Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
5. The Transition from Microscopic to Macroscopic: Statistical Thermodynamics
As indicated in Chapter 1, in liquids a natural reference state is absent because the structure is highly irregular. Therefore, in the description of liquids and solutions, we essentially will need statistical thermodynamics, which provides the link between the microscopic considerations and concepts to the macroscopic – in particular thermodynamic – aspects. In this chapter we introduce and outline statistical thermodynamics in a concise way. First, we deal with the basic formalism and apply that to noninteracting molecules. Thereafter, the effect of intermolecular interactions is introduced, leading to the virial description. While being a reasonable approach for low- and medium-density fluids, the virial approach does not work for liquids. The reasons for this are briefly discussed.
5.1. Statistical Thermodynamics
While classical or phenomenological thermodynamics provides relations between macroscopic properties, statistical thermodynamics relates the macroscopic properties of a system (set of molecules) to the microscopic phenomena – that is, to molecular behavior.
5.1.1 Some Concepts
Let us try to describe a system by classical mechanics (CM). If the system is relatively small, say a single molecule or particle, one may proceed as follows. In classical mechanics the system is described by Hamilton's equations (see Section 2.2). Each system can be characterized by a set of n generalized coordinates (degrees of freedom) and the associated momenta. A state of a system thus can be depicted as a point in a 2n-dimensional (Cartesian) space whose axes are labeled by the allowed momenta and coordinates of the particles of the system. This space is called the (molecule) phase space or μ-space. Example 2.2 provides a possible choice of generalized coordinates for the water molecule. If we have a system containing many (not or weakly interacting) particles, such as molecules in a volume of gas, a collective of points describes the gas in μ-space, each point describing a molecule. Similar considerations hold in quantum mechanics (QM), which yields the energy levels and associated wave functions of individual quantum systems. However, in many cases we need the time average behavior of a macroscopic system. There are four problems to this:
· The size of the macroscopic system, which contains a large number of particles. The large number of degrees of freedom present renders a general solution (both in CM and in QM) highly unlikely to be found.
· The initial conditions of such systems are unknown so that, even if a general solution was possible in principle, a particular solution cannot be obtained.
· Even if the initial conditions were known, they have a limited accuracy. Note that, for example, Avogadro's number is “only” known with an accuracy of 10−7. Since it appears that the relevant equations are extremely sensitive to small changes in initial conditions, this rapidly leads to chaotic behavior.
· It is difficult to incorporate interaction between the molecules and the interaction with environment.
To overcome the aforementioned problems, in 1902 Gibbs made a major step and used the ensemble – a large collection of identical systems with the same Hamilton function but different initial conditions. By taking a 2nN-dimensional space, where N is the total number of particles in the system, we can make a similar representation as before. The axes are labeled with the allowed momenta and coordinates of all the particles. This enlarged space is called the (gas) phase space or Γ-space. To each system in the ensemble there corresponds a representative point in the 2nN-dimensional Γ-space. Since the system evolves in time, the representative point will describe a path as a function of time in Γ-space, and this path is called a trajectory. From general considerations of differential equations it follows that through each point in phase space (or phase point) can pass one and only one trajectory so that trajectories do not cross. The ensemble thus can be depicted as a swirl of points in Γ-space, and the macroscopic state of a system is described by the average behavior of this swirl. Since the macroscopic system considered has a large number of degrees of freedom, the density of phase points can be considered as continuous in many cases; this density is denoted by ρ(p,q), where p and q denote the collective of momenta and coordinates, respectively. Obviously, ρ(p,q) ≥ 0 and we take it normalized, that is, ∫ρ(p,q)d p d q = 1. One can prove from the equations of motion that this density is constant if we follow along with the swirl, a general theorem called Liouville's theorem. A function F(p,q) in phase space representing a certain property is called a phase function, the Hamilton function providing an important example. The time-average of a phase function is given by
However, as stated before, this average cannot be calculated in general since neither the solution p = p(t) and q = q(t) nor the initial conditions p = p(0) and q = q(0) are known. To obtain nevertheless estimates of properties the phase-average
is introduced. The assumption, originally introduced by Boltzmann in 1887 for μ-space and known as the ergodic theorem, is now that
and essentially implies that each trajectory visits each infinitesimal volume element of phase space. This assumption has been proven false but can be replaced by the quasi-ergodic theorem, proved by Birkhoff in 1931 for metrically transitive systems1). It states that trajectories approach all phase points as closely as desired, given sufficient time to the system. In practice, this means that we accept the equivalence of the time- and phase-average. For details (including the conditions for which the theorem is not valid) we refer to the literature.
5.1.2 Entropy and Partition Functions
We will develop statistical thermodynamics using a quantum description, since that approach is conceptually simpler, and in a later stage make the transition to classical statistical thermodynamics. However, the terminology of classical statistical mechanics is frequently used in the literature, even when dealing with quantum systems.
The first question is how to characterize a macroscopic state in terms of molecular parameters. It might be useful to recall that a macroscopic system is considered to possess a thermodynamic or macro-state, characterized by a limited set of macroscopic parameters, such as the volume V and the temperature T. However, such a system contains many particles, for example, atoms, molecules, electrons and photons. A macroscopic system is thus also a quantum system – that is, an object that contains energy and particles and is identified by a large set of distinct quantum or micro-states of the system. Identical particles – for example, the electrons of a molecule – are objects that have access to the same subset of micro-states. From these considerations we conclude that each macro-state contains many micro-states, and describe the macroscopic state of the system by the fraction pi of each possible quantum state i in which the macroscopic system remains. With each quantum state i an energy Ei and a number of particles Ni is associated. It should be clear that the simple label i implicates the complete description of the microstate while the distribution over microstates i describes the macroscopic system, and thus the whole hides a considerable complexity.
Further, it should be clear that the number of quantum states increases rapidly with increasing energy. For large systems, the states of the system will form a quasi-continuum that can be described by the density of states g, indicating the number of quantum states dn for a certain energy range δE, that is,
where δE represents the so-called accessibility range, that is, an energy range that is small as compared to E but large when compared to the Heisenberg uncertainty ΔE. For a macroscopic system g is astronomically large, is related to the temperature (see Problem 5.4), and increases extremely rapidly with the size of the system (see Example 5.1).
Example 5.1: The density of states for translation*
First, we calculate the volume G of a hypersphere with radius R and of dimension D, given by G = ARD = A(R2)D/2. To calculate A we consider the integral
with Γ(t) the gamma function. The volume can also be written as G = ∫ h(R2−xTx) dx with h(t) the Heaviside function and . So, dG/dR2 = ∫δ(R2−xTx)dx where the Dirac function δ(t) = dh(t)/dt is used. For the functions used, see Appendix B. Therefore, the integral I also becomes
Combining yields A = 2πD/2/DΓ(D/2) = πD/2/Γ(½D + 1). Consider now N particles of mass m in a three-dimensional box with edges a so that the energy E can be written as where n = ni covers the set n1x, n1y, … , nNz with ni ∈ (±1, ±2, ±3, … ). The degeneracy of energy levels is given by the number of ways the integer nTn = 8ma2E/h2 can be written as a sum of squares of integers. To estimate the degeneracy, consider a D-dimensional space (D = 3N) with the axes labeled n. The number of lattice points inside the hypersphere with radius squared nTn = 8ma2E/h2, gives the number of energy levels. Equating nTn to R2, we have
with D = 3N. The final step is to realize that we need only one “octant” of the hypersphere, so we have to divide by 2D, and that the particles are indistinguishable, so we have to divide by N! (see Section 5.2). The final result for the volume G and density of states g(E) thus becomes, respectively,
For N = 1, using E = 3kT/2, T = 300 K, m = 10−22 g, and a = 10−2 m, we obtain g ≅ 1030, which is a large number. For N particles, using again T = 300 K, m = 10−22 g and a = 10−2 m, but now with N = 6 × 1023 and E = 3NkT/2, we obtain g ≅ 10N, which is an extremely large number. Finally, the expression for g shows that it increases extremely rapidly with the number of particles in the system.
Since in thermodynamics equilibrium is reached at maximum entropy S(U,V,N), given energy U, volume V and number of particles N, it seems natural in statistical thermodynamics to start with defining the entropy. For the statistical representation of S various expressions can be given2). Since the macroscopic system is characterized by the pis, we will have S = S(pi;U,V,N), or S(pi) for short. Probably the most direct way to obtain S for a system having n states is to require the following properties:
· S(pi) ≤ S(1/n). This statement implies that equi-probability for states with the same energy yield the maximum entropy and thus the equilibrium state.
· S(pi,0) = S(pi). This statement requires that if a state cannot be occupied it will not contribute to the entropy.
· S = SA + SB for noninteracting systems A and B. This condition delivers the usual additivity of entropy. For interacting systems the condition changes to , where denotes the conditional entropy of system B, that is, the entropy of system B given the entropy of system A.
These requirements have a unique solution (see Justification 5.1) apart from a multiplicative constant. This constant is chosen in such a way that the statistical entropy thus defined corresponds to the conventional thermodynamic entropy, and this constant appears to be Boltzmann's constant k. The entropy S, sometimes called Gibbs entropy, is then defined by
Justification 5.1: The Gibbs entropy*
To show that, given the three properties indicated above, Eq. (5.5) represents the unique solution for the entropy, we argue as follows . First, as an abbreviation, we set
Using properties 1 and 2 we have
so that L(n) is a nondecreasing function of n. Now, taking m and r as positive integers, consider m mutually independent systems A1,A2, … ,Am, each containing r equally likely states. So, we have S(Ak) = S(1/r, … ,1/r) = L(r) for 1 ≤ k ≤ m. Using property 3, since the systems are independent, we have
Since the combined (product) system A1A2 … Am contains rm equally likely events, we have for its entropy also L(rm) and therefore
This relation holds for any other pair of integers m and r. As a next step, take three arbitrary integers r, s and n and a number m determined by
Since L(n) is monotonous we have also
By combining Eqs (5.6) and (5.7) we have
Equation (5.8) is independent of m, and since n can be arbitrarily large we obtain
which implies, since r and s are arbitrary, that L(n) = λ lnn, with λ ≥ 0 a non-negative constant as L(n) is monotonous.
So far, the special case pi = 1/n (1 ≤ k ≤ n) is proved and now we turn to the case where pk can be any rational number. Suppose we have pk = gk/g with g = Σkgk where the gk are all positive integers. Further suppose that we have a system A consisting of n states with probabilities p1,p2, … ,pn. We also have a specially devised system B, dependent on A and consisting of n groups of states such that the kth group contains gk states. If state Ak is realized in system A, then in system B all the gk events of the kth group have the same probability 1/gk, while all the events of the other groups have probability zero. In this case the system B reduces to a system of gk equally probable states with conditional entropy . This implies that
We turn now to the product system AB containing AkBl states (1 ≤ k ≤ n, 1 ≤ l ≤ g). A state in this scheme is only possible if Bl belongs to the kth group, so that the number of possible states AkBl for a given k is gk. The total number of states in system AB is then Σkgk = g, and the probability of each possible state AkBl is pk/gk = 1/g. Thus, the system AB consists of g equally likely states and thus S = L(g) = λ lng. We now use property 3, S = SA + SB(A), to obtain
Since the entropy must be continuous, Eq. (5.9) is not only valid for the rational numbers p1,p2, … ,pn but for any value of its arguments. Note that, although we used a specially devised system B, the result is independent of system B. Finally, the constant λ can be identified with Boltzmann's constant k by calculating the pressure for a perfect gas, Eq. (5.33). This leads to λ = k, which we use from now on.
A short route to obtain the relevant thermodynamic expressions runs as follows. We assume an open system3), that is, a system of volume V to be in contact with a thermal bath characterized by the temperature T, and a particle bath characterized by the chemical potential μ4). Alternatively, we may assume that the system is in contact with many other, similar systems. For such a system we may assume that we are always near equilibrium. The probabilities of the system ought to be normalized5), that is,
In addition, we require that the average energy of the system6) is constant, that is,
and that the average number of particles 〈N〉 of the system is constant, that is,
We use the expression for the entropy considered before, S = −kΣi pilnpi, and for this expression we seek, conform thermodynamics, the maximum given the above-mentioned constraints. This maximum can be obtained by using the Lagrange method of undetermined multipliers (see Appendix B). We take these multipliers as −kα, −kβ, and −kγ. The maximum is now obtained from
and after some calculation this expression leads to the solution
The normalization condition, Eq. (5.10), yields
so that we may define
For the probabilities we thus obtain
The function Ξ is the macro-canonical or grand canonical partition function, often just called the grand partition function. We find for the entropy, meanwhile using the normalization condition ,
We compare this expression with the thermodynamic expression for the entropy
as solved from U = TS − PV + μN (see Section 2.1) with T, μ, and P as before, and U the internal energy, N the number of particles, and V the volume. If we identify the average microscopic energy with the internal energy U, the average number of particles 〈N〉 with the macroscopic number N, we obtain
so that we finally have the solution
Since we are only using the values of pi corresponding to the maximum entropy, in the sequel we omit the asterisk and just write pi. For future reference, we also write the expressions7) for pi and Ξ as
where λ is the absolute activity (see Section 2.1). The grand partition function is directly related to the grand potential Ω, since we have −PV = Ω and thus Ω = −kT ln Ξ. The thermodynamic properties can be calculated in the usual thermodynamic way by realizing that the natural variables for Ξ are T, V and μ, that is, Ω = Ω(T,V,μ). Hence, we have S = −∂Ω/∂T, P = −∂Ω/∂V, and N = −∂Ω/∂μ.
If the number of particles is fixed at, say N, we have a closed system, the constraint characterized by γ is removed, and one can take γ = 0. Therefore,
Here, ZN denotes the canonical partition function for N particles, often labeled as just Z and addressed as the partition function. For the entropy we have in this case
which, upon comparison with
results again in β = 1/kT, as expected, and F = − kT ln Z. From the Helmholtz energy F we obtain the internal energy U, the entropy S, pressure P, and chemical potential μ
We note that using ZN we can write Ξ = ΣNλNZN, where the original index i has been replaced by the pair (N,j) with j the index used in ZN.
Finally, if also the energy is fixed, we have an isolated system, the constraint characterized by β is also removed and one can take β = 0 as well. In this case the probabilities pi reduces to the micro-canonical distribution
where W = Σi(1) is the number of accessible states for the macroscopic system at fixed energy E8). In this case, the entropy S(pi) = −k Σi pi lnpi becomes the Boltzmann relation
Writing this expression as −TS = −kT ln W renders the expressions for the relation between the potentials (Ω = −PV = U − TS − μN, F = U − TS, −TS) and the partition functions (Ξ, Z, W) all similar. Once having calculated the pis, the mean or average value for any property X of the system can be calculated using
where Xi denotes the value in microstate i. This expression is valid for all three distributions discussed. Note that adopting this view we can write for the entropy S = −k Σipilnpi = −k<lnp>.
To show that the entropy just defined (for all situations) can be identified as the thermodynamic entropy (defined for equilibrium situations only), we calculate dS for a closed system using lnpi = −Ei/kT − lnZ and obtain
Here, we have used the definition of the Boltzmann distribution and twice the fact that ∑i dpi = 0. Hence, for reversible heat flow we have δQ = T dS = Σi Ei dpi. The total energy of the system of interest and its differential d. From thermodynamics we know that the internal energy U is given by the first law of thermodynamics dU = δW + δQ, where δW and δQ denote the increment in work and heat, respectively. If we again identify U with , we can associate Σi pidEi with δW since Σi Eidpi corresponds to TdS = δQ. Finally, from Eq. (5.21) one easily can show that ∂S/∂U = 1/T (see Problem 5.1).
One property not discussed so far is that, for the thermodynamic entropy S, we have dS ≥ 0 for an isolated system. One can show that for the statistical entropy we also have dS ≥ 0 always (see Justification 5.2). Finally, one might wonder what the partition function represents. In fact, it just provides the average number of states that is thermally accessible for the system. This is particularly clear for the micro-canonical ensemble (S = k lnW).
Justification 5.2: dS ≥ 0 always*
One property of the thermodynamic entropy is that for arbitrary processes it will increase unless the process is reversible. To prove that our statistical entropy behaves similarly, we first have to make the meaning of micro-states somewhat more precise. In fact, for any macroscopic system it is impossible to obtain the exact eigenstates of the system. Moreover, even if we would know them at a certain moment, soon afterwards, because of the interaction between the particles in the system, they would be unknown. For example, for an ideal gas the eigenstates are the described by the momentum eigenfunctions of structureless point particles. During equilibration the particles collide and exchange momentum. The uncertainty in energy E thus becomes ΔE ≥ ħ/2Δt, where Δt is the mean time between collisions. As indicated in footnote 7, experimentally we are limited to δEwith δE >> ΔE. Hence, generally we have groups of approximate eigenstates associated with a small energy range, the accessibility range δE, small as compared to E but large as compared to the Heisenberg uncertainty ΔE and in which the density of states is essentially constant. The probabilities pi therefore refer to these approximate states and, when δE is small enough, can be considered as constant within the accessibility range. It then makes no difference whether we consider transitions from exact state i to exact state j as restricted by ΔE or transitions from one approximate state (group) i to another state (group) j as restricted by δE: in either case, the value of dpi/dt will be the same. This assumption is equivalent to the ergodic assumption as it also realizes mixing of all states. Sometimes it is referred to as the accessibility or equal a priori probabilityassumption.
To discuss equilibration we need the jump rate νij from state j to state i. For the rate at which a state i with probability pi is changing, we have two contributions. First, we have the contribution from state i to any state j, given by Σjνji pi and, second, the contribution from any state j to state i, given by Σjνij pj. Quantum mechanics tells us that νij = νji, that is, the principle of jump rate symmetry (see Section 2.4), and therefore the total rate becomes dpi/dt = Σjνij(pj − pi), referred to as the master equation.
We now consider the entropy differential dS/dt = −k Σilnpi (dpi/dt). For an isolated system we have for the effect of jumps between two particular states i and j a contribution to dpi/dt reading νij(pj − pi) as well as a contribution νij(pi − pj) to dpj/dt. The total rate is thus, taking in to account all jumps, dS/dt = kΣi,jνij(pj − pi)(lnpj − lnpi). Since the terms in brackets have the same sign, dS/dt cannot be negative. Moreover, if pi = pj, dS/dt = 0, and so we conclude that always dS ≥ 0. We refer to the literature for a further discussion of the foundations of statistical thermodynamics .
We note that the difference between the results from the grand partition function Ξ and the partition function Z is usually small for macroscopic systems, that is, fluctuations in the number of molecules and therefore energy, are generally not terribly important. This can be seen as follows. The grand partition function is
where ZN is the canonical partition function for N particles, so that the probability to have N particles in the system is
Hence, it follows that
Noting that since 〈N〉 = ∂lnΞ /∂lnλ, differentiation of 〈N〉 with respect to μ yields
where the variance σX2 ≡ 〈X2〉 − 〈X〉2 is used. We also have
Since the derivative (∂N/∂P)V,T is at constant V, we have (∂N/∂P)V,T = V[∂(N/V)/∂P]V,T = V[∂(N/V)/∂P]N,T = NV[∂(1/V)/∂P]N,T = NκT, where the second step can be made because N/V is an intensive quantity. Therefore, we have . Hence, the relative fluctuation in number of molecules in the system is given by
and therefore for large N is negligible. The fluctuation in energy for an open system can be derived similarly, although the process is somewhat more complex, and we quote (leaving the derivation for Problem 5.6)
Show, using Eq. (5.21), that ∂S/∂U = 1/T.
Verify from the thermodynamic expression for dU and the statistical expression for dS that δW in the grand canonical ensemble is given by
Problem 5.3: The two-state model
A very simple model in statistical mechanics is the two-state model with (obviously) two states, state 1 with energy 0 and state 2 with energy ε.
a) Give the expression for the partition function Z.
b) For temperature T = ε/k, where k is Boltzmann's constant, calculate the occupation probabilities of state 1, p1, and of state 2, p2.
c) Calculate the internal energy U.
d) Calculate the heat capacity CV and sketch the behavior of CV(T).
Problem 5.4: Temperature and the density-of-states
Show that the temperature T is related to the density of states g(E) for a system with energy Es via .
Show in detail for macro-systems, although we have S = g(E)δE, that S is essentially independent of δE. Consider to that purpose the value of δE based on the quantum uncertainty relations and a typical experimental uncertainty.
Derive for a closed system in a way similar as the relation (σN/<N>)2 = kTκT/V is derived for the open system. For an open system, show that .