Statistical Thermodynamics

1. Statistical Thermodynamics Dr. Henry Curran NUI Galway

2. 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. Can we have unlimited power

3. Common mathematical functions Kinetics, Arrhenius • k = A exp{-E/RT} Vapour pressure, Clausius-Clayperon • p = p¥ exp{-DHV/RT} Viscosity, Andrade • h = A exp{+E/RT} • But re-define as inverse: F= F¥ exp{-E/RT}

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

5. 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.

6. 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

7. 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.

8. 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:

9. 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!

10. 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

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

12. 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

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

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

15. 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 0

16. Molecular partition function, q • If e1-e0 (De) is large (De >> kT) • q 1 (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.

17. 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

18. 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

19. The partition function explored 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

20. The partition function explored 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)

21. The partition function explored • 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

22. 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

23. Canonical partition function • 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 1, 2, 3, 4, …, N Molecular states a, b, c, d, …

24. Canonical partition function 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)

25. 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:

26. 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

27. 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.

28. 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

29. 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:

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

31. Distinguishable and indistinguishable particles 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

32. Distinguishable and indistinguishable particles 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!

33. 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

34. 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:

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

36. T dependence of populations but total number of particles is constant:

37. T dependence of populations (Fig. 6.2)

38. 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 = 1 + 0.82 q2L = 1 + 0.14 q2L2 q2L1 Both states equally only lower state accessible accessible

39. 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

40. 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.

41. Two-level heat capacity, CV • 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

42. Two-level heat capacity, CV

43. 0.44Nk 0.42 Variation of CV and Energy of a two-level system as a function of reduced Temperature

44. 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:

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

46. Toolkit equations

47. 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.

48. 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.

49. 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 we 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.

50. 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.