Physical Chemistry ACS QM

What is the order of the electromagnetic spectrum from high to low energy?

Gamma rays, x-rays, UV, visible, IR, microwaves, radio

What transitions occur in the microwave spectrum?

Rotational

What transitions occur in the IR spectrum?

Vibrational with accompanying rotational fine structure bands

What transitions occur in the UV-vis spectrum?

Electronic

In quantum mechanics, energy levels exist as multiples of what number?

Planck's constant, h

What is the ultra-violet catastrophe?

The classical mechanics equation for frequency distribution predicts that heating an object should create blackbody radiation of gamma rays of infinite intensity.

How was the ultra-violet catastrophe solved?

It was assumed that energy exists in discrete units as multiples of Planck's constant, h.

What does classical mechanics predict about the photoelectric effect?

Both the number of electrons emitted from a material and their energy should be proportional to intensity of the incident light beam

What does quantum mechanics reveal about the photoelectric effect?

Energy is proportional to frequency, and number of emitted electrons is proportional to intensity.

What types of energy do single atoms lack?

rotational and vibrational

How does increasing bond strength affect vibrational energy?

It increases it

What is the order of gaps between energy levels?

From greatest to least, electronic, vibrational, rotational, nuclear spin.

What do electronic transitions depend upon?

Electron kinetic energy, coulombic attractions and repulsions, and magnetic interactions

What is rotational energy dependent upon?

Geometry

What are the four claims of the Bohr Model?

i. Electrons orbit nuclei

ii. atoms exist as integer energy levels

iii. energy of a transition is equal to h(photon frequency)

iv. allowed orbits are given in integer multiples of h/(2pi)

What does the wavefunction contain?

Information on an object's location and momentum

What are the requirements for a wave function?

It must be finite, continuous, single valued, normalizable

What does integrating the square of the wavefunction give?

The probability of finding a particle over the integrated space

What is the position operator?

Multiply by x

What is the momentum operator?

-i(hbar)d/dx

What is the kinetic energy operator?

-[{(hbar)^2}/2m]d^2/dx^2

What is the Hamiltonian?

Gives the total energy of a system

When is an operator linear?

If operating upon the sum of the two functions is equivalent on the sum of operating upon the two individual functions

What are the boundary conditions for a particle in a box?

(0,L)

When x=0 or x=L, V is infinite

What is the energy equation for PIB?

E= (n^2h^2)/(8mL^2)

What is the wavefunction for PIB?

psi(x)= sqrt(2/L)sin(npix/L), n= 1, 2, 3, ...

What is the number of nodes for a wavefunction?

nodes = n-1

What is the normalization constant?

1 over the square root of the square of the wavefunction over all space.

When are two wavefunctions orthogonal?

When the integral of multiplying the two gives 0

When are two wavefunctions equal?

When the integral of multiplying the two gives 1

What is the expectation value, ?

The average of an observable for a wavefunction.

It is calculated by taking the integral of psi* M psi

What is for PIB?

Independent of n

What must operators be in quantum mechanics?

Linear and Hermitian

When is an operator Hermitian?

When int(fMg) = int(gMf)

What is the Heisenberg Uncertainty Principle?

it is impossible to simultaneously determine, with perfect accuracy, the momentum and the position of a particle. The product of their uncertainties must be at least their commutator

When is the commutator zero?

AB pis(x) is equivalent to BA psi(x) where A and B are operators

What is the commutator for momentum and position?

hbar/2

How does the Hamiltonian change for PIB in multiple dimensions?

The operator adds a term for each dimension

What is the energy for PIB in multiple dimensions?

The sum of the energies in each dimension

What is the wavefunction for PIB in multiple dimensions?

The product of the wavefunction in each dimension

How do the number of nodes follow the previous rule?

nodes = n-1, but nodes are dependent upon dimension

When are two states degenerate?

When they have equal energies

How does the separation between energy levels change as L increases?

Separation decreases

What is the variance in thermal energy for an individual particle?

E= (Boltzmann constant)(T)

Why does energy appear continuous between quantum states at high temperatures?

The variance in thermal energy is greater than the gaps between their energy levels

When do energy levels appear continuous?

Longer L, higher T, higher mass

What is the Boltzmann distribution?

Ni/Nj = gi/gj e^[-(Ei-Ej)/kT]

What happens as T increases?

Particles become distributed evenly across all energy levels according to degeneracy

What are the bounds for a harmonic oscillator?

(-infinity, +infinity)

Why can the bounds be infinite?

The probability goes to zero as the harmonic oscillator is stretched because the PE goes to infinity

What is the Hamiltonian for a harmonic oscillator?

-[(hbar)^2]/2md (d^2/dx^2) + .5kx^2

What is the Energy of a harmonic oscillator?

E = hbar sqrt(k/m) (n+1/2)

What is the wavefunction for a harmonic oscillator?

Needlessly complicated

Is the zero point energy for a harmonic oscillator equal to zero?

No

What is the spacing between energy levels for a harmonic oscillator?

h(frequency), even spacing

What is the symmetry of an even function?

symmetric across the y-axis

What is the symmetry of an odd function?

antisymmetric across the y-axis

What is the integral of an odd function over all space?

zero

What is the integral of an even function over all space?

double the integral from zero to infinity

What is an even function times an even function?

even

What is an odd function times an odd function?

even

What is an odd function times an even function?

odd

How many nodes are observed in a harmonic oscillator?

nodes = n

How does the range of bond lengths change with n?

Increases as n increases

What happens when total energy is less than potential energy?

Kinetic energy is negative, quantum tunneling

What is quantum tunneling for a harmonic oscillator?

The wavefunction for a distribution of bond lengths lies outside of the classically allowed region

How does quantum tunneling change with n?

It decreases as n increases

How does the wavefunction change as n increases?

The probability becomes more likely everywhere, appearing classical

What are the requirements for a molecule to absorb a photon in the microwave spectrum?

It must be able to interact with an electromagnetic field, so it must have a permanent or temporary dipole moment

What is the probability that a vibrational transition is observed proportional to?

intensity of the radiation at that wavelength, number of molecules in the absorbing or emitting state, the dipole moment which is the integral of the electric dipole operator

When is a transition forbidden?

When its dipole moment is zero

For a heteronuclear molecule, this is true unless the change in n = +- 1

When molecules aren't in the state to begin with

What is the spacing of energy levels for vibrations?

h(frequency)

What is the equation for vibrational energy?

E= h(frequency)(n +1 /2)

As a consequence, what must frequency equal?

f = (1/2pi)sqrt(k/meu)

What is meu?

meu = (m1m2)/(m1+m2)

What is k?

The force constant

What determines the number of peaks for a vibrational spectrum?

Number of energy levels

What determines the intensity of the peaks?

The number of molecules at each energy level, Boltzmann distribution and degeneracy

What determines the spacing between peaks?

The energy difference between vibrational states

What changes when treating a molecule like an anharmonic oscillator?

The change in n = +-1 is not strictly observed, so echoes are seen for other transitions, energy level spacing also decreases as the energy levels increase according to the quantum number J

What is the typical model used for anharmonic oscillators which allows bonds to break?

the Morse Potential

What coordinates are used in the particle on a ring model?

Cylindrical, (r, phi, z)

where r = (0, infinity), phi = (0, 2pi), z = (-infinity, infinity)

What is the equation for angular momentum and kinetic energy?

L=I(omega), KE=(L^2)/(2I)

What is the kinetic energy operator for this system?

-[(hbar)^2]/(2m)(laplacian)^2

What terms are allowed to vary in the particle on a ring model?

Only phi

What is the wavefunction for particle on a ring?

psi(phi) = 1/sqrt(2pi) e^(im_lphi)

What is the energy for particle on a ring?

[(m_l)^2(hbar)^2]/(2mr^2)

What is the angular momentum operator for the z direction?

-i(hbar)d/dphi (partial derivative)

What is the eigenvalue of the angular momentum z operator?

hbar(m_l),

consequently, rotational energy levels are evenly spaced for particle on a ring

What coordinates are used for modeling molecular rotation?

Spherical, (r, theta, phi)

theta = (0, pi)

What is the Hamiltonian for the rigid rotor?

-(hbar)^2/(2mr^2) (legendrian)

What is the energy for a rigid rotor?

E= (hbar)^2/(2I) l(l+1), I= ur^2

What is the shrodinger equation for the rigid rotor model?

A combination of Theta and Phi functions

What is the degeneracy for molecular rotation?

degenerate states = 2(l) +1

What is the number of nodes in the rotational wavefunction?

nodes = l

What is the total angular momentum operator?

l(l+1)(hbar)^2

What is m_l in relation to this?

The z component of l, this is why m_l cannot be greater than l

What is the observable for the Lz operator?

hbar,

as a consequence, rotational energy levels are multiples of hbar and evenly spaced

What is the energy for the rigid rotor model?

(E~)= h/(8pi^2cI) J(J+1) = (B~e)J(J+1)

What are the requirements for pure rotational transitions to be observed?

The molecule must have a permanent dipole, and if it is a heteronuclear diatomic, J = +-1