ACS PHYSICAL CHEMISTRY
THERMOCHEMISTRY TEST PAPER 2026
COMPREHENSIVE ANSWER KEY
◉ What is the expectation value, <m>? Answer: The average of an
observable for a wavefunction.
It is calculated by taking the integral of psi* M psi
◉ What is <m> for PIB? Answer: Independent of n
◉ What must operators be in quantum mechanics? Answer: Linear
and Hermitian
◉ When is an operator Hermitian? Answer: When int(fMg) =
int(gMf)
◉ What is the Heisenberg Uncertainty Principle? Answer: 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? Answer: AB pis(x) is equivalent to
BA psi(x) where A and B are operators
,◉ What is the commutator for momentum and position? Answer:
hbar/2
◉ How does the Hamiltonian change for PIB in multiple dimensions?
Answer: The operator adds a term for each dimension
◉ What is the energy for PIB in multiple dimensions? Answer: The
sum of the energies in each dimension
◉ What is the wavefunction for PIB in multiple dimensions? Answer:
The product of the wavefunction in each dimension
◉ How do the number of nodes follow the previous rule? Answer:
nodes = n-1, but nodes are dependent upon dimension
◉ When are two states degenerate? Answer: When they have equal
energies
◉ How does the separation between energy levels change as L
increases? Answer: Separation decreases
◉ What is the variance in thermal energy for an individual particle?
Answer: E= (Boltzmann constant)(T)
, ◉ Why does energy appear continuous between quantum states at
high temperatures? Answer: The variance in thermal energy is
greater than the gaps between their energy levels
◉ When do energy levels appear continuous? Answer: Longer L,
higher T, higher mass
◉ What is the Boltzmann distribution? Answer: Ni/Nj = gi/gj e^[-
(Ei-Ej)/kT]
◉ What happens as T increases? Answer: Particles become
distributed evenly across all energy levels according to degeneracy
◉ What are the bounds for a harmonic oscillator? Answer: (-infinity,
+infinity)
◉ Why can the bounds be infinite? Answer: 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? Answer: -
[(hbar)^2]/2md (d^2/dx^2) + .5kx^2
THERMOCHEMISTRY TEST PAPER 2026
COMPREHENSIVE ANSWER KEY
◉ What is the expectation value, <m>? Answer: The average of an
observable for a wavefunction.
It is calculated by taking the integral of psi* M psi
◉ What is <m> for PIB? Answer: Independent of n
◉ What must operators be in quantum mechanics? Answer: Linear
and Hermitian
◉ When is an operator Hermitian? Answer: When int(fMg) =
int(gMf)
◉ What is the Heisenberg Uncertainty Principle? Answer: 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? Answer: AB pis(x) is equivalent to
BA psi(x) where A and B are operators
,◉ What is the commutator for momentum and position? Answer:
hbar/2
◉ How does the Hamiltonian change for PIB in multiple dimensions?
Answer: The operator adds a term for each dimension
◉ What is the energy for PIB in multiple dimensions? Answer: The
sum of the energies in each dimension
◉ What is the wavefunction for PIB in multiple dimensions? Answer:
The product of the wavefunction in each dimension
◉ How do the number of nodes follow the previous rule? Answer:
nodes = n-1, but nodes are dependent upon dimension
◉ When are two states degenerate? Answer: When they have equal
energies
◉ How does the separation between energy levels change as L
increases? Answer: Separation decreases
◉ What is the variance in thermal energy for an individual particle?
Answer: E= (Boltzmann constant)(T)
, ◉ Why does energy appear continuous between quantum states at
high temperatures? Answer: The variance in thermal energy is
greater than the gaps between their energy levels
◉ When do energy levels appear continuous? Answer: Longer L,
higher T, higher mass
◉ What is the Boltzmann distribution? Answer: Ni/Nj = gi/gj e^[-
(Ei-Ej)/kT]
◉ What happens as T increases? Answer: Particles become
distributed evenly across all energy levels according to degeneracy
◉ What are the bounds for a harmonic oscillator? Answer: (-infinity,
+infinity)
◉ Why can the bounds be infinite? Answer: 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? Answer: -
[(hbar)^2]/2md (d^2/dx^2) + .5kx^2
