Physical Chemistry II - FInal Examination

What is classical mechanisms able to predict that quantum mechanics struggles to do?

CM may predict exact trajectories, locations, and momenta of particles at any time.

What are the three failures of classical mechanics?

1. Blackbody radiation

2. Planck distribution

3. Atomic-molecular spectra

What is a blackbody?

An idealised object that absorbs all EM radiation that strikes it. When in equilibrium, it solely emits radiation based on temperature and not its material.

What is blackbody radiation?

The EM radiation emitted by an idealised blackbody above absolute zero. The maximum intensity wavelength of blackbody emitted radiation is governed by the Wien Displacement Law (Tλ_max = 0.2c₂).

How does blackbody radiation demonstrate the failure of classical mechanics compared to quantum mechanics?

Classical mechanics assumes continuous (unquantised) energy, and predicts that a blackbody should emit infinite energy at shorter wavelengths (ultraviolet catastrophe).

Experimentally, blackbody spectra have a peak (λ_max) and then a decrease in intensity. This can only be explained by energy being quantised. Energy is emitted in discrete amounts called quanta, as given by Planck’s Law, E = hv. This prevents the ultraviolet catastrophe problem and matches real-world observations.

Provide the Wien Displacement Law formula and context..

Tλ_max = 0.2c₂

• T is temperature (K).

• λ_max is the peak wavelength (cm).

• c₂ is 1.44 cm K.

The Wien Displacement Law describes the real-world observations seen with blackbody radiation spectra, where the peak wavelength is dependent solely on temperature and not material.

Describe the Planck distribution, and how it represents a fault in classical mechanics.

The Planck distribution describes how radiant EM radiation must possess discrete energy levels (quanta). It is governed by the equation E = hv = hc/λ.

It encourages the suppression of high-frequency radiation through quantisation.

Describe how atomic/molecular spectra represent a fault in classical mechanics and support the quantisation of energy.

Absorbance and emission of radiation (E = hv) results in a unique spectrum for the atom or molecule, where there are sharp lines (peaks) at specific wavelengths.

This demonstrates that the wavelengths at those peaks are discrete in value rather than continuous, falling in line with the idea of energy quantisation governed by E = hv = E₂ - E₁.

Describe wave-particle duality.

Radiation (hv) has both wave- and particle-like properties. It is supported by the mathematically-derived De Broglie Postulate, which posits that all matter exhibits wave-particle duality (λ = h/p = h/mv).

Describe the photoelectric effect and its conditions.

The photoelectric effect describes the emission of electrons from a material when EM radiation of sufficient threshold frequency v₀ strikes it.

• The number of electrons ejected is proportional to the intensity of radiation.

• The kinetic energy of electrons ejected is proportional to the frequency of radiation.

Provide and describe De Broglie’s Postulate.

De Broglie’s Postulate posits that all matter exhibits wave-particle duality, and that particle matter possesses a wavelength λ inversely proportional to its momentum p.

λ = h/p = h/mv

• λ is the De Broglie wavelength.

• h is the Planck constant.

• p is the momentum.

• m is the particle mass.

• v is the particle velocity.

In other words, a particle of mass m moving at velocity v will possess wave-like properties of wavelength λ.

Provide the (simplified) Schrödinger Wave Equation and describe its components.

Ĥψ = Eψ

• Ĥ is the Hamiltonian operator, which performs an operation on the wavefunction ψ to extract the energy eigenvalue E.

• ψ is the wavefunction, which indicates the probability that a particle is at some position.

• E is the energy eigenvalue, which scales the ψ without changing its fundamental shape.

Define the following terms: operator, eigenfunction, and eigenvalue.

Operator:

A “machine” function that converts a function f into another function g. Examples are the derivative dy/dx or the Hamiltonian operator.

Eigenfunction:

A function that does not change its fundamental character after being acted upon by an operator, though this action may result in the extraction of a coefficient called an eigenvalue. An example includes e^x, which when acted upon by the derivative dy/dx, results in e^x (possesses the same character as before, with an eigenvalue of one).

Eigenvalue:

A coefficient that is extracted from an eigenfunction following action by an operator. This value scales the eigenfunction without changing its fundamental character.

Should the wavefunction ψ be an eigenfunction, what does Ĥψ allow for? What about if the wavefunction ψ is not an eigenfunction?

Should wavefunction ψ be an eigenfunction, then Ĥψ will yield a specific energy eigenvalue E that is known for certain.

Should wavefunction ψ not be an eigenfunction, then Ĥψ will not allow for determining E directly. Other probabilistic determination strategies will need to be used to solve for E.

What rule does a Hermitian operator obey?

∫ψ_i*Ω̂ψ_jdx = [∫ψ_j*Ω̂ψ_idx]*

What rules does a linear operator obey?

Ω̂(f+g) = Ω̂(f) + Ω̂(g)

Ω̂(cf) = cΩ̂(f)

What rule do two orthogonal wavefunctions obey?

∫ψ_i*ψ_jdτ = 0

Describe superposition.

Superposition describes how the total wavefunction is composed of more than one wavefunction, all of which add up to form the total.

ψ_tot = Σc_nψ_n

Describe expectation values and provide the equation to determine a wavefunction’s expectation value.

Given some observable wavefunction (i.e., possesses a known eigenvalue) with an operator, each wavefunction will correspond to a definite expectation value that represents the average value that would be obtained if a measurement was repeated multiple times.

<Ω> = ∫ψ*Ω̂ψdx

Describe the Heisenberg Uncertainty Principle and provide its equation.

The Heisenberg Uncertainty Principle posits that it is impossible to specify simultaneously the position and momentum of a particle with arbitrary precision.

ΔxΔp ≥ h/4π

Given a particle-in-a-box, what is the “base” equation used that needs to be manipulated to obtain the ψ(x) and what is the final ψ(x)?

ψ(x) = Asin(kx) + Bcos(kx)

ψ(x) = √(L/2)sin(nπx/L)

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

ψ(x) = 0 for:

x ≤ 0

x ≥ L

Describe quantum tunnelling and its requirement to occur.

Quantum tunnelling is when a particle has a non-zero probability of being found in a region that it does not have enough classical energy to reach. Quantum ‘particles’ are described by wavefunctions, which do not stop abruptly but rather decay smoothly upon reaching the barrier region.

The wavefunction and its slope must be continuous (equivalent first derivatives) at the barrier’s edges.

What is line width?

The “thickness” of spectral lines or energy levels, indicating the uncertainty in values (given the Heisenberg Uncertainty Principle).

What are the two equations important for line width?

dE = h dv

dv = dt / 4π

Describe the Doppler effect.

The frequency of emitted radiation is dependent on the atom’s velocity relative to the detector.

What are two equations key to the Doppler effect? Describe their components.

v_receding = v₀ / (1 + s/c)

• v_receding is the frequency moving away (receding) from the detector.

• v₀ is the frequency of radiation emitted.

• s is the velocity of the atom.

• c is the speed of light.

v_approaching = v₀ / (1 - s/c)

• v_approaching is the frequency moving towards (approaching) the detector.

Provide the formula to determine the wavenumber of hydrogenic atoms undergoing a transition.

ṽ = R_H(1/n₁² - 1/n₂²)

• ṽ is the wavenumber.

• R_H is the Rydberg constant.

• n₁ and n₂ are principal quantum numbers of the two states, where n₂ > n₁.

What do high and low principal quantum number values indicate?

A high n indicates that the electron is farther from the nucleus, and is higher in energy.

A low n indicates that the electron is closer to the nucleus, and is lower in energy.

Given the principal quantum number integer values n = 1, 2, 3, and 4, provide the shell designation letters.

n = 1 is K

n = 2 is L

n = 3 is M

n = 4 is N

What is the formula to determine the momentum quantum number? What are its value-dependent designations (i.e., the letter values)?

ℓ = n - 1

ℓ = 0 is s

ℓ = 2 is p

ℓ = 3 is d

ℓ = 4 is f

What is the formula to determine the magnitude of the angular momentum given the momentum quantum number?

J = (h/2π)√(ℓ(ℓ +1))

What are the possible values of the magnetic quantum number?

m_ℓ = -ℓ, -ℓ + 1, …, 0, …, +ℓ

What are the possible values of the spin magnetic quantum number?

m_s = ±1/2

For hydrogenic atoms, what are the rules for permitted spectroscopic transitions?

Δℓ = ±1

Δm_ℓ = 0, ±1

ΔS = 0 (S = Σm_s)

ΔJ = 0, ±1 (J = 0 cannot transition to J = 0)

What is the Born-Oppenheimer approximation, and why may it be made?

Nuclei may be treated as stationary while electrons move in their field, due to the nuclei’s much larger masses.

What is the order of orbital filling for polynuclear molecules should the “average” of the molecule either be less than or round down from oxygen?

σ_2s < σ*_2s < π_2py = π_2px < σ_2pz < π*_2px = π*_2py < σ*_2pz

“π_2p < σ_2pz”

What is the order of orbital filling for polynuclear molecules should the “average” of the molecule either be greater than or round up to oxygen?

σ_2s < σ*_2s < σ_2pz < π_2py = π_2px < π*_2px = π*_2py < σ*_2pz

“σ_2pz < π_2p”

What is the Bragg equation for diffraction?

nλ = d(sinθ - sinφ)

• n is the diffraction order (n = 1, 2, 3, …)

• λ is the radiation wavelength

• d is the distance between gratings

• θ is the incident angle of radiation

• φ is the emerging angle of radiation

What is the entire four-part Beer-Lambert relation?

log(I₀/I) = -log(T) = A = εℓc

• I₀ is incident intensity of radiation

• I is emerging intensity of radiation

• T is transmittance (T ≤ 1.00)

• A is absorbance (0 ≤ A ≲ 3.00)

• ε is molar extinction coefficient

• ℓ is path length

• c is concentration

What is the moment of inertia equation for a diatomic molecule?

I = μR²

• I is moment of inertia

• μ is reduced mass

• R is bond length

What is the moment of inertia equation for a polyatomic molecule?

I = Σmr²

• I is moment of inertia

• m is nonreduced mass of atom

• r is distance from rotational axis

What are the two formulae for the rotational constant?

B = h/8π²I

B = h/8π²Ic

What is the formula for calculating the induced dipole moment?

μ = αE

• μ = induced dipole moment

• α = polarisability

• E = electric field

What is the difference between the two models for vibrations of diatomic molecules?

Harmonic quantum oscillator model:

• energy levels (v) are evenly spaced

• the dissociation energy is infinite

• for infrared spectroscopy, Δv = ±1

Morse model:

• energy level spacing decreases as energy level number increases

• dissociation energy is finite

• for infrared spectroscopy, Δv > 1 is weakly permitted

What are the two formulae for calculating the degrees of freedom for a molecule?

if linear, DOF = 3N - 5

if non-linear, DOF = 3N - 6

What are the components of a molecular term symbol?

Λ

• “orbital angular momentum along internuclear axis”

• central character

M = 2S+1

• “multiplicity”

• placed in the top-left.

±

• “reflectional symmetry along plane containing internuclear axis”

• placed in the top-right.

Ω

• “total angular momentum along internuclear axis”

• placed in bottom-right

g/u

• “parity”

• placed in the bottom-right, immediately to the right of Ω

What are the three steps of fluorescence?

1. absorption (excitation)

2. excited-state lifetime (nonradiative transition)

3. fluorescent emission

What are the four steps of phosphorescence?

1. absorption (excitation)

2. excited-state lifetime (nonradiative transition)

3. intersystem crossing

4. phosphorescent emission

What are the six steps of laser beam formation?

1. pumping (excitation)

2. nonradiative relaxation (metastabilisation)

3. population inversion

4. stimulated emission

5. light amplification

6. optical activity

Describe the two laser pulsation methods.

1. Q-switching (continuous pumping plus keeping the cavity “closed” results in a very large population inversion that when the cavity is “opened”, light is allowed to resonate which results in intense stimulated emission)

2. Mode locking (a modulator forces all allowed modes to have the same phase relationship such that they line up periodically, where constructive interference causes pico/femtosecond-long pulses and destructive interference causes silence)

What are the shortcut rules for determining which ΔJ values are ascribed to a spectroscopy?

ΔJ = ±1 for pure rotational (microwave) spectroscopy

ΔJ = 0, ±2 for all Raman spectroscopy

ΔJ = 0, ±1 for all (non-infrared) other spectroscopy

What are the shortcut rules for determining which Δv values are ascribed to a spectroscopy?

Δv = ±1 for all vibrational (infrared, rot-vib, Raman rot-vib) spectroscopy

Δv = irrelevant for all other spectroscopy

What are the shortcut rules for determining which equation to use to “solve” for a spectroscopy?

ṽ = ṽ₀ - ΔX for all spectroscopy except pure rotational (microwave) and rot-vib spectroscopy.

• ΔX = F(J’) - F(J) for Raman rotational spectroscopy

• ΔX = S(v’, J’) - S(v, J) for rot-vib Raman spectroscopy

• ΔX = F(J) - F(J’) for electronic excitation spectroscopy

ṽ = 2B(J+1) for pure rotational (microwave) spectroscopy.

ṽ = S(v’, J’) - S(v, J)’ for rot-vib spectroscopy.