A Statistical mechanics
A Statistical mechanics
The main objective of statistical mechanics is to calculate macroscopic (thermodynamic) properties from a knowledge of microscopic information like quantum mechanical energy levels. The purpose of the present appendix is merely to present a selection1 of the results that are most relevant in the context of reaction dynamics.
Consider a macroscopic system with N particles, a volume given by V, and an energy given by E. This is the so-called microcanonical system. There will be an enormous number of quantum states consistent with the fixed macroscopic properties N, V, and E. The energy E must be one of the eigenvalues of the N-body Hamiltonian H N, and the number of states of energy E is equal to the degeneracy, which we denote by Ω(N, V, E). This number is also called the microcanonical partition function. A basic postulate of statistical mechanics is that all states at the same energy are equally probable (the assumption of equal a priori probabilities). That is, the probability of finding the system in any of the states is P = 1/Ω(N, V, E).
For most practical applications, we consider not isolated systems but systems in which the temperature is fixed. This is the canonical system consistent with the fixed macroscopic properties (N, V, T), where T is the absolute temperature.
We consider a canonical system of N interacting particles described by the N-body Hamiltonian H N. In such a system the energy is not fixed. We shall not give the detailed derivation here but merely state that the probability that the system is found in the ith energy state εi, given as an eigenvalue of the N-body Hamiltonian, is
(p.292) A.1 A system of non-interacting molecules
Now, at sufficiently high temperatures, for N non-interacting identical and indistinguishable particles (e.g., an ideal gas), the partition function can be written in the form
The probability of finding a molecule in the energy level E i is
A.1.1 The molecular partition function
The evaluation of the molecular partition function can be simplified by noting that the total energy of the molecule may be written as a sum of the center-of-mass translational energy and the internal energy, E = E trans + E int, which implies
These partition functions can be evaluated quite readily. In the following it turns out to be convenient to choose the zero of energy such that it coincides with the zero-point level of the quantized energy levels. To that end, we note that, if we subtract (p.293) the zero-point energy e 0 from all energy levels, the partition function Eq. (A.5) takes the form
Here the sum is the partition function with the energy measured relative to the zero-point level and Q is obtained after multiplication by exp(–E 0/k B T). The Boltzmann distribution Eq. (A.4) can be written in the form
Thus, the partition functions differ by the factor exp(–E 0/k B T), whereas the Boltzmann distribution is invariant to such a shift of the energy scale in the standard expressions for the energy levels.
For the translational partition function, we first consider a particle in a one-dimensional box of length l. The energy levels, with the zero of energy as the zero-point level, are E n = (n 2 – 1)h 2/(8ml 2), with n = 1,2,…, and degeneracy ωn = 1. The partition function takes the form
This sum cannot be evaluated analytically. However, when the energy difference between subsequent levels can be considered as small, then the sum can be replaced by an integral. Thus, when E n+1 – E n = (2n + 1)h 2/(8ml 2) ≪ k B T, that is, at high temperatures,2 we have
The energy levels for a particle in a three-dimensional box are given as the sum of the energies for each dimension, and the partition function for the three-dimensional box is simply a product of the partition functions for each dimension; that is,
In order to evaluate the vibrational partition function we consider a single harmonic oscillator. The energy levels, with the zero of energy as the zero-point level, are E n = hν s n, with n = 0,1,…, and degeneracy ωn = 1. The partition function takes the form (p.294)
For the rotational partition function, we first consider a linear rigid rotor. The energy levels are E J = J(J + 1)ħ 2/(2I), with J = 0,1, …, and I is the moment of inertia. Each energy level has a degeneracy of m J = 2J + 1. The partition function takes the form
This sum cannot be evaluated analytically. However, when the energy difference between subsequent levels can be considered as small (E J+1 – E J = ħ 2(J+1)/I ≪ k B T, that is, at high temperatures) then the sum can be replaced by an integral:
The last integral is easily evaluated, and
This is the correct expression for the rotational partition function of a heteronuclear diatomic molecule. For a homonuclear diatomic molecule, however, it must be taken into account that the total wave function must be either symmetric or antisymmetric under the interchange of the two identical nuclei: symmetric if the nuclei have integral spins or antisymmetric if they have half-integral spins. The effect on Q rot is that it should be replaced by Q rot/σ, where σ is a symmetry number that represents the number of indistinguishable orientations that the molecule can have (i.e., the number of ways the molecule can be rotated ‘into itself). Thus, Q rot in Eq. (A.19) should be replaced by Q rot/σ, where σ = 1 for a heteronuclear diatomic molecule and σ = 2 (p.295) for a homonuclear diatomic molecule. When Eq. (A.19) is extended to a non-linear molecule with moments of inertia I a, I b, and I c about its principal axes, the rotational partition function takes the form
Finally, the electronic partition function is considered. The zero of energy is chosen as the electronic ground-state energy. The spacings between the electronic energy levels are, normally, large and only the first term in the partition function will make a significant contribution; that is,
Thus, the partition function is simply the degeneracy of the electronic ground state.
A.1.2 Macroscopic properties
When we know the partition function, we can calculate thermodynamic quantities from a knowledge of the quantum mechanical energy levels. Consider, as an example, the energy E. A basic postulate of statistical mechanics is that such an energy is the average value of all the quantum mechanical energy levels with the weights given by the Boltzmann distribution; that is,
In a similar manner, the equilibrium constant of a chemical reaction can be related to the quantum mechanical energy levels of the reactants and products. Consider, as an example, a mixture of A and B molecules in equilibrium:
The equilibrium constant is
According to the Boltzmann distribution, Eq. (A.4), the probability of finding an A molecule in the energy level E a is
For a general equilibrium of the form
(p.297) A.2 Classical statistical mechanics
It is often impossible to obtain the quantized energies of a complicated system and therefore the partition function. Fortunately, a classical mechanical description will often suffice. Classical statistical mechanics is valid at sufficiently high temperatures. The classical treatment can be derived as a limiting case of the quantum version for cases where energy differences become small compared with k B T.
The state of a classical system can be completely described by coordinates and momenta, i.e., a point in phase space; see Fig. 1.2.1. For a system with s degrees of freedom (that is, s coordinates are required to completely describe its position), the phase space has the dimension 2s. When the system evolves in time, its dynamics is described by the trajectory of the phase-space point through phase space. The trajectory is given by Hamilton’s equations of motion, Eq. (4.63), and the total energy for the molecule is given by the classical Hamiltonian H. The summations over states (or energy levels) in quantum statistical mechanics are replaced by integrations over regions of phase space.
The problem now is how to count the number of states in this continuous description. For proper counting, the use of quantum mechanics (Planck’s constant) is unavoidable. For a system with one degree of freedom, the number of states in the area element dqdp is
The Boltzmann statistics of particles described by classical mechanics is obtained from Eqs (A.6) and (A.34):
The expectation value of the energy in Eq. (A.22) is given by
Finally, the classical partition function for N interacting molecules, that is, the classical limit of Eq. (A.2), takes the form
D.A. McQuarrie, Statistical mechanics (University Science Books, 2000)
(1) For a detailed discussion of this subject see, e.g., D.A. McQuarrie, Statistical mechanics (University Science Books, 2000).
(2) The inequality cannot hold as n increases; but, by the time n is large enough to contradict this, the terms are so small that they make no contribution to the sum