Statistical Mechanics Flashcards
population of a state
the average number of molecules that occupy each state
there are on average ni molecules in a state of energy εi
Boltzmann distribution
gives the number of molecules ni in a particular energy state εi at temperature T
k - Boltzmann constant
using the Boltzmann distribution
used to predict the populations of states in a system at thermal equilibrium
yellow bars: population
black lines: energy states available to that system
shows the distribution of molecules across energy states of increasing energy at different temperatures
at T=0K - all molecules occupy the lowest energy state
as T increases - more molecules have eneough energy to populate higher states
as T → ∞ - all the energy states become equally populated
relative population of states
considers the ratio of the number of molecules in state i, ni with energy εi to the number of molecules in state j, nj with energy εj
removes the ned for a proportionality constant
TEMPERATURE is the only parameter that governs the population of the available energy states
the total number of molecules, N
N is equal to the sum of the number of molecules in each energy state (Avogradro’s number NA or L)
equation for ni (rewrite Boltzmann distribution)
lowest energy state = ε0
number of molecules in that state = n0
assume ε0 = 0
rewrite Boltzmann distribution, assuming εj is the lowest energy state, ε0 and that ε0 = 0
substitute equation for ni into equation for N
partition function, q
an abbreviation for the sum over all energy states of e^-βεi
it measures how the total number of molecules is distributed (partitioned) over all available states
- contains the information required to calculate the bulk properties of a system of independent (non-interacting) molecules
principle of equal a priori probabilities
states that all possibilities for the distribution of energy are equally probable. - hence, the treatment of energy states can be used for any type of molecular energy
allows us to assume that a translational, roatational and vibrational energy level of the same energy have equal probability of being occupied
energy states concerned with molecular motion: translationa, rotational and vibrational energy levels
vibrational energy levels
solve Schrodinger equation for SHO:
evenly spaced non-degenerate energy levels
E = (v+1/2)ħω
energy difference between levels
Δ = ħω
vibrational levels in spectroscopy:
G(v) = ωe(v+1/2) determined in wavenumbers
spacing between levels:
ΔG(v) = ωe
rotational energy levels
solve Schrodinger equation for a particle on a sphere or rigid rotor
E = (ħ^2/2I)J(J+1)
separation increases with increasing energy
rotational levels in spectroscopy
F(J) = BJ(J+1) determined in wavenumbers
difference between consecutive levels increases from 2B to 4B to 6B etc.
typical values for B range from 0.1 to 60 cm-1
unlike vibrational energy levels, rotational energy levels exhibit multiple degeneracy with (2J+1) states at each energy level
translational energy levels
derived and described using the particle in a box model
energy levels can be typically treated as a continuum (translational motion of gases can be treated classically)
exceptions to this: very light molecules at low temperatures in confined spaces
energy levels and states
a number of states (gi) may be available at an energy (εi)
- hence, that energy level is gi-fold degenerate
diagram shows two energy levels, ε0 and ε1, each of which is 2-fold degenerate (g0 = 2 and g1 = 2)
modify parition function, q equation to accound for the degeneracy and sum over other levels
contributions to the molecular partition function
the Born-Oppenherimer approximation allows us to sum the contributions from modes of energy to obtain the total energy of an isolated molecule
T = translation
R = rotation
V = vibration
we can therfore rewrite expression for the molecular partition function
consider q^V, q^R and q^T
for translational energy levels, the energy separation between them is far smaller than the thermal energy available to a molecule at room temperature
Δε^T «_space;kT
hence, q^T is large
for vibrational energy levels, the energy separation between them is largr than the thermal energy available at room temperature
Δε^V»_space; kT
hence, most molecules occupy the lowest energy level (vibrational ground state) at 298 K
q gives us an indication of the number of states that are thermally accessible to a molecule at the temperature of the system
total energy of molecules in a system
assume that we have a total of N molecules that are independent (non-interacting)
if each molecule occupies a state with energy εi
the total energy E is:
mean energy of a molecule, <ε></ε>
substitute expression for ni into equation for <ε></ε>
assumptions made for this expression:
- by convention, we have assumed that ε0 = 0 when determining q
- this is NOT always the case if εgs ≠ 0
εgs is the zero-point energy for the SHO
- hence, the true mean energy would be:
<ε> + εgs
2. in some case, partition function might depend on the volume as well as temperature (as in the case for q^T)
</ε>
expression for <ε> as a partial derivatie</ε>
relates the partition function to the mean energy of a molecule
internal energy of non-interacting molecules
continuing with the assumption that molecules are non-interacting
total energy for N molecules, E_N
the equation:
E_N = N<ε></ε>
is for non-interacting molecules
- assumes that the total energy of the system scales by multiplying the mean energy of each molecule by N
if molecules interact (electrostatic attractions/repulsions, H-bonding etc.), additional energy terms arise that do not scale linearly with N
the canonical ensemble
to account for interactions between molecules:
consider a closed collection of N interacting molecules at constant volume and temperature
- these molecules can be distributed across a set of energy states
- each molecule has a total energy Ei
these energy states adjust to any intermolecular interaction
this system is replicated multiple times
in forming this system, we have partitioned the molecules among the available energy states, which gives rise to the canonical partition function, Q
the canonical partition function, Q
(just like q)
Q can be useed to calculate the mean energy of an entire system composed of molecules that may be interacting with eachother
the sum is over all members of the ensemble
Q is more versatile than q because it accounts for molecular interactions
- can be applied to condensed phases and real gases
energy of an ensemble of interacting molecules
scaling factor N is not required
Q already considers the partitioning of N molecules as a whole and not as the product of N separate molecular contributions
relationship between Q and q: when molecules are independent and distinguishable
relationship between Q and q: when molecules are independent and indistinguishable
determining internal energy of a system
does not matter which relationship between Q and q is used for internal energy
the mean energy of a gas composed of N indistinguishable molecules is N times the mean energy of a single molecule
from quantum mechanics, energy level of a system containing molecule of mass m free to move in a 1-dimensional container of length X
partition function can be factorised into 3-dimensions
lowest energy level, ε0 is defined as zero energy
therefore consider the subsequent energy levels relative to the n=1 level
modify summation to fnd qX^T
two reasonable assumptions are made in order to modify the summation:
- the translational energy levels are so close together under typical conditions that we replace the sum with an integral (the continuum approximation)
- extending the lower limit from 1 to 0 introduces neglible error and therefore allows the solution of a standard integral
solution to qX^T
translational partition function in 3-dimensions
partition function increases with volume of the container and mass of the molecule
- an increase in each of these results in the separation between translational energy levels becoming smaller
- more levels become thermally accessible
β = 1/kT
- an increase in temperature increases the value of the partition function
large values of q^T are typical for light molecules in small volumes at room temperature
equation q^T can be rewritten
Λ - thermal wavelength of molecule: has dimensions of length
continuum approximation led to this expression - therefore it is only valid if q^T is very large
for this to be true, the thermal wavelength of the molecule must be far smaller than the linear dimensions of the container
V»Λ
using expression for q^T to determine the mean molecular energy
consider a monatomic perfect gas
this allows us to:
1. consider only translational motion (no rotational or vibrational states)
2. assume the atoms are independent and non-interacting
using expression for q^T to determine the internal energy
consider a monatomic perfect gas
using expression for q^T to determine the pressure
consider a monatomic perfect gas
using expression for q^T to determine the Gibbs energy
consider a monatomic perfect gas
this result allows us to understand Gibbs’ energy on a statistical level
q - the number of thermally accessible states
N - number of moleucles
q/N = average number of thermally acessible states per molecule
G(T) - G(0) is proportional to the log(ln) of the average number of thermally accessible states available to the molecules
as q/N increases,
- log(ln) term becomes larger
- G(T) - G(0) becomes more negative
the thermodynamic tendancy to lower Gibbs’ energy is driven by the tendency to maximise the number of thermally accessible states
the rotational partition function
q^R evaluated by summation
if many rotational states are occupied, kT»_space; ε^R
at 298K, a considerable number of rotational states are occupied
the continuum approximation can be used and the sum in the partition function is replaced with an integral
evaluating integral for the roational partition function to show that:
this equation is only valid for linear non-symmetricsl diatomic molecules (i.e. A-B)
validity of using this equation (continuum approximation) for rotational states is less clear cut than for translational states
- when moleucles have a large value for B, they have a small reduced mass
- these molecule are borderline for the assumption that kT»_space; ε^R
- they have the form: H-X
to assess the temperature above which continuum approximation equation is valid, use the characteristic rotational temperature of the molecule
number of thermally accessible rotational states for a symmetrical linear molecule
half the number available to an asymmetric linear molecule
a homonuclear diatomic molecule (A-A) or symmetrical linear molecule (O=C=O) has a roatation through 180° that results in an indistinguishable state of the molecule
continuum approxmation for rotational partition function is no longer valid for symmetrical linear molecules
expression for the rotational partition function beocmes
symmetry number, σ
heteronuclear diatomic: σ =1
homonuclear diatomic/linear symmetric molecule: σ = 2
fermions
nuclei with half-integer nuclear spin
1H
19F
I=1/2
bosons
nuclei with integer or zero spin
16O
14N
I=0
The Pauli principle states: when the labels of an two identical FERMIONS are exchanged, the sign of the total wavefunction CHANGED
Ψ(2,1) = - Ψ(1,2)
The Pauli principle states: when the labels of any two identical BOSONS are exchanged, the sign of the total wavefunction STAYS THE SAME
Ψ(2,1) = Ψ(1,2)
Born-Oppenherimer approximation: the total wavefunction for the molecule
consider C16O2 (CO2)
rotation through 180° results in exchange of two identical bosons
(16O, I=0)
hence:
Ψtot(2,1) = Ψtot(1,2)
consider each contribution to the total wavefunction in turn:
ELECTRONIC WAVEFUNCTION
electronic ground state for CO2 is 1Σg - hence symmetric
Ψele(2,1) = Ψele(1,2)
VIBRATIONAL WAVEFUNCTION
vibrational ground state (v=0) has a symmetric wavefunction
Ψvib(2,1) = Ψvib(1,2)
NUCLEAR WAVEFUNCTION
I=0, so exchaging the 16O nuclear spins is symmetric
Ψnuc(2,1) = Ψnuc(1,2)
ROTATIONAL WAVEFUNCTION
symmetries of rotational wavefunctions depend on J
Ψrot for even J are symmetrical wrt rotation through 180°
Ψrot(2,1) = Ψrot(1,2)
Ψrot for odd J are asymmetrical wrt rotation through 180°
Ψrot(2,1) = -Ψrot(1,2)
to obey Pauli principle for a boson, only J values that do NOT change sign on exchange are allowed
- hence for CO2, only rotational levels with even J occupied
why are only half of the thermally available rotational states allowed to be occupied for symmetrical linear molecules
the rotational partition function is only half of the value that is obtained for asymmetrical linear molecules
the electronic partition function, q^E
q^E has been neglected from the molecular partition function as it is not associated with molecular (or atomic) motion
calculating q^E
summation over electronic levels i
typical energy gaps between ground and excited electronic states are about 10^-19 to 10^-18 J
more than 10x that between the vibrational ground and first excited state
in many cases, q^E = 1
unless a molecule/atom has a particularly low lying electronic state and T»298K, we only need to consider the occupancy of the ground electronic state
such that: q^E = g0
the ground electronic state of most simple diatomic molecules is non-degenerate, so q^E = 1
- q^E does not contribute to the overall partition function of the molecule
determining degeneracy from the molecular term symbol of the electronic state
a common ground state term symbol for a diatomic molecule is: 1Σ
degeneracy is determined from the superscript (2S+1) value
- hence, g0 = 1
notable exceptions:
O2 (g0 = 3) - molecular ground state term symbol is 3Σ
NO (g0 =2) - molecular ground state term symbol is 2Π
q^E for atomic species
atoms frequently have degenerate electronic ground states
- degeneracy is determined by the total angular momentum quantum number J
Li ground state term symbol is:
2S 1/2
subscript 1/2 denotes J (J=1/2)
degeneracy is given by 2J +1
2(1/2)+1 = 2
g0=2
hence, if not excited electronic states are occupied q^E=2
in contrast to diatomic molecules, it is relatively common for atoms to have low-lying electronic states that are accessble at elevated temperatures
examples of two-level (2L) chemical systems
two non-degenerate energy levels separated by an energy difference:
ε1 - ε2 = ε
- two conformations of a molecule with different energy (proteins with different tertiary structure conformations)
- an electorn spin in a magnetic field (Zeeman effect)
- protons in a magnetic field (NMR)
- atoms or molecules with a low-lying electronic excited state
when kT «_space;ε, the temperature dependence of q2L and population (ni/N) is
when kT»_space; ε, the temperature dependence of q2L and population (ni/N) is
the general temperature dependence of q2L is:
partition function for a two-level system
energy for a sample size, N
at T = 0K
N<ε> = 0</ε>
as T → ∞
N<ε> → Nε/2</ε>
heat capacity for two-level systems
plot of dimensionless Cv against dimensionless T
- illustrates how head capacity changes with temperature for a 2L system
heat capacity is the change in internal energy as temperature is changes
- for a 2L system, it tells us the rate of population of the upper level as a function of temperature
at very low temperatures (kT «_space;ε)
- Cv is small as promotion to the higher level is not possible
when T increases
- there is sufficient thermal enegry to populate the upper level
- Cv increases rapidly as the internal energy increases
a maximum is observed for graph of heat capacity for a 2L system
there is a tendency for upper and lower levels to become equally populated
rate of change of internal energy slows down as you are reaching equal population of the two states
rate of internal energy (dU/dT) slows down, hence Cv foes through a maximum
internal energy tends towards a constant value of Nε/2
hence, Cv drops to zero
- there cna be no further change in internal energy as T increases
Schottky anomaly
instantaneous configuration
consider N molecules that can occupy a range of states, s0, s1, s2…etc
the number of molecules in s0 is n0
the number of molecules in s1 is n1
the population of these states at a given time (instantaneous configuration) is expressed in the form:
{n0, n1, n2…}
instantaneous configuration fluctuates continuously with time as the relative populations of the states change
for a large number of N molecules, the populations of the different energy states are changing but the total energy remains unchanged
- i.e. continuous gas collisions cause atoms to change energy state
assume that all states available to the molecule have exactly the same energy
- allows us to assume that there is no restriction on the number of molecules that occupy each state
- overall energy will be the same for every possible configuration
N = 3 molecules
- each molecule can occupy one of three degenerate states, s0, s1, s2
instantaneous configuration: {n0, n1, n2}
there are three possible configurations of this system when N = 3
{3, 0, 0} - only 1 way this configuration can be achieved
{2, 1, 0} - 3 ways this configuration can be achieved
{1, 1, 1} - 6 ways this configuration can be achieved
{2, 1, 0}
label 3 molecules A, B and C
{1, 1, 1}
label 3 molecules A, B and C
the system is statistically more likely to adopt confirmation 3 over 2 and 1 in the ratio
6:3:1
for larger systems of N molecules, (with many available degenerate states) system more skewed towards one dominating configuration
weight (of the configuration)
the number of ways in which a configuration can be achieved
for a large system of N molecules, the dominating configuration is found by determining for which values of ni the value of W is a maximum
- achieved by varying ni and determining those values for which dW=0 (maxima of a function)
constraints on real systems
the total numer of molecules and thier total energy remain constant
for a real system, the available states can have different energies
value of ni for the dominating (most probable) configuration
total energy for an independent (non-interacting) molecule
each of the energy terms is determined by summing over all occupied states i
εtot is independent of how many molecules are present
total energy of a collection of molecules
determined by multiplication of the mean total molecule energy <εtot> by the number of molecules N</εtot>
total energy for interacting molecules
additional term for molecule interacting energy, which is a function of N
Ei(N) is an energy state (of all forms of energy) that can be occupied by a molecule in the canonical ensemble we have defined
canonical ensemble therefore consists of a range of energy states, Ei that molecules can occupy
- the energy of these states can change as a function of how many molecules are present N
fixing T in the canonical ensemble
in the canonical ensemble, we define a constant N and V
- so the energies of the states are fixed
if T is also fixed, then the total energy remains constant
- as the number of energy states that are accessible to the molecules only depends on T
if states are equally thermally accessible
there will be a dominating configuration which determines the thermodynamic properties of the ensemble
the number of states at a particular energy level rise sharply as a function of energy
as the same time, the occupany of energy levels will decrease sharply as a function of energy
(Boltzmann distribution)
to detemrine the distribution of ensemble members across the available states:
multiply occupancy of energy level by the number of states at tat energy level
this product is a sharply peaked function at the mean energy
- shows that most members of the ensemble have an energy that deviates little from the mean energy
the Boltzmann formula
S - statistical entropy
situation A
- increasing T at constant V
- energy levels are unchanged but populations change
- W will also increase
- therefore, S will increase
as T → 0, every atom must occupy the lowest energy level
- hence, W=1 and S=0
as T → 0, S → 0 (Third Law of thermodynamics: the entropies of all perfect crystals approach zero as T → 0)
situation B
- decreasing V at constant T (compression)
- as volume decreases, spacing between energy levels increases and so fewer energy levels are occupied
- W will decrease
- therefore, S will decrease
deriving an expression that relates q (the partition function) to S (entropy)
for independent, indistinguishable molecules
W is reduced by a factor of N! because there are N! permutations among the energy states that would result in the same system
lnW and therefore S are reduced by klnN! from the distinguishable value
for interacting molecules
insert canonical partition function in place of the molecular partition function
residual entropy
where the entropy at T=0K is greater than zero (the value predicted by the third law of thermodynamics)
origin of residual entropy
consider a crystal composed of A-B molecules
A and B have similar sizes and electronegativity, hence molecule has a small dipole moment
there is little energy difference between
A-B A-B A-B A-B
A-B B-A B-A A-B
and other possible arrangements
during crystallisation, molecules adopt random A-B and B-A arrangements in the solid
- assuming the two orientations are equally probable and the system consists of N molecules
the total number of ways of achieving the same energy is: W = 2^N
hence, molar residual entropy for molecule A-B that can randomly adopt 2 orientations at T=0 K is 5.8 J K-1 mol-1
is molecule A-B has a large dipole moment, there is a strong enthalpic drive for the molecules to order A-B A-B A-B A-B in the solid
this results in a single orientation for all the molecules
W = 1 AND S = 0
