Statistical Thermodynamics

Statistical Thermodynamics
Dr. Henry Curran NUI Galway

Background Thermodynamic parameters of stable molecules can be found. However, those for radicals and transition state species cannot be readily found. Need a way to calculate these properties readily and accurately.

The Boltzmann Factor Boltzmann law for the population of quantised energy states:

Average basis of the behaviour of matter
Thermodynamic properties are concerned with average behaviour. The instantaneous values of the occupation numbers are never very different from the averages.

Distinct, independent particles
Consider an assembly of particles at constant temperature. These particles are distinct and labelled (a, b, c, … etc) They are independent interact with each other minimally enough to interchange energy at collision Weakly coupled Sum of individual energies of labelled particles

Statistical weights At any instant the distribution of particles among energy states involve n0 with energy e0, n1 with energy e1, n2 with energy e2 and so on. We call the instantaneous distribution the configuration of the system. At the next moment the distribution will be different, giving a different configuration with the same total energy. These configurations identify the way in which the system can share out its energy among the available energy states.

Statistical weights A given configuration can be reached in a number of different ways. We call the number of ways W the statistical weight of that configuration. It represents the probability that this configuration can be reached, from among all other configurations, by totally random means. For N particles arriving at a configuration in which there are n0 particles with energy e0, n1 with e1 etc, the statistical weight is:

Principle of equal a priori probabilities
Each configuration will be visited exactly proportional to its statistical weight We must find the most probable configuration How likely is this to dominate the assembly? For an Avogadro number of particles with an average change of configuration of only 1 part in 1010 reduces the probability by: A massive collapse in probability!

Maximisation subject to constraints
The predominant configuration among N particles has energy states that are populated according to: where a and b are constants under the conditions of constant temperature

The lowest state has energy e0 = 0 and occupation number n0
which identifies the constant a, and enables us to write:

this is a temperature dependent ratio since the occupation number of the states vary with temperature The constant b can be stated as: where k is the Boltzmann constant

Molecular partition function
Any state population (ni) is known if: ei, T, and n0 are known

Molecular partition function
If the total number of particles is N, then:

Molecular partition function, q
Determines how particles distribute (or partition) themselves over accessible quantum states. An infinite series that converges more rapidly the larger both the energy spacing between quantum states and the value of b is. Convergence is enhanced at lower temperatures since b = 1/kT. when be >> 0, e-be

Molecular partition function, q
If e1-e0 (De) is large (De >> kT) q (lowest value of q) If e1-e0 (De) ≈ kT (thermal energy) q large number magnitude of q shows how easily particles spread over the available quantum states and thus reflects the accessibility of the quantum energy states of the particles involved.

Energy states and energy levels
However, quantum states can be degenerate with a number (g) of states all sharing the same energy. States with the same energy comprise an energy level and we use the symbol gj to denote the degeneracy of the jth level

The partition function explored
The total number of particles in our assembly is N or, expressed intensively, NA per mole The partition function is a measure of the extent to which particles are able to escape from the ground state

The partition function q is a pure number which can range from a minimum value of 1 at 0 K (when n0 = NA and only the ground state is accessible) to an indefinitely large number as the temperature increases Fewer and fewer particles are left in the ground state and an indefinitely large number of states become available to the system

We can characterise the closeness of spacing in the energy manifold by referring to the density of states function, D(e), which represents the number of energy states in unit energy level. If D(e) is high (translational motion in gas): particles find it easy to leave ground state q will rise rapidly as T increases If D(e) is low (vibrations of light diatomic molecules) small value of q ( 1)

If q/NA (the number of accessible states per particle) is small few particles venture out of the ground state If q/NA is large there are many accessible states and molecules are well spread over the energy states of the system q/NA >> 1 for the valid application of the Boltzmann law in gaseous systems

Canonical partition function
Molecular  molar level assume value of an extensive function for N particles is just N times that for a single particle true for energy of non-interacting particles but not so for other properties (e.g. entropy) Particles do interact! Thus we consider: every system has a set of system energy states which molecules can populate these states are not restricted by the need for additivity but can adjust to any inter-particle interactions that may exist

Molar sum over states each possible state of the whole system involves a description of the conditions experienced by all the particles that make up a mole Suppose N identical particles each with the set of individual molecular states available to them. Particle labels , 2, 3, 4, …, N Molecular states a, b, c, d, …

Any given molar state can be described by a suitable combination of individual molecular states occupied by individual molecules. If we call the ith state Ψi, we can begin to give a description of this molar state by writing: with energy: there is no restriction on the number of particles that can be in the same molecular state (e.g. particles 5 & 7 are both in molecular state c)

Canonical partition function (QN)
State Ψi, with energy Ei is just one of many states of the whole system. The predominant molecular configuration is called the canonical distribution applies to states of an N-particle system at constant amount, volume, and Temperature with energy:

The Molar energy The canonical partition function (QN) is much more general than “the product of N molecular partition functions q” since there is no need to consider only independent molecules

The Molar energy If we are able to calculate thermodynamic properties for assemblies of N independent particles using q and for N non-independent particles using QN, then, in the limit of the particles of QN, becoming less and less strongly interdependent, the two methods should eventually converge.

The Molar energy Note that the expressions:
are compatible if we assume that the two different partition functions are related simply by: where Q is a function of an N-particle assembly at constant T and V

Distinguishable and indistinguishable particles
for the canonical partition function we can write: In every one of the i system states, each particle (1, 2, 3, …) will be in one of its possible j molecular states (a,b, c, d …) just once in each system state. If we factorise out each particle in turn from the summation over the system states and then gather together all the terms that refer to a given particle, we get:

If all molecules are of the same type and indistinguishable by position they do not need labelling

If particles are indistinguishable the number of accessible system states is lower than it is for distinguishable ones. A system Ψi differs from a state Ψj if particles are distinguishable, because of the interchange of particles 2 and 3 between states b and h. However, Ψi is identical to Ψj if particles are indistinguishable

In systems which are not at too high a density and are also well above 0 K, the correction factor for this over-counting of configurations is 1/N!

Two-level systems The simplest type of system is one which comprises particles with only two accessible states in the form of two non-degenerate levels separated by a narrow energy gap De: At temperatures that are comparable to De/k only the ground state and first excited state are appreciably populated

Effect of increasing Temperature
The average population ratio of the two levels is an assembly of such two-level (or two-state) particles is given by: where the two-level temperature (q2L) is defined as:

Temperature Dependence of the populations
Comparing energy gap with background thermal energy Comparing characteristic T (q2L) with Temperature (T)

T dependence of populations
but total number of particles is constant

T dependence of populations (Fig. 6.2)

Two-level Molecular partition function
The effect of increasing T only two energy states to consider High T Low T T = 5q2L T = 0.5q2L q2L = q2L = q2L q2L Both states equally only lower state accessible accessible

The energy of a two-level system
At high T, half the particles occupy the upper state and the total energy takes the value ½ N De

Two-level heat capacity, CV
The spacing of energy levels in discussing two-level systems is not affected by changes in volume, so the relevant heat capacity is CV not CP Variation of CV with T is a measure of how accessible the upper states becomes as T increases.

Low T kT is small small DT has little tendency to excite particles overall energy remains constant => CV low Intermediate T kT is comparable to De small DT has larger effect in exciting particles CV is somewhat larger High T Almost half particles in excited state small DT causes very few particles excited to upper Overall energy remains constant => CV low

0.44Nk 0.42 Variation of CV and Energy of a two-level system as a function of reduced Temperature

The effect of degeneracy
Consider a two-level system with degenerate energy levels. If the degeneracy of the lower level is g0 and that of the upper level is g1 then the two-level molecular partition function is:

The effect of degeneracy
Energy of the degenerate two-level system: and the heat capacity:

Toolkit equations

Toolkit equations In order to relate the partition function to classical thermodynamic quantities, the equations for internal energy and entropy are needed. Once these have been expressed in terms of the canonical partition function, Q, the Massieu bridge can be derived. This in turn provides the most compact link to classical thermodynamics.

Ideal monatomic gas Consider an assembly of particles constrained to move in a fixed volume. This system consists of many, non-interacting, monatomic gas particles in ceaseless translational motion. The only energy that these particles can possess is translational kinetic energy.

Translational partition function, qtrs
In classical mechanics, all kinetic energies are allowed in a system of monatomic gas particles at a fixed volume V and temperature T. Quantum restrictions place limits on the actual kinetic energies that are found. To determine the partition function for such a system to need to establish values for the allowed kinetic energies. We consider a particle constrained to move in a cubic box with dimensional box with dimensions lx, ly, and lz.

Particle in a one-dimensional box
The permitted energy levels, ex, for a particle of mass m that is constrained by infinite boundary potentials at x = 0, and x = lx to exist in a one-dimensional box of length lx are given by: and similarly for the y- and z-directions. The translational quantum number, nx, is a positive integer and the quantum numbers in the y- and z-directions are ny and nz, respectively.

One-dimensional partition function
The one-dimensional partition function, qtrs,x, is obtained by summing over all the accessible energy states. Thus: an expression that is exact but cannot be evaluated except by direct and tedious numerical summation.

One-dimensional partition function
For all values of lx in any normal vessel, these energy levels are very densely packed and lie extremely close to each other. They form a virtual continuum, so the summation can be replaced by an integration with the running variable nx resulting in the expression:

Extension to three dimensions
We can factorise the translational partition function: product of three dimensions gives the volume, V:

Extension to three dimensions
For the canonical partition function for N indistinguishable particles

Testing the continuum approximation
At its boiling temperature of 4.22 K, one mole of He occupies 3.46 x 10-4 m3. How many translational energy states are accessible at this very low temperature, and determine whether the virtual continuum approximation is valid.

Ideal monatomic gas: thermodynamic functions
Since all classical thermodynamic functions are related to the logarithm of the canonical partition function, we start by taking the logarithm of Qtrs:

Ideal monatomic gas: thermodynamic functions

The translational energy, Etrs
For a monatomic gas, the translational kinetic energy, Etrs, is the only form of energy that the monatomic particles possess, so we can equate it directly to the internal energy, U, and substitute the value of the derivative: (for one mole N = NA, NAk = R)

The equation of state This can be derived using:
(for one mole N = NA, NAk = R)

The heat capacity, CV This can be derived using:
(for one mole N = NA, NAk = R)

Entropy of an ideal monatomic gas
All of the simplifying factors that result from taking partial derivatives of ln Qtrs no longer hold lnQtrs is a direct term and so all variables appear

Next, using Stirling’s approximation (lnN! ≈ NlnN – N) One mole: (N = NA, NAk = R, NAm = M and V = RT/p

gathering together all experimental variables (M. T, p) The Sackur-Tetrode Equation The first term contains all of the experimental variables, the second consists of constants

Using the Sackur-Tetrode equation
The second term has a value of J K-1 mol-1 or R. Thus we can write: Some calculated and measured entropies Argon Krypton Tb / K Scalc / R Scalor / R

Using the Sackur-Tetrode equation
The Sachur-Tetrode equation can also be written as using ln p-1 = ln V – ln R – ln T where the variables are expressed in SI units (V/m3, T/K, and M/kg mol-1)

Significance of Sackur-Tetrode equation
volume dependence temperature dependence the first two terms are known from classical thermodynamics the last two terms (3/2 R lnM and R) could not have been foreseen from classical thermodynamics

Change in conditions Effect of increase in T, V, or M on:
D(e) Q U p CV S T nil nil V nil nil nil M nil nil nil

Ideal diatomic gas: internal degrees of freedom
Polyatomic species can store energy in a variety of ways: translational motion rotational motion vibrational motion electronic excitation Each of these modes has its own manifold of energy states, how do we cope?

Internal modes: separability of energies
Assume molecular modes are separable treat each mode independent of all others i.e. translational independent of vibrational, rotational, electronic, etc, etc Entirely true for translational modes Vibrational modes are independent of: rotational modes under the rigid rotor assumption electronic modes under the Born-Oppenheimer approximation

Thus, a molecule that is moving at high speed is not forced to vibrate rapidly or rotate very fast. An isolated molecule which has an excess of any one energy mode cannot divest itself of this surplus except at collision with another molecule. The number of collisions needed to equilibrate modes varies from a few (ten or so) for rotation, to many (hundreds) for vibration.

Thus, the total energy of a molecule j:

Weak coupling: factorising the energy modes
Admits there is some energy interchange in order to establish and maintain thermal equilibrium But allows us to assess each energy mode as if it were the only form of energy present in the molecule Molecular partition function can be formulated separately for each energy mode (degree of freedom) Decide later how individual partition functions should be combined together to form the overall molecular partition function

Imagine an assembly of N particles that can store energy in just two weakly coupled modes a and w Each mode has its own manifold of energy states and associated quantum numbers A given particle can have: a-mode energy associated with quantum number k w-mode energy associated with quantum number r

The overall partition function, qtot: expanding we would get:

but e(a+b) = ea.eb, therefore: each term in every row has a common factor of in the first row, in the second, and so on. Extracting these factors row by row:

the terms in parentheses in each row are identical and form the summation:

If energy modes are separable then we can factorise the partition function and write:

Factorising translational energy modes
Total translational energy of molecule j: which allows us to write:

Factorising internal energy modes
Total translational energy of molecule j: using identical arguments the canonical partition function can be expressed: but how do we obtain the canonical from the molecular partition function Qtot from qtot? How does indistinguishability exert its influence?

When are particles distinguishable (having distinct configurations, and when are they indistinguishable? Localised particles (unique addresses) are always distinguishable Particles that are not localised are indistinguishable Swapping translational energy states between such particles does not create distinct new configurations However, localisation within a molecule can also confer distinguishability

When molecules i and j, each in distinct rotational and vibrational states, swap these internal states with each other a new configuration is created and both configurations have to be counted into the final sum of states for the whole system. By being identified specifically with individual molecules, the internal states are recognised as being intrinsically distinguishable. Translational states are intrinsically indistinguishable.

Canonical partition function, Q
and thus: This conclusion assumes weak coupling. If particles enjoy strong coupling (e.g. in liquids and solutions) the argument becomes very complicated!

Ideal diatomic gas: Rotational partition function
Assume rigid rotor for which we can write successive rotational energy levels, eJ, in terms of the rotational quantum number, J. where I is the moment of inertia of the molecule, m is the reduced mass, and B the rotational constant.

Another expression results from using the characteristic rotational temperature, qr, 1st energy increment = 2kqr 2nd energy increment = 4kqr

Rotational energy levels are degenerate and each level has a degeneracy gJ = (2J+1). So: If no atoms in the atom are too light (i.e. if the moment of inertia is not too small) and if the temperature is not too low (close to 0 K), allowing appreciable numbers of rotational states to be occupied, the rotational energy levels lie sufficiently close to one another to write:

This equation works well for heteronuclear diatomic molecules. For homonuclear diatomics this equation overcounts the rotational states by a factor of two.

When a symmetrical linear molecule rotates through 180o it produces a configuration which is indistinguishable from the one from which it started. all homonuclear diatomics symmetrical linear molecules (e.g. CO2, C2H2) Include all molecules using a symmetry factor s s = 2 for homonuclear diatomics, s = 1 for heteronuclear diatomics s = 2 for H2O, s = 3 for NH3, s = 12 for CH4 and C6H6

Rotational properties of molecules at 300 K
qr/K s T/qr qrot H CH HCl HI N CO CO I

Rotational canonical partition function
relates the canonical partition function to the molecular partition function. Consequently, for the rotational canonical partition function we have:

Rotational Energy this can differentiated wrt temperature, since the second term is a constant with no T dependence

Rotational heat capacity
this equation applies equally to all linear molecules which have only two degrees of freedom in rotation. Recast for one mole of substance and taking the T derivative yields the molar rotational heat capacity, Crot, m. Thus, when N = NA, the molar rotational energy is Urot,m

Rotational entropy Srot is dependent on (reduced) mass (I = mr2), and there is also a constant in the final term, leading to:

Rotational entropy Typically, qrot at room T is of the order of hundreds for diatomics such as CO and Cl2. Compare this with the almost immeasurably larger value that the translational partition function reaches.

Extension to polyatomic molecules
In the most general case, that of a non-linear polyatomic molecule, there are three independent moments of inertia. Qrot must take account of these three moments Achieved by recognising three independent characteristic rotational temperatures qr, x, qr, y, qr, z corresponding to the three principal moments of inertia Ix, Iy, Iz With resulting partition function:

Conclusions Rotational energy levels, although more widely spaced than translational energy levels, are still close enough at most temperatures to allow us to use the continuum approximation and to replace the summation of qrot with an integration. Providing proper regard is then paid to rotational indistinguishability, by considering symmetry, rotational thermodynamic functions can be calculated.

Ideal diatomic gas: Vibrational partition function
Vibrational modes have energy level spacings that are larger by at least an order of magnitude than those in rotational modes, which in turn, are 25—30 orders of magnitude larger than translational modes. cannot be simplified using the continuum approximation do not undergo appreciable excitation at room Temp. at 300 K Qvib ≈ 1 for light molecules

The diatomic SHO model We start by modelling a diatomic molecule on a simple ball and spring basis with two atoms, mass m1 and m2, joined by a spring which has a force constant k. The classical vibrational frequency, wosc, is given by: There is a quantum restriction on the available energies:

The diatomic SHO model The value is know as the zero point energy Vibrational energy levels in diatomic molecules are always non-degenerate. Degeneracy has to be considered for polyatomic species Linear: 3N-5 normal modes of vibration Non-linear: 3N-6 normal modes of vibration

Vibrational partition function, qvib
Set e0 = 0, the ground vibrational state as the reference zero for vibrational energy. Measure all other energies relative to reference ignoring the zero-point energy. in calculating values of some vibrational thermodynamic functions (e.g. the vibrational contribution to the internal energy, U) the sum of the individual zero-point energies of all normal modes present must be added

The assumption (e0 = 0) allows us to write: Under this assumption, qvib may be written as: a simple geometric series which yields qvib in closed form: where qvib = hw/k = characteristic vibrational temperature

Unlike the situation for rotation, qvib, can be identified with an actual separation between quantised energy levels. To a very good approximation, since the anharmonicity correction can be neglected for low quantum numbers, the characteristic temperature is characteristic of the gap between the lowest and first excited vibrational states, and with exactly twice the zero-point energy,

Ideal diatomic gas: Vibrational partition function
Vibrational energy level spacings are much larger than those for rotation, so typical vibrational temperatures in diatomic molecules are of the order of hundreds to thousands of kelvins rather than the tens of hundreds characteristic of rotation. Species qvib/K qvib 300 K) H2 5987 1.000 HD 5226 D2 4307 N2 3352 CO 3084 Cl2 798 1.075 I2 307 1.556

Light diatomic molecules have: high force constants low reduced masses Thus: vibrational frequencies (wosc) and characteristic vibrational temperatures (qvib) are high just one vibrational state (the ground state) accessible at room T the vibrational partition function qvib ≈ 1

Heavy diatomic molecules have: rather loose vibrations Lower characteristic temperature Thus: appreciable vibrational excitation resulting in: population of the first (and to a slight extent higher) excited vibrational energy state qvib > 1

Situation in polyatomic species is similar complicated only by the existence of 3N-5 or 3N-6 normal modes of vibration. Some of these normal modes are degenerate (1), (2), (3), … denoting individual normal modes 1, 2, 3, …etc. Species qvib/K ∏(qvib) 300 K) CO2 3360 1.091 1890 954(2) NH3 4880(2) 1.001 4780 2330(2) 1360 CHCl3 4330 2.650 1745(2) 1090(2) 938 523 374(2)

As with diatomics, only the heavier species show values of qvib appreciably different from unity. Typically, qvib is of the order of ~3000 K in many molecules. Consequently, at 300 K we have: in contrast with qrot (≈ 10) and qtrs (≈ 1030) For most molecules only the ground state is accessible for vibration

High T limiting behaviour of qvib
At high temperature the equation gives a linear dependence of qvib with temperature. If we expand , we get: High T limit

T dependence of vibrational partition function
As T increases, the linear dependence of qvib upon T becomes increasingly obvious

The canonical partition function, Qvib
Using we can find the first differential of lnQ with respect to temperature to give:

The vibrational energy, Uvib
This is not nearly as simple as: linear molecules

The vibrational energy, Uvib
This does reduce to the simple form at equipartition (at very high temperatures) to: (equipartition) Normally, at room T:

The zero-point energy So far we have chosen the zero-point energy (1/2hw) as the zero reference of our energy scale Thus we must add 1/2hw to each term in the energy ladder For each particle we must add this same amount Thus, for N particles we must add U(0)vib, m = 1/2Nhw

Vibrational heat capacity, Cvib
The vibrational heat capacity can be found using: The Einstein Equation This equation can be written in a more compact form as:

Vibrational heat capacity, Cvib
FE with the argument qvib/T is the Einstein function The Einstein function

The Einstein heat capacity
low T High T

The Einstein function The Einstein function has applications beyond normal modes of vibration in gas molecules. It has an important place in the understanding of lattice vibrations on the thermal behaviour of solids It is central to one of the earliest models for the heat capacity of solids

The vibrational entropy, Svib
We know and N = NA for one mole, thus:

Variation of vibrational entropy with reduced temperature

Electronic partition function
Characteristic electronic temperatures, qel, are of the order of several tens of thousands of kelvins. Excited electronic states remain unpopulated unless the temperature reaches several thousands of kelvins. Only the first (ground state) term of the electronic partition function need ever be considered at temperatures in the range from ambient to moderately high.

It is tempting to decide that qel will not be a significant factor. Once we assign e0 = 0, we might conclude that: To do so would be unwise! One must consider degeneracy of the ground electronic state.

The correct expression to use in place of the previous expression is of course: Most molecules and stable ions have non-degenerate ground states. A notable exception is molecular oxygen, O2, which has a ground state degeneracy of 3.

Atoms frequently have ground states that are degenerate. Degeneracy of electronic states determined by the value of the total angular momentum quantum number, J. Taking the symbol G as the general term in the Russell—Saunders spin-orbit coupling approximation, we denote the spectroscopic state of the ground state of an atom as: spectroscopic atom ground state = (2S+1)GJ

spectroscopic atom ground state = (2S+1)GJ where S is the total spin angular momentum quantum number which gives rise to the term multiplicity (2S+1). The degeneracy, g0, of the electronic ground states in atoms is related to J through: g0 = 2J+1 (atoms)

For diatomic molecules the term symbols are made up in much the same way as for atoms. Total orbital angular momentum about the inter-nuclear axis. Determines the term symbol used for the molecule (S, P, D, etc. corresponding to S, P, D, etc. in atoms). As with atoms, the term multiplicity (2S+1) is added as a superscript to denote the multiplicity of the molecular term.

In the case of molecules it is this term multiplicity that represents the degeneracy of the electronic state. For diatomic molecules we have: spectroscopic molecular ground state = (2S+1)G for which the ground-state degeneracy is: g0 = 2S + 1 (molecules)

Species Term Symbol gn qel/K Li 2S1/2 g0 = 2 C 3P0 g0 = 1 N 4S3/2 g0 = 4 O 3P2 g0 = 5 F 2P3/2 2P1/2 g1 = 2 590 NO 178 O2 3S-g g0 = 3 1Dg g1 = 1 11650

Where the energy gap between the ground and the first excited electronic state is large the electronic partition function simply takes the value g0. When the ground-state to first excited state gap is not negligible compared with kT (qel/T is not very much less than unity) it is necessary to consider the first excited state. The electronic partition function becomes:

For F atom at 1000 K we have: For NO molecule at 1000 K we have:

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