantum number

how to calculate E_J?

 

E_J = hBJ(J+1) (B has units s^-1)

what are the conditions needed for rotational spectroscopy?

the molecule must have a permanent electric dipole

why must a molecule have a permanent electric dipole to have a rotational spectrum?

the oscillating electric dipole during rotation can interact with oscillating electric and magnetic fields of the EM radiation at the same frequency

which molecules cannot be probed for a rotational spectrum?

homonuclear molecules have no permanent electric dipole

what is the specific selection rule for rotational spectroscopy?

transitions can only take place between adjacent energy levels

ΔJ = ± 1

what is the equation to calculate the ratio of populations in energy levels?

what is the wavefunction for J=0?

spherical symmetric

• similar to s orbital

how many wave functions for J=1?

what are they like?

three wave functions

• similar to p orbitals of H atom

how many wavefunctions for J=2?

what are they like?

five wave functions

• similar to d orbitals of H atom

why are higher rotational energy levels more populated than lower?

higher energy levels have higher degeneracy (2J+1)

• higher potential for higher occupancy

how does population change as you go up rotational energy levels?

increases until energy gaps eventually become ≥ k_BT and then population drops off

how can you find ΔE from microwave spectrum?

find from wavenumbers of spectral peaks

• peak intensities means energy level occupancies can be calculated

• assigning peaks to ELs means can find degeneracy

• temperature is only unknown

what is centrifugal distortion?

rotational energy increases, molecule spins faster, bond length increases

what is the equation for E_J accounting for centrifugal distortion?

E_J = hBJ(J+1) - DJ²(J+1)²

how does the gaps between ELs change as rotational energy increases for non-rigid rotors?

as rotational energy increases, the gaps between the energy levels get smaller

how does vibrational spectroscopy probe molecular motions?

due to movement of atoms in individual chemical bonds

how are chemical bonds modelled for IR spec?

what is the equation?

bonds modelled as mechanical springs

F = -kx

k is bond force constant

x is displacement

F is restoring force

how to calculate displacement from equilibrium position (bond length)?

x = r - r₀

how does the force work when the atoms are pushed farther apart? if displacement is inwards (pushed together?)

the more you push the atoms further apart, the greater the force pushing back together

if pushed inwards, the force is pushing them back out

what is the graph for potential energy of bond vibrations?

how to calculate force?

force is negative gradient

what does vibrational frequency depend on (2)?

the mass of the atoms (heavier atoms vibrate at a slower frequency)

the bond force constant (stiffer springs vibrate at higher frequency)

what is the equation for E_V?

E_V = (v + ½ )hv

what two rules pertain to infrared spectroscopy?

gross selection rule (must be change in dipole moment)

specific selection rule Δv=±1.

what does +1 and -1 values of Δv mean (vibrational spec)?

+1 is adsorption

-1 is emission

how to calculate dipole moment of vibration of bond?

charge x distance (r)

how does dipole moment change as bond stretches?

r increases so dipole moment increases

when is there no change in dipole moment for CO₂?

when is there?

for symmetric (vibrate both inwards/outwards), there is no change

for asymmetric there is a change as the electrons are pulled towards O differently

• dipole moment for bending modes

why do you only expect one peak for diatomic molecules in a vibrational spectrum?

energy levels are evenly spaced

only one frequency that a diatomic molecule absorbs at

what is the bond force constant a measure of?

stiffness and therefore strength of the bond

why is k independent of isotopes?

doesn’t change significantly by adding neutrons to nucleus

why is the shift in infrared spectra most significant for H→D isotopic substitution?

lightest atom moves much more than the heavy atom and dominates the value of the reduced mass

how is anharmonic oscillation different from harmonic oscillation?

real energy levels aren’t equally spaced

potential energy curve isn’t a parabola

if the atoms are pulled far enough apart, the bond breaks

what transitions are observed in real molecules (vibrational)? why?

what are these called? when do they occur?

selection rule is only for harmonic

transitions v = 0 → v = 2 and v = 0 → v - 3

• overtone bands (occur at twice and three times the fundamental frequency)

how many vibrational modes do linear molecules have?

3N-5 (N is number of atoms)

how many vibrational modes do non-linear molecules have?

3N-6 (N is number of atoms)

how does number of modes of vibration change with number of atoms?

more atoms means more possible modes of vibration

how many degrees of freedom do N atoms have?

3N