6.1 The initial state of the universe

We start at the beginning – or shortly after the beginning. Immediately after the Big Bang, the universe was in a rather uniform state in which the density of energy exceeded the density of matter. The rather (though not completely) uniform density corresponded to a much more ordered state of the universe than the one we observe now: an extremely inhomogeneous clumping of atoms into molecules and compounds and planets and stars (as well as plants and animals), and stars into galaxies, and galaxies into galactic clusters. From this highly ordered initial state sprang the natural tendency for systems to develop from whatever state they are in to less-ordered states. This is a tendency, not an irrefutable law, and work can be done to overcome this tendency. Experimentalists can use lasers to create initial states that will develop and return to their initial ordered state. But the recurrence time in these so-called ‘spin-echo’ experiments is extremely short and the number of atoms that will cooperate decays rapidly with time. Thus, even with the most care and best lasers, such an exceptional state of affairs can only be maintained for a fraction of a second.

In order to understand the tendency of isolated systems to equilibrate – to understand how systems evolve chemically – we need to understand the physics that controls and describes the interactions between atomic and subatomic particles. But even this is not enough. We need to understand that the material world around us is composed of microscopic atoms and molecules. Further, there are an enormous number of these molecules. A mere 10 ml of liquid water contains on the order of 10²³ water molecules. Their microscopic nature (in particular their minute masses and distances between them) means that they do not always play by the rules of the macroscopic world. We will discuss much more about that when we treat quantum mechanics, but for now we concentrate on the effect of the enormous number of molecules.

The sheer number of molecules in a macroscopic sample leads to emergent behavior. The equations that describe the dynamics of individual atoms and molecules are symmetric in time and space. Nonetheless, we know that samples made up of atoms and molecules irreversibly evolve toward equilibrium. In addition, even if we have a complete description of the properties and dynamics of one molecule, there is no way for us to describe the phase transitions that a collection of these molecules undergoes. Both, the irreversible approach to equilibrium and phase transitions are examples of emergent behavior: behavior that cannot be predicted from the properties of one molecule but that emerges from the interactions of collections of molecules.

6.2 Microstates and macrostates of molecules

In order to derive the macroscopic behavior of matter from the microscopic properties of molecules, we need to derive the principles of statistical mechanics. Statistical mechanics starts with the conception of matter as being composed of N particles occupying volume V. This is not how thermodynamics was founded but, as we shall see, it allows us to derive the principles of thermodynamics on a molecular basis. The thermodynamic state of these N particles is called a macrostate. The macrostate is a description of the system in terms of collective thermodynamic parameters. We choose our macrostate to be isolated, that is, its internal energy U is held constant, as is N. To specify our system fully, we need only identify its composition in addition to N, V, and U. Should it be more convenient, we can use the methods of thermodynamics to specify our system in terms of other variables, such as N, the pressure p, temperature T, and composition. We will develop these thermodynamic methods in subsequent chapters.

If we want to know details of the individual particles – let us call them molecules though they may be molecules or atoms – then we need to define the microstate of the system. The microstate of a system is the description of the system in terms of the characteristics of each individual molecule. Each of the N particles is described quantum mechanically by a wavefunction ψ_k, and the total wavefunction of the system ψ contains a product of all the molecular wavefunctions ψ₁ ψ₂ …ψ_N (it may also contain some other terms to correctly account for the statistics of the system, as we shall see below). At this point we do not need to define what the wavefunction is mathematically. Each individual wavefunction is simply a mathematical function that describes a molecule.

At any finite temperature the molecules are constantly changing their states. The molecules collide, and these collisions change the velocities as well as the rotational and vibrational states of the molecules. Nonetheless, the number of molecules, their internal energy, as well as the volume remain constant. Thus, there is a large number Ω of microstates that correspond to the macrostate defined by N, U, and V. The fundamental postulate of statistical mechanics is the Gibbs postulate:

All possible microstates of an isolated assembly are equally probably.

Being a postulate, there is no general proof of this statement, though proofs for specific cases exist. Support of this postulate is derived from the vast number of correct results derived from it.

To describe our system more specifically, we need to know whether the particles are distinguishable or indistinguishable. This makes a difference in the statistics of the system. A one-component gas contains molecules that are identical and indistinguishable. All molecules are able to explore the entirety of the container, and there is no way to label individual molecules. Molecules arranged in a solid lattice are also identical, but because each is fixed to a lattice site, they can be labeled and distinguished.

We allow the N particles to equilibrate in a fixed volume V. In order to equilibrate, the molecules must interact; however, these interactions are not so strong as to interfere with identifying the molecules as individual particles. No reactions occur and for now we ignore any intermolecular interaction energy (essentially we will think in terms of ideal gases rather than real gases). Each of the N particles has an energy ϵ_j. Anticipating the results of quantum mechanics, we say that there is a level (state) that corresponds to each allowed value of energy and that each molecule occupies one of these levels. The total occupancy of level j is called the population n_j. Another term for population is the occupation number. The total number of particles N is given by the sum of energy level populations over all allowed states

(6.2)

and the total internal energy is the sum of the energies of all molecules

(6.3)

For any collection of N particles there are different ways for the n_j particles to populate the ϵ_j energy states. That is, there are numerous microstates described by the different energy arrangements. An example is shown in Fig. 6.1. The number of ways that N distinguishable molecules can be distributed into j energy levels is given by:

(6.4)

The total number of microstates Ω is the sum of t(n) for all allowed sets of populations that have a total internal energy U according to

(6.5)

The sum is over all distributions satisfying the constraint on internal energy. The total number of microstates is sometimes referred to as the weight.

What types of distribution satisfy the internal energy constraint? Consider a system with N = 3 and levels with energies ϵ₀ = 0, ϵ₁ = ϵ, ϵ₂ = 2ϵ, ϵ₃ = 3ϵ, …. If the total energy is U = 3ϵ, then we could put all of the particles in energy level ϵ₁. Or we could put one particle in ϵ₀, which requires that the other two go into ϵ₁ and ϵ₂, such that the sum of the energies 0 + ϵ + 2ϵ = 3ϵ. Or we could put two particles in ϵ₀, which requires that the third occupy ϵ₃. If the particles were indistinguishable, these would be the three allowed distributions. However, if the three particles are distinguishable – that is, if we have particles that are identifiable as A, B and C – then we also have to keep track of which particle has which energy. In this case, all of the distributions listed in Table 6.1 are possible.

There are a total of ∑t(n) = 10 microstates in Table 6.1. That is, there are 10 distinct ways to write the populations of the energy levels. We also see that there are three different classes of distributions (call them classes I, II, and III), but that each class has a certain number of permutations of how it can be written. Since each individual microstate is equally probable, the class with six different permutations is the most likely of the three classes to be present. Let us call this maximum value t_max. In this example, there is a 60% chance of finding the macrostate with U = 3ϵ in one of the microstates of class II. As N increases, the concentration of probability into the class that corresponds to t_max increases. For macroscopic quantities of molecules N is of the order of the Avogadro constant (10²³ or so). For such astronomically high values of N, to a very good approximation, we can simply take

(6.6)

Therefore, we can rewrite Eq. (6.5) as

(6.7)

where the factorial in the denominator must be for the distribution that maximized t(n). Maximization is performed by using the method of undetermined multipliers and Stirling’s theorem. The method of undetermined multipliers, also called Lagrange multipliers, can be found in Appendix 1. From it we obtain the useful expression

(6.8)

Stirling’s theorem is used over and over again in statistical mechanics. It is derived in Appendix 2, and allows us to approximate the value of the logarithm of a very large number according to

(6.9)

With these methods we find

(6.10)

We will determine the values of α and β later. The distribution described by Eq. (6.10) is the Boltzmann distribution. It describes the most probable constellation of the macrostate, which is the equilibrium state. This distribution is a set of population values – a set of values of n_j – for which the value of Ω is a maximum. Other distributions are possible. These represent fluctuations away from the equilibrium distribution. In the example above, classes I and III represent fluctuations away from the equilibrium distribution of class II. As shown in Fig. 6.2, these fluctuations become increasingly less likely or, at least, the values of the fluctuations become increasingly small compared to the mean values of thermodynamic parameters as the number of particles approaches 10²³ (a mole of atoms). However, fluctuations are substantial for particles on the nanoscale (<100 nm), and this is one reason why the behavior of nanoparticles can differ from our expectations of macroscopic samples.

Note a subtle difference between the definitions of distribution and microstate. The microstate identifies the state of each molecule. The distribution specifies how many molecules (regardless of their identity) populate each of these microstates.

6.3 The connection of entropy to microstates

In 1872, at a time when the very existence of molecules was still very much in doubt, Boltzmann related the entropy S of a gas to the probability of a gas being in a particular microstate. He did so in terms of the distribution function of molecular velocities f(v_x, v_y, v_z). Planck’s quantum hypothesis – the idea that the energy states of matter are discreet (quantized) levels rather than continuous distributions – allowed Planck to generalize Boltzmann’s results to all thermodynamic systems.

From a thermodynamic viewpoint (as we shall define later), equilibrium in an isolated system is obtained when the entropy is a maximum. Statistical mechanics defines the equilibrium state as the most probable macrostate, the macrostate that corresponds to the maximum number of microstates. Therefore, the link between thermodynamics and statistical mechanics must lie in an equation that links the entropy S to the number of microstates Ω. The link, as derived by Boltzmann and Planck, is the exceedingly compact and important Boltzmann formula:

(6.11)

This equation is so fundamental and so groundbreaking that it is literally etched in stone – it appears on Boltzmann’s tombstone. The constant k_B is the Boltzmann constant.

An a posteriori derivation of Eq. (6.11) is amazingly simple, but only because we know the answer. The deep insight of the work of Boltzmann and Planck should not be underestimated. We start with the idea that there is a ground energy state, a perfect crystal in internal equilibrium. At T = 0, all of the particles must enter this lowest energy state. Thus, each molecule has the same energy and there is only one microstate that describes the system – that is, Ω = 1. This single state is a state of perfect order for which, according to the third law of thermodynamics, S = 0. Note that ln 1 = 0, suggesting but not proving a logarithmic relationship

(6.12)

S is an extensive property. It depends on how much of a system we have. Take two independent assemblies with entropies S₁ and S₂ and corresponding numbers of microstates Ω₁ and Ω₂. The total entropy S of the two systems is given by the sum

(6.13)

The total number of microstates of the combined system Ω is obtained by pairing each of the Ω₁ microstates of system 1 with each of the Ω₂ microstates of system 2. The value of Ω is given by the product

(6.14)

Using the Ansatz that Eq. (6.11) is satisfied by simply multiplying ln Ω by a constant, which we call k_B, we can show that Eqs (6.13) and (6.14) are confirmed. That is, assuming that Eq. (6.11) holds, then on substitution from Eq. (6.14)

(6.15)

we arrive at the result of Eq. (6.13). We will leave it to mathematicians to prove uniqueness. That is, not only are Eqs (6.11), (6.13) and (6.14) consistent, but also the only expression that makes Eqs (6.13) and (6.14) consistent is Eq. (6.11). If the statistical mechanical definition of entropy had been developed first, k_B could have been assigned any arbitrary value. However, to ensure consistency with thermodynamics, its value is set by derivation of the ideal gas law, which we do below, and to satisfy an equation derived from the study of heat, which is shown in Chapter 9 to be a mathematical expression of the second law of thermodynamics,

(6.16)

6.3.1 Example

Calculate the entropy change when a mole of CO molecules in a perfect crystal at T = 0 K is transformed into an ensemble in which the orientation of the CO dipoles flip randomly between parallel and antiparallel alignments.

There are N_A = 6.02 × 10²³ molecules in a mole. A perfect crystal at 0 K has but one microstate, Ω = 1. The randomness introduced by the two orientations of the CO dipole introduces on the order of microstates. Thus, the entropy change is

Note that this value of ΔS is the value for a sample with 1 mole of molecules, the value of the molar entropy change is ΔS_m = 5.76 J K⁻¹ mol⁻¹. This extra entropy that is present due to disorder even at 0 K is known as residual entropy.

6.4 The constant α: Introducing the partition function

The population of level j is given by

(6.17)

The exponential term involving β is called the Boltzmann factor. The total number of molecules is given by the sum of the populations, thus,

(6.18)

The exponential term in the summation is sufficiently important that we give it the name partition function and the symbol Z,

(6.19)

The partition function tells us how the population is partitioned into the various levels in accord with their respective Boltzmann factors. Solving for e^α and substituting from Eq. (6.19) yields,

(6.20)

Thus, the populations of the individual levels are given by

(6.21)

The value of α is, therefore,

(6.22)

Its meaning is related to the chemical potential, which we will define later.

6.4.1 The value of β

Take a system with constant N and V. Now change the internal energy by adding an infinitesimal amount of heat dq. No pV work is done and the change in internal energy is related to the added heat by

(6.23)

The energy level structure does not change at constant volume as shown for example in Eq. (6.57) below. Therefore, dϵ_j = 0 and the heat transfer is related to a change in populations according to

(6.24)

The system will equilibrate by maximizing Ω, which will change according to

(6.25)

Using an expression found during the derivation of the Boltzmann distribution, Eq. (6.8), we substitute for dln Ω/dn_j, which gives

(6.26)

However at constant N, and then upon substitution from Eq. (6.24)

(6.27)

Solving for β and substituting from the second law of thermodynamics Eq. (6.16) dq = T dS,

(6.28)

Then from Eq. (6.11)

(6.29)

Proof that the value of k_B truly is what is known as the Boltzmann constant is actually given unambiguously by the derivation of the equation of state of the ideal gas, which we will perform below.

6.5 Using the partition function to derive thermodynamic functions

For any one arrangement of N identical distinguishable molecules (such as molecules localized at lattice points), the partition function is

(6.30)

This assumes that the energy levels ϵ_j are nondegenerate. We will deal with degeneracy (more than one level at the same energy) in a moment. The localized particles can be arranged in many different ways. For the assembly as a whole, the partition function is

(6.31)

where the values of E_N include all possible values of the total energy of the assembly. E_N is the sum of all values of ϵ and the summation is over all possible values of E_N.

Consider two distinguishable particles A and B, in which case

(6.32)

(6.33)

and

(6.34)

Thus

(6.35)

However, since A and B are identical Z_A = Z_B = Z and Z_A Z_B = Z². This argument is easily generalized to N particles, for which

(6.36)

Until now we have considered the energy levels to be nondegenerate; that is, there is one and only one level at each value of energy. However, it is possible for more than one wavefunction – the quantum mechanical description of an energy level – to have the same energy. If more than one level has the same energy, then we say that the levels are degenerate. Another term for degeneracy is statistical weight, and this gives you an idea of how to account for degeneracy should it occur. If the number of levels at the same energy is g_j, we say the degeneracy of the jth level is g_j and we need to count each degenerate level g_j times.

Including the effects of degeneracy, we can now summarize the results we have derived so far.

(6.37)

(6.38)

(6.39)

(6.40)

(6.41)

(6.42)

(6.43)

To these results derived from statistical mechanics we can now add the following results from thermodynamics, which will allow us to derive all thermodynamic functions directly from statistical (molecular) arguments. These thermodynamic results will be derived in Chapters 7–11.

Helmholtz energy

(6.44)

Gibbs energy

(6.45)

Enthalpy

(6.46)

Entropy

(6.47)

Pressure

(6.48)

Temperature

(6.49)

6.5.1 Example

Take the derivative of the Helmholtz energy at constant volume with respect to temperature and derive an equation that relates it to the entropy.

Taking the derivative of Eq. (6.44), we obtain

(6.50)

The definition of the heat capacity at constant volume is

(6.51)

therefore,

(6.52)

From the second law of thermodynamics, Eq. (6.16), we know¹ that

(6.53)

The heat exchanged along a reversible path at constant volume is dq_{V, rev} = C_V dT. Upon substitution into Eq. (6.53), we obtain

(6.54)

Therefore

(6.55)

For purists, having specified the path as reversible and isochoric, we can equate dS/dT with (∂S/∂T)_V. Substituting for C_V in Eq. (6.52), yields the desired expression.

(6.56)

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