422
J. Phys. Chem. 1996, 100, 422-432
Length Scale for the Constant Pressure Ensemble: Application to Small Systems and Relation to Einstein Fluctuation Theory Ger J. M. Koper and Howard Reiss* Department of Physical and Macromolecular Chemistry, Leiden UniVersity, Gorlaeus Laboratories, P.O. Box 9502, 2300 RA Leiden, The Netherlands ReceiVed: June 30, 1995; In Final Form: September 29, 1995X
In this paper we address the constant pressure ensemble and the volume scale that must be introduced in order to represent the corresponding partition function as a dimensionless integral. The volume scale or length scale problem arises quite generally when it is necessary (for whatever reason) to apply semiclassical statistical mechanical theory in configuration space alone, rather than in the full phase space of the system. We find that the length scale, derived by earlier workers concerned primarily with systems in the thermodynamic limit, is not suitable for application of the constant pressure ensemble to small systems such as clusters in nucleation theory or mesodomains in microemulsion theory. We discuss some of the well-known deficiencies of the conventional representation of the constant pressure ensemble and some which are not so well-known. Also the close connection between the constant pressure ensemble and Einstein fluctuation theory is emphasized, and we clarify the two types of fluctuation that are relevant to both developments but which are not always understood and distinguished by workers in the field. We derive the proper length scale applicable to systems of any size and remark that when it is used for small systems, the constant pressure ensemble partition function can no longer be derived from that for the canonical ensemble by simple Laplace transformation. We emphasize the fact that although the constant pressure ensemble has only found modest application in the statistical thermodynamics of macroscopic systems, it is being increasingly applied in the theory of small systems that may be conceptual rather than real, and that, for this reason, the ensemble should be placed on a firm fundamental foundation. In particular, we illustrate the relevance of the so-called “shell molecule”. Finally we apply our development to fluctuations in small systems to illustrate the qualitative and quantitative differences between small and large systems.
1. Introduction The constant pressure ensemble was suggested in 1939 by Guggenheim1 largely on the basis of analogy and the apparent correctness of the thermodynamic relations that could be derived from it. In spite of the operational success of the ensemble, its specification contained some disturbing features. Not the least of these was the fact that the relevant partition function involved a sum over an unspecified set of discrete volumes (“eigenvolumes” in the language of Guggenheim). Since the volume of a system is regarded as a continuous variable, this represented a situation that was physically unacceptable in the absence of the precise specification of the volumes, reinforced by an underlying rationale. In later years, Hill,2 Munster,3 Byers Brown,4 and Sack5 examined this problem as part of a more ambitious investigation dealing with the so-called generalized ensemble. Every ensemble is constrained by a characteristic set of thermodynamic parameters, and it was the goal of these authors to provide a universal expression for a partition function that could be adapted, by means of a simple well-defined procedure, to the choice of any thermodynamically acceptable set of parameters. Their approaches, at least in the semiclassical limit, were reminiscent of the Gibbsian method based on the examination of the properties of classical phase space,6 but they were also interested in quantal systems. In general, their considerations were limited to systems in the thermodynamic limit, and a * To whom correspondence is to be sent. Permanent address: Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, CA 90024-1569. X Abstract published in AdVance ACS Abstracts, December 1, 1995.
0022-3654/96/20100-0422$12.00/0
review of the literature gives the impression of a cooperative evolutionary procedure in which later authors criticized noncatastrophic errors in the work of earlier authors with the help and acquiescence of those earlier authors. One of the goals of their work was the placement of systems constrained by intensive variables on a better foundation. The grand ensemble furnishes an example, but a relatively clear cut one, since the extensive variable conjugate to chemical potential (one of the intensive variables of the ensemble) is particle number, a discrete integer quantity. In contrast, one of the intensive variables of the constant pressure ensemble is pressure, whose conjugate is volume, a continuous variable that increases the complexity of the physical analysis. Nevertheless, the constant pressure ensemble was the subject of considerable attention. Hill2 showed that, in the thermodynamic limit, the use of a sum over discrete unspecified volumes could lead to only a completely negligible error. In the same limit, Sack5 was able to derive a volume scale, in a sense a quantum of Volume, which allowed him to replace the sum over discrete volumes by a integral over continuous volumes and to represent the constant pressure ensemble partition function as a Lapalce transform of the canonical ensemble partition function. Still, in recent years, workers have not seemed to realize the importance of the volume scale and have opted to simply replace the sum by an integral, thereby arriving at an unenviable result in which the partition function acquires a dimension, namely that of volume.7 The proper use of the volume scale would correct this deficiency. Until recently, the constant pressure ensemble has not found widespread use in statistical thermodynamics, and its sometimes poorly defined representation has not been viewed as a serious © 1996 American Chemical Society
Constant Pressure Ensemble problem. However, in the past few years, the ensemble and variations of it have become important in the theory of small systems, e.g. of “clusters”,8 microemulsions,9 etc. This indicates that it is of value to place the constant pressure ensemble on firm ground, even if its principal application involves entities, e.g. single small clusters, on which direct measurements are not possible but for which theoretical analysis is imperative. It turns out that the above-mentioned work on the generalized ensemble cannot be rigorously applied to small systems. For example, we shall show, among other things in this paper, that for a small system the constant pressure ensemble cannot be obtained from the canonical ensemble by the simple device of Laplace transformation. The developments in refs 2-5, although they appear to lead to rigorous results, are primarily devoted to the global problem of the generalized ensemble. As such, they do involve some new axioms and are couched in somewhat abstract mathematical language. Consequently, weaker arguments are not clearly seen and the focus on a particular ensemble is not easy to implement. Also, as we have indicated, the results are limited to macroscopic systems. In this paper we extend the theory to small systems, but we also relinquish the goal of treating a generalized ensemble. In this way, we are able to avoid a “global” Gibbsian type argument that explores the features of phase space and concentrate on a more specialized (and more concrete) approach involving a “hands on” thermodynamic model. Hopefully the argument will then be transparent to a wider audience. Furthermore, we shall consider fluctuations and identify two classes of fluctuation that we denote as type 1 and type 2, respectively. These types are connected with two sorts of partition functions, defined by Sack5 as cumulatiVe or differential, but we prefer the type 1 or 2 classification, since fluctuations can be considered without invoking partition functions directly. Indeed Einstein, in his famous paper10 on fluctuations, did not resort to partition functions. Einstein’s approach, although eighty-five years old, really contains the germ of the approach that will allow the constant pressure ensemble for small systems to be placed on a firm physical foundation. A discussion of his method and some of the ideas surrounding it therefore serves as an excellent entry to the method that we shall use. We therefore discuss Einstein’s work in somewhat descriptive terms in the following section. 2. Einstein Theory and Two Kinds of Fluctuation In 1910 Einstein introduced a theory of fluctuations10 designed to deal with quasimacroscopic systems and involving an approximation whose error declined as the system was enlarged toward the thermodynamic limit. In his paper Einstein concentrated on volume or density fluctuations (although the theory was, in principle, more general), and it is convenient to discuss his ideas in terms of such fluctuations, but we shall not proceed exactly in the manner of his paper. Thus consider a system, e.g. a fluid, confined to a container of fixed volume V. There is always a chance that all of the molecules in the system would be found instantaneously in a volume V′ smaller than V. Denote this chance by P(V′|V) and note that P(V|V) is not the simple sum of P(V′|V) over all V′, since the “events” corresponding to different V′ are not mutually exclusive. In fact, P(V′|V) is clearly the probability that the system (i.e. the fluid) will be found to occupy a volume no larger than V′ and not the probability that it will occupy precisely that volume. Notice that if we wanted to force this condition on the system, we could easily do so by merely reducing the volume of the container from V to V′, i.e. by controlling the thermodynamic variable V. On the other hand, if we wanted to assure that the volume was
J. Phys. Chem., Vol. 100, No. 1, 1996 423 no smaller than some volume V′, we would literally have to provide “handles” on each of the molecules in the system. Although we could imagine such a detailed method of control, it could hardly be considered thermodynamics. The system itself might impose a lower limit on the volume, for example when the repulsive cores of the molecules overlap, but this limit is not subject to macroscopic thermodynamic control. Einstein theory evolves from a concern with fluctuations (of the above type) that can be forced into existence by a thermodynamic (one sided) control, i.e. by a constraint corresponding to a thermodynamic variable like volume that only bounds the system on one side. In the thermodynamic limit, however, the one-sided bound will pretty much determine the state of the system. For instance in the case of volume an upper bound will pretty much determine the volume, since fluctuations are relatively so small. The hallmark of the Einstein theory is that the probability of such a fluctuation, in say an isothermal system, is given by the exponential of the reversible work (divided by kT) that the constraint must perform in order to force the fluctuation into existence. For example, in the above case involving volume, Einstein theory would give
{
P(V′|V) ) exp -
∫V′Vp(V′′) dV′′}
1 kT
(2.1)
where k is the Boltzmann constant, T is the temperature, and p is the pressure of the system. P(V′|V) is not a normalized probability density, but P(V|V) is clearly equal to unity, and P(V′|V) is the probability of a bounded fluctuation, i.e. the probability that the system will occupy a volume V′ or less. For convenience, we denote this type of fluctuation as a type 1 fluctuation. We can ask for the probability Φ(V′|V) dV′ that the occupied volume lies between V′ and V′ + dV′, which is simply given by
Φ(V′|V) )
dP(V′|V) dV′
(2.2)
Evaluation of the derivative in this equation, making use of eq 2.1, yields
Φ(V′|V) )
{
∫V′Vp(V′′) dV′′} )
p(V′) 1 exp kT kT
p(V′) P(V′|V) kT (2.3)
We denote the more limited fluctuation (confined to the range V′ to V′ + dV′) as a type 2 fluctuation. It is not a fluctuation that could be controlled by thermodynamic variables, since we would literally require handles on every molecule in order to constrain it into existence. Note that since P(V′|V) is unity, eq 2.2 assures that Φ(V′|V) is normalized. The two types of fluctuation, though not always distinquished and enunciated (and not defined by Einstein), are recognized by most workers in the field, although some of the more subtle consequences of the distinction are not always fully appreciated. The integral in the exponent of eq 2.1 represents the volume work that must be expended in order to further limit the system, already constrained to a volume equal to V, to a volume equal to V′. In the thermodynamic limit this is the only work that need be considered, since surface effects (and therefore changes in surface area, etc.) are entirely negligible. Thus the integral in eq 2.1 is the proper measure of the relevant reversible work. However as the system is reduced in size, corrections to this work are obviously necessary. Einstein did not really address this issue and characterized his theory as limited to systems that are essentially macroscopic.
424 J. Phys. Chem., Vol. 100, No. 1, 1996
Koper and Reiss
In eq 2.3 the preexponential factor p/kT confers the proper dimensions on Φ, and its reciprocal represents some sort of natural volume scale (a quantum of volume) that resolves states of different volume within the spatial continuum. This concept will be further illuminated when we discuss the constant pressure ensemble. For future reference when dealing with the constant pressure ensemble, we note that p depends on V and is the equilibrium pressure within the system, although, of course, in the performance of reversible work the external pressure must equal that within the system. Suppose that, instead of considering a system limited to a volume V, we consider an isothermal system in a container of variable volume but subject to a constant external pressure p. Under this condition, the system will occupy a mean (equilibrium) volume V h , about which its actually occupied volume will fluctuate. If the system is macroscopic, these fluctuations will be very small. Taking a hint from eq 2.3, we might then ask for the probability Ψ(V) dV that the occupied volume will fluctuate into the range V to V + dV and write this function as
Ψ(V) dV ) f(V) P(V|V h ) dV
(2.4)
where f(V) is again the reciprocal of some quantum of volume, chosen to render eq 2.4 correct. The important difference with eq 2.3 is that here the occupied volume V can be both smaller and larger than the mean value V h . Nevertheless, the exponential factor P(V|V h ) is the same as in eq 2.3. Einstein arrived at the above expression by considering a very large total system of constant volume in which the volume under consideration was that of one of the many systems into which the total system was subdivided. Although Einstein did not make an issue out of it, one could well imagine that such a subsystem would be subject to a constant external pressure. Einstein expanded the Helmholtz free energy A for the whole system in terms of the volumes of the subsystems. This expansion begins with quadratic terms because A is minimized for the most probable distribution of subsystem volumes, i.e. for the average or equilibrium distribution. He then assumed the subsystems to behave independently, so that the quadratic cross terms in the expansion would sum to zero and the free energy exponent would be diagonalized, leaving a product of Gaussian exponentials to give the impression that each subsystem did indeed have a Gaussian fluctuation, even for the volume. As Tolman11 indicates, the philosophy behind eq 2.4 is that, in the thermodynamic limit, the probability is dominated by the exponential in P(V|V h ) and that f, like p/kT in eq 2.3, must be a slowly varying function. Tolman derives eq 2.4 by showing how, for one particular system, the reversible work in the exponent of P(V|V h) may be expressed in terms of A - A h , the Helmholtz free energy difference between the states at V and V h . Actually, Tolman works with a general variable x bounded on one side (e.g. x could be the degree of advancement of some chemical reaction) and not necessarily V (as we discuss below, identifying x with V raises some difficulty). Since the fluctuations are small, A -A h may be expanded in powers of x - xj, with the quadratic term, the first to survive because he assumes the first derivative of A vanishes under the equilibrium condition x ) xj. f(xj) may also be expanded in powers of x - xj with leading term f(xj) constant. Working only with these leading terms, the right side of eq 2.4, written for x, is reduced to a Gaussian in x - xj, and f(xj) is then determined by normalization. If, after the expansion is completed, one arbitrarily sets x ) V, the normalization of Ψ(V) between V ) -∞ and V ) ∞ leads to an f(V h ) proportional to p/kT, where p is the external pressure to which the system is subjected. Because Ψ(V) is a probability density, this result is not surprising, since, on a dimensional basis, p/kT is really the
only constant with the dimensions of density that can be constructed from the available thermodynamic variables. The constant of proportionality is of course dimensionless. Later we small show that, in the thermodynamic limit, it is unity. However, the identification of x with V in the unadjusted preceding analysis is not allowable. The reason for this is that A does not have a minimum along a path of constant temperature and pressure at the point of equilibrium. Instead, the derivative of A is equal to -p at this point. Thus the expansion of A A h begins with the linear rather than the quadratic term that led to the Gaussian. Einstein’s approach is closely related to the constant pressure ensemble, but the theory is approximate in application, because, aside from the special problem with V, Ψ(V) can only be determined under the assumption that fluctuations are small enough for the truncated expansion to be accurate. However it should be noted that the theory is approximate when used for type 2 fluctuations but exact (at least in the thermodynamic limit) for fluctuations of type 1. When the system is small enough, Einstein theory runs into difficulty for the several reasons enunciated above, so that more exact statistical mechanical methods must be used to characterize fluctuations. As far as fluctuations in volume are concerned, the most expeditious method involves the constant pressure ensemble on which the remainder of this paper is concentrated. As indicated in the previous section, for small systems several fundamental issues concerning this ensemble have not been resolved. The problems with macroscopic systems are both interesting and instructive but they are academic, since it can be shown that, in this limit, the use of an incorrectly specified ensemble leads to a correct answer, the error being of negligible consequence. The root problem in both the Einstein theory of volume fluctuations and the constant pressure ensemble is the rigorous determination of the volume or length scale. This appears to be part of a more general problem that arises when it is necessary (for whatever reason) to apply semiclassical statistical mechanical theory in configuration space alone rather than in the full phase space of the system.12 In phase space the length (of action) scale, i.e. the smallest distance over which two physical states can be resolved, has the magnitude of Planck’s constant. In configuration space the scale is not so obvious. Recently, Ellerby,8 extrapolating upward from a theory of physical clusters, involving the concept of a “shell molecule,” impacted the problem of the constant pressure ensemble and presented an estimate for the volume scale. At the same time, he discussed the connection between the shell molecule and the rigorous basis for the ensemble. Ellerby’s estimate was based on the assumption of the equivalence of ensembles in the thermodynamic limit, and because of this assumption, his estimate was not fundamental in the molecular sense. In any event, it is possible to develop a simple theory for the constant pressure ensemble that places it on a firm physical foundation and concomitantly reveals the proper length scale. At the same time, interesting and useful byproduct information relevant to clusters, shell molecules, etc. is generated. We begin this study in the following section. 3. Constant Pressure EnsemblesA Derivation Hill13 provides a useful tutorial on the constant pressure ensemble. The ensemble is summarized by its partition function and the thermodynamic potential to which it corresponds. The relation is
G(N,p,T) ) -kT ln ∆(N,p,T)
(3.1)
where G is the Gibbs free energy of the system to which the
Constant Pressure Ensemble
J. Phys. Chem., Vol. 100, No. 1, 1996 425
ensemble corresponds and ∆ is the ensemble partition function, given by
∆(N,p,T) )
∑V Q(N,V,T) exp
{ } pV
-
kT
(3.2)
where Q(N,V,T) is the canonical ensemble partition function for a system containing N molecules in the volume V at the temperature T. In this equation p is the constant external pressure to which the system is supposed to be subject. The summation is over some set of discrete volumes V, but the members of the set are not specified, although they are usually assumed to differ only by small increments of volume. This lack of specification of the volumes over which the sum extends is one of the unsatisfactory features of the ensemble, a point mentioned in section 1. However, if one assumes the validities of eqs 3.1 and 3.2, all of the thermodynamic properties of the system can be obtained by the manipulation of eq 3.1 via partial differentiation. An important application of the constant pressure ensemble is the calculation of the probability that the system, under constant pressure and temperature, will fluctuate to a certain volume. It is never clearly stated whether the fluctuation in question is of type 1 or of type 2. However a usual example is that of a system in a container, say a rectangular box, in which one wall is moveable and subjected to a constant external pressure p. In this case, it is implied that the fluctuation is of type 2, since the wall would follow the actual instantaneous volume occupied by the system. As we indicated in section 2, the two types of fluctuation would be hardly distinguishable in the thermodynamic limit. Nevertheless, it is important to be clear about which type is intended, not only in the interest of sound scientific argument but also because, in the case of small systems, the two types of fluctuation do become distinct. Whatever type of fluctuation is implied, the formula that the conventional framework of the constant pressure ensemble gives for its probability is
Π(V) )
Q(N,V,T) exp{-pV/kT} ∆
(3.3)
where we use the noncommital symbol Π(V) to indicate the probability of the fluctuation whose type we do not know. However Π(V) cannot describe type 2 fluctuations, since it is not a probability density; i.e., it does not have the dimension of inverse volume. Typically one uses Π(V) to obtain the moments of V, summing over the set of V’s appearing in eq 3.2, and in the thermodynamic limit, the correct first two moments are obtained. Yet as we shall show, Π(V) is a poorly defined quantity. The goal of the present section is to demonstrate that Π(V), as given by eq 3.3, is the probability of neither a type 1 nor a type 2 fluctuation, although in the thermodynamic limit these fluctuations are negligibly different from one another. Our demonstration involves the precise derivation of eq 3.3 by means of an analysis in which it is clear that Π(V) is not the probability of a type 1 fluctuation. Then, since, as we have already indicated, it could not be the probability of a type 2 fluctuation, it is not the probability of either. As a result the usual formulation and application of the constant pressure ensemble, as summarized by eqs 3.1, 3.2, and 3.3, are seen to rest on a poorly defined foundation. We begin the analysis by considering a thermodynamically isolated composite system consisting of a closed subsystem containing N molecules in contact with a heat bath of size infinitely larger than that of the subsystem. Although the
volume of the total system is fixed, that of the subsystem and therefore that of the bath can vary. However, because of the infinite ratio of the size of the bath to that of the subsystem, the pressure p of the bath will be effectively constant, and this will be the constant external pressure to which the subsystem is subject. The bath will have an average (equilibrium) volume V0 and an average (equilibrium internal energy U0 while the subsystem will have the corresponding average (equilibrium) quantities V h and U h . We note that because the composite system is isolated,
dV ) -dV0 and dU ) -dU0
(3.4)
At fixed U, V, and N, the number of microstates Ω of the subsystem is related to its entropy S, using the familiar formula of the microcanonical ensemble, by
Ω(U,V,N) ) exp{S(U,V,N)/k}
(3.5)
A similar relation holds for the number of microstates Ω0 of the bath,
Ω0(U0,V0,N0) ) exp{S0(U0,V0,N0)/k}
(3.6)
The total number of microstates Ωt of the composite system is given by
Ωt )
Ω(U,V,N) Ω0(U0,V0,N0) ∑V ∑ U
(3.7)
where it is understood that U0 ) Ut - U and V0 ) Vt - V, where Ut and Vt are the fixed total energy and fixed total volume of the composite system, so that each term in the sum is completely specified by the indices V and U. We now expand S0 in powers of U0 - U0 and V0 - V0, retaining only linear terms because it can be shown that the higher order terms in the expansion are at least inversely proportional to the size of the heat bath and can be neglected in the thermodynamic limit where we begin our analysis. The result is
{( ) } {( ) }
S0 ) S0(U0,V0,N0) + (U0 - U0)
∂S0 ∂U0
+
V0 U ,V 0 0
(V0 - V0)
∂S0 ∂V0
+ ...
U0 U ,V 0 0
1 p ) S0(U0,V0,N0) + (U0 - U0) + (V0 - V0) + ... T T p 1 h ) + ... h ) - (V - V ) S0(U0,V0,N0) - (U - U T T
(3.8)
In this equation, the curly brackets and their subscripts indicate that the derivatives within them are to be evaluated at the equilibrium state of the bath. The derivatives themselves are, according to straightforward thermodynamics, 1/T and p/T, respectively, where T is the temperature and p is the pressure of the bath. The last step in eq 3.8 is based on the conservation of both the total energy and volume of the composite system, expressed also by eq 3.4. Equation 3.8 may be substituted into eq 3.6, and the result into eq 3.7 to yield
426 J. Phys. Chem., Vol. 100, No. 1, 1996
{
U h + pV h + TS0(U0,V0,N0)
Ωt ) exp
{ }[∑ kT
∑V exp
pV
-
kT
}
×
Koper and Reiss
{ }]
Ω(U,V,N) exp -
U
U
kT
(3.9)
At this point, we may observe that Ω(U,V,N) is the number of microstates of the subsystem consistent with a fixed V. Therefore, included among these microstates are those for which the actual volume occupied by the system may be less than V. Thus the fluctuated state of the subsystem to which Ω corresponds is a fluctuation of type 1! As explained in section 1, its probability is a cumulative one; i.e., it is not a sum of probabilities of mutually exclusive events. Also, at this point we will begin to refer to the subsystem as the system, since it is the entity to which the constant pressure ensemble will refer. We might ask for the total number of microstates of the composite system under the condition that the system is restricted to the volume V′. Clearly, this is derivable from eq 3.9 by retaining only that term in the sum over V for which V ) V′. Denoting the restricted number of microstates by Ω′t, we find
{
U h + pV h + TS0(U0,V0,N0)
Ω′t ) exp
{ }[∑
kT
exp -
) exp
{
pV′ kT
{ }]
Ω(U,V′,N) exp -
U
U h + pV h + TS0(U0,V0,N0) kT
}
×
}
U
kT
×
{ }
exp -
pV′ Q(N,V′,T) (3.10) kT
The quantity Q, representing the sum in square brackets in eq 3.10, is clearly the canonical ensemble partition function. Now our previous discussion has identified Ω(U,V′,N) as representing the number of microstates having volumes no larger than V′, and Ω′t, because of its relation to Ω(N,V′,T) in eq 3.10, must also have the same character. We can now try to show that Π(V′) in eq 3.3 is the probability that the system experiences a fluctuation such that its volume is no larger than V′, i.e. that it is the probability of a fluctuation of type 1. In this connection we write
Π(V′) )
Ω′t(V′) Ωt
(3.11)
based on the idea that, since Ωt is the sum of Ω′t over the full set of V′, the fraction in eq 3.11 represents the fraction of all microstates in which the occupied volume is V′ or less. This supposition is false because the restricted partition functions associated with two different V’s can include microstates of the same volume; i.e., fluctuations of type 1 are not mutually exclusive, so that the denominator in eq 3.11 contains redundancy. Thus Π(V′), given by eq 3.11, does not represent the probability of a fluctuation of type 1. Nevertheless, it leads immediately to eq 3.3! To see this we need merely substitute eqs 3.9 and 3.10, respectively, into the numerator and denominator of eq 3.11, after replacing the quantity in square brackets in eq 3.9 by Q(N,V,T). The first exponential factor in both equations cancels in the ratio, and the remaining denominator
Figure 1. (a) System and bath at volumes V and V0. The boundary between the two components is indicated by the thick black line. Shaded regions represent redundant counting in parts a and b. (b) Same as part a except that the system has expanded to the volume V + dV.
is seen to be exactly ∆, as prescribed by eq 3.2 and appearing in the denominator of eq 3.3. At the same time the numerator in eq 3.11 is seen to become exactly the numerator in eq 3.3. Thus Π(V′) in eq 3.3, the probability that the system will be found to have the volume V′sthe probability prescribed by the theory of the constant pressure ensemblesis not the probability of a fluctuation of type 1. It has already been determined that it cannot be the probability of a fluctuation of type 2, so it is indeed a poorly defined quantity. Nevertheless, there is good evidence, as mentioned above, that it can be used in the thermodynamic limit to evaluate the first two moments of V in the fluctuating constant pressure system. As we shall show, the reason for its success in the thermodynamic limit is due to the fact that (even though only a type 1 upper bound on V is envisioned), in that limit, the chance of a fluctuation in which the volume is less than V is vanishingly small, so that the fraction of microstates common to different Ω′t(V′) becomes vanishingly small. For systems not in the thermodynamic limit, however, the formula of eq 3.3 must be used with great caution. Its derivation to date rests on little more than an argument by analogy, and it is not surprising that the probability it prescribes is of a poorly defined nature. The analysis presented above, beginning with eq 3.7 and ending with eq 3.3, can be conducted in reverse so that, in addition to the necessity of the result, its sufficiency may also be demonstrated. 4. Repairing the Constant Pressure Ensemble The first step in placing the constant pressure ensemble on a rigorous foundation involves the provision of an unambiguous definition for Π(V′). The analysis of the previous section has shown that the conventional definition involves a quantity that is neither a probability of a type 1 fluctuation nor one of type 2. Judging from the manner in which the constant pressure ensemble has been applied, a type 2 probability is intended. Therefore we will modify the analysis of the preceding section in an attempt to formulate an ensemble capable of dealing rigorously with type 2 fluctuations. We deal first with the problem in the thermodynamic limit and begin with a modification of eq 3.7 that removes the unwanted redundancy. Figure 1 is helpful in following the
Constant Pressure Ensemble argument. The figure shows two states of the composite system. Figure 1a shows the first state in which the volume of the system (subsystem) is V while Figure 1b shows a second state in which the volume is V + dV. The volume of the bath is automatically determined by the volume of the system, since the composite volume is fixed. The unshaded region in the depictions of both states is dV. In Figure 1a the shaded region of the system covers the whole volume V and is meant to indicate the extent of all of the system microstates consistent with the volume V, i.e. Ω(U,V,N) microstates. For simplicity of notation we will now denote this number of microstates by the abbreviated symbol Ω(V). At the same time the shaded region in the bath indicates the extent of the bath microstates consistent with the volume V + dV of the system, i.e., according to eq 3.4, consistent with a bath volume V0 + dV0 ) V0 - dV. Figure 1b illustrates a similar situation, except that the volume of the system is now V + dV and that of the bath is V0 - dV, and the shading in the bath indicates the extent of the number of microstates consistent with what is now the full volume of the bath, namely V0 - dV, while the shading in the system indicates the extent of the number of system microstates consistent with the now partial volume V of the system. The total number of microstates in the composite system of Figure 1a is
Ωt(V) ) Ω(V) Ω0(V0)
(4.1)
{
[
][
]
∂Ω0 ∂Ω dV Ω0 + dV Ωt + dΩt ) Ω + ∂V ∂V0 0
[
]
∂Ω0 dV Ω Ω0 + ∂V0 0
kT
(4.3)
V
exp -
pV kT
dV ×
∂Ω ∂V
U
exp -
{
Ωt ) exp
U h + pV h + TS0(U0,V0,N0) kT
∂Ω dΩt ) Ω0 dV ∂V
(4.4)
From this equation it is clear that, to avoid redundancy, eq 3.7 should be replaced by
Ωt )
∂Ω Ω0(U0,V0,N0) dV V ∂V
∫VdΩt ) ∑∫ ∑ U U
(4.5)
where it is understood that V0 is determined by V in view of the fixed volume of the composite system. Aside from the replacement of the sum over V by an integral, the only difference between eqs 4.5 and 3.7 is the appearance of ∂Ω/∂V in place of Ω. Therefore, repeating the steps between eqs 3.7 and 3.9 yields, in place of eq 3.9,
kT
}∫ ( ) { } V
dV
∂Q ∂V
N,T
exp -
(4.6)
×
pV kT
(4.7)
Now
(∂Q ∂V )
N,T
)Q
(∂ ∂Vln Q)
N,T
)
p′′(V) Q kT
(4.8)
(where p′′(V) is the pressure of the system at Volume V) and substitution of this equation into eq 4.7 yields
{
U h + pV h + TS0(U0,V0,N0) kT
}∫
V
dV ×
{ }
p′′(V) pV Q(N,V,T) exp kT kT
(4.9)
Now eq 4.9 sums the microstates of the composite system over all V in a nonredundant manner, since we have taken care to subtract any redundancy between eqs 4.2 and 4.3. It then follows that the number of microstates lying between V′ and V′ + dV′ is given by that portion of the integral corresponding to dV′. Thus we can write
dΩ′t ) exp
{
}
U h + pV h + TS0(U0,V0,N0) p′′(V′) Q(N,V′,T) × kT kT
{ }
exp and should be subtracted from the number given in eq 4.2 if redundancy is to be avoided. If we ignore higher order infinitesimals, this subtraction yields, for the number of additional nonredundant microstates associated with the change dV of system volume, the quantity
U
The differential operator behind the summation sign in this equation can be moved in front of the sign, since U in the sum is independent of V, so that the equation may be rewritten in the form
(4.2)
However, this number recounts states that have been counted in eq 4.1. A little thought shows that the recounted states correspond to the product of the numbers of microstates associated with the shaded regions in either Figure 1a or Figure 1b. This number is
}∫ { }∑ { }
U h + pV h + TS0(U0,V0,N0)
Ωt ) exp
Ωt ) exp while in Figure 1b it is
J. Phys. Chem., Vol. 100, No. 1, 1996 427
pV′ dV′ (4.10) kT
In complete analogy to the definition of Π(V′) as the ratio of eq 3.10 to eq 3.9, we can now define the probability π(V′) dV′ that the system will be found in the interval V′ to V′ + dV′ as the ratio of eq 4.10 to eq 4.9, yielding
p′′(V′) Q(N,V′,T) exp{-pV′/kT} dV′ kT π(V′) dV′ ) p′′(V) Q(N,V,T) exp{-pV/kT} dV V kT
∫
(4.11)
Since this equation is meant to replace eq 3.3, the denominator (which is now a true normalizing denominator, since it includes no volume redundancy) represents ∆ the partition function for the constant pressure ensemble. We thus write, in place of eq 3.2
∆)
∫V
{ }
p′′(V) pV Q(N,V,T) exp dV kT kT
(4.12)
In eqs 4.11 and 4.12 p′′/kT is the density whose reciprocal constitutes the natural volume scale. Thus, since eq 4.11 gives
428 J. Phys. Chem., Vol. 100, No. 1, 1996
Koper and Reiss
the probability of a type 2 fluctuation of volume, we now see the basis of the remark, in section 2, that the constant of proportionality between the density (whose reciprocal is the quantum of volume) and p/kT is unity. We also note that π(V′), like Ψ(V), is properly a probability density. At this point, we observe that p′′ in eq 4.12 is the pressure within the system and depends on V. In contrast, p is the external pressure that is maintained constant under the definition of the ensemble. In Sack’s paper,5 the constant external pressure p appears where p′′ appears in eq 4.12, and since p/kT is constant, it can be moved outside of the integral, leaving the integral as the Laplace transform of Q with transform parameter p/kT, an important feature of the theory developed in refs 2-5. However, since p′′ depends on V, it cannot be removed from the integral. As a result, the unifying idea of Laplace transformation does not hold. We shall come back to this issue later. But the situation is saved in the thermodynamic limit to which the analysis of refs 2-5 applies. In that case, as we show later, p′′ ) p where the integrand in eq 4.12 has an overwhelming maximum, so that p′′ may be replaced by p without error. On the other hand, when the system is small, this replacement is not allowable. Also, when the system is small, the meaning of (∂ ln Q/∂V)N,T in eq 4.8 is not necessarily that of a pressure, since there are usually additional independent thermodynamic variables such as surface area, curvature, etc. that may have to be considered. Nevertheless, we will continue to refer to p′′ as a pressure even though it may only be a symbol for some derivative of Q. We discuss this issue further, below. It is still necessary to show how ∆ given by eq 4.12 is related to G, i.e. to derive the analog of eq 3.1. We begin by taking the derivative of ∆, in eq 4.12, with respect to T. We find
1 p′′ pV ) - ∫ Q exp{- } dV + (∂∆ ∂T ) T kT kT 1 pV ∂ ln Q p′′ Q exp{- } dV + kT ( ∫ ) ∂T kT kT kT p′′ pV p V Q exp{- } dV ∫ kT kT kT
( )
U h H h pV h 1 1 ∂ ln ∆ )- + 2+ 2)- + 2) ∂T p T kT T kT kT h /T) 1 1 ∂(G - T k ∂T
( )
2
2 V
V
2 V
)-
p′′ pV Q exp{- } dV + ∫VU kT kT
(
p′′ pV Q exp(- ) dV ∫VV kT kT
p kT2
(4.13)
In this equation there are two steps that might need additional explanation. The first concerns how the derivative (∂ ln Q/∂T)V appears instead of the corresponding derivative at constant p. The explanation is that the derivative at constant p may be written as
(∂ ln∂TQ) ) (∂ ln∂TQ) + (∂ ∂Vln Q) (∂V∂T) p
V
T
( )
p
p
G ) -kT ln T - kT ln ∆ + K(p)
(4.17)
Now, from this equation, we have
(∂G∂p ) ) V ) -kT(∂ ln∂p∆) ) kT dK(N,p) dp T
T
(4.18)
But from eq 4.12 we have kT(∂ ln ∆/∂p)T ) -V h ) -V, and substitution of this relation into eq 4.18 yields
dK(N,p) )0 dp
(4.19)
which shows that K is a constant, independent of p. Thus, in eq 4.17, K can be eliminated by the choice of reference level for the energy, and we have
G ) -kT ln T - kT ln ∆
(4.20)
Now, we recall that we are working in the thermodynamic limit and that both G and ln ∆ are proportional to N. As a result, as N f ∞ in this limit, kT ln T becomes absolutely negligible in eq 4.20, and we arrive, finally, at the result
(4.21)
which is identical, in form, with eq 3.1 except that, now, ∆ has the meaning of eq 4.12. However, unlike eq 3.1, eq 4.21 has a rigorous fundamental basis. At this point it is of interest to compare eq 4.11 with the expression (eq 2.4) that originates from Einstein theory. For this purpose we rewrite eq 4.11 as
Π(V′) dV′ )
{
p′′(V′) 1 exp - [A(N,V′,T) + kT kT pV′ - (A(N,V h ,T) + pV h )]
(4.14)
However, in the integrand in which this derivative appears, V must be constant for any interval dV. Thus the last term in eq 4.14 must vanish, so that the derivative of ln Q with respect to p must be the same as the derivative with respect to V. The second step that might need explanation is the appearance of U. This comes from the use of the standard formula for U in the canonical ensemble.
(4.16)
Integration of this relation with respect to T gives
)
If eq 4.13 is divided by ∆, the result is
)
1 ∂(G/T) 1 ∂ ln ∆ + )k ∂T p T ∂T
G ) -kT ln ∆
∆ 1 + T kT2
(4.15)
where the barred quantities are again average (equilibrium) quantities and H h is the enthalpy. The last term in eq 4.15 results from the use of the Gibbs-Helmholtz equation. We can now drop the bars and write eq 4.15 as
V
p
p
{
∫V′Vh (p′′(V′′) - p) dV′′}
p′′(V′) 1 exp kT kT
}
(4.22)
where in the first step V h ) (∂G/∂p)N,T. In the exponent we find the reversible work needed to bring the volume from V h to V′, and the factor f in eq 2.4 is indeed exactly equal to p′′(V′)/kT and not only proportional as suggested by Tolman.11 In the next section, we look at the problem from a more microscopic perspective and consider the new features that arise when the system is small. We also comment on the relation of
Constant Pressure Ensemble
J. Phys. Chem., Vol. 100, No. 1, 1996 429
this work to recent theoretical studies performed in connection with physical clusters. 5. Wall Molecule and Volume Scale The semiclassical form of the canonical ensemble partition function is
Q(N,V,T) )
Z(N,V,T)
(5.1)
Λ3NN!
where Λ is the thermal deBroglie wavelength and Z is the configuration integral. We now consider Q(N,V+dV,T) such that dV > 0 and the boundary of V lies entirely within V + dV. Then Q(N,V+dV,T), in addition to counting new configurations, repeats the count of all configurations in Q(N,V,T). Consider the part of Q(N,V+dV,T) that is nonredundant. These configurations are characterized by the fact that at least one of the N molecules is found in the volume dV outside of V. The partition function corresponding to these configurations is
dQ ) Q(N,V+dV,T) - Q(N,V,T) )
(∂ ∂Vln Q)
)Q
T,N
dV )
(∂Q ∂V )
T,N
p′′(V) Q(N,V,T) dV kT
dV (5.2)
where, again, p′′(V) is not the pressure p of the heat bath but rather the pressure of the system confined to the particular volume V. Using eq 5.2 in the integrand of eq 4.12 we obtain
∆)
∫VdQ exp{- pV kT }
(5.3)
Comparison of this equation with eq 3.2 shows that the only difference between it and eq 3.2 is that the sum over V has been replaced by an integral and that Q has been replaced by dQ. The fact that the dQ’s summed in the integral do not contain common configurations is responsible for the elimination of the redundancy and simply provides another view of the process that we introduced in section 4. However, the view through this “window” illustrates that redundancy is avoided by the device of having at least one molecule in dV. Put in another way, there must be at least one molecule near the wall of the container that establishes the volume V + dV. This reveals a close connection to some of the ideas introduced by Byers Brown,4 who, in effect, attached the wall to molecules in order to bring pV into the Hamiltonian of the system. At the same time, it shows that an actual attachment is unnecessary and that, when freed from that requirement, the formalism is simpler, more transparent, and easier to apply. As we shall see below, the molecule near the wall is very similar to the “shell molecule” introduced in a molecular theory of nucleation14-16 and in the characterization of physical clusters.8 In these studies, which involved small systems, the constant pressure ensemble was not directly at issue, but there remains a close connection, and a fundamental understanding of that ensemble is important for continued development in that area. It should be noted that, in the thermodynamic limit, the overwhelming majority of configurations that give rise to Q will have at least one molecule near the wall. Thus, in one sense, Q and dQ are not so different in that limit, and this accounts, in part, for the accuracy of the conventional formulation of the constant pressure ensemble. With small systems, the failure of the conventional approach has several causes. One of these is the occurrence of relatively large fluctuations and the increasing difference between type 1
and type 2 fluctations, a subject that we have already discussed and which we discuss further below. Another is the appearance of additional independent variables, so that the state of the system will have to be defined by surface area, curvature, etc., as well as pressure and temperature. Indeed, derivatives of Q with respect to volume will then have to be taken with these additional variables, as well as pressure, held constant. One can easily imagine a case in which the boundary of the larger volume (V + dV) differs from that of the smaller volume in such a manner that dV is a small protuberance on the smaller volume. In this case the system near dV would be highly curved and the interpretation of dQ on the molecular level would become complicated. If, for example, surface area was important, the elimination (in the theory) of redundant molecular configurations would be more difficult. Furthermore, if we focused on volume fluctuations at constant surface area, it would be necessary to restrict the system to a well defined set of shapes, e.g. ellipsoids of varying volume but constant surface. Also, when we are forced to deal with surface phenomena, the specific material nature of the “wall” may have to be taken into account. To illustrate a few more points, we will continue to ignore additional variables such as surface area even though we will consider systems too small to be considered to be in the thermodynamic limit, bearing in mind that in the treatment of a real problem such additional variables may have to be dealt with. To further simplify matters we will assume that V is always a spherical volume of radius R, so that dV ) 4πR2 dR. In adopting this approach we eliminate surface as an independent variable by insisting that surface area be a unique function of volume. Without going into detail, it can be shown that for this case p′′ would be the pressure outside of the system, regarded as a drop, if the surface behavior could be summarized by a surface tension. However this pressure would still not necessarily be the constant external pressure p. Now dQ can be expressed as follows
dQ )
1 × (N - 1)!Λ3N
∫dVdr1 ... ∫V+dV exp{-ω(r1,...,rN)/kT} dr2 ...drN dV ) ∫ exp{-ω(r1,...,rN)/kT} dr12 ... dr1N (N - 1)!Λ3N V+dV (5.4) where r1, ..., rN are the coordinates of the N molecules in the volume V + dV while molecule 1 is confined to the spherical shell of volume dV and r12, ..., r1N are the coordinates of the remaining N - 1 molecules relative to the position of molecule 1 (molecule 1 is considered to be at a fixed position in the spherical shell). Its motion, during integration, is, because of symmetry, accounted for by the inclusion of dV in the last member of eq 5.4. ω(r1, ..., rN) is the N particle interaction potential. Similarly, we can write Q as
Q)
∫V...∫Vexp{-ω(r1,...,rN)/kT} dr1 ... drN
1 N!Λ3N
The ratio of the two partition functions is
[
∫
]
(5.5)
N Vexp{-ω(r1,...,rN)/kT} dr12 ... dr1N dQ ) dV ) Q ... exp{-ω(r ,...,r )/kT} dr ... dr 1 N 1 N V V
∫ ∫
F(1)(wall) dV (5.6)
where F(1)(wall) is the singlet density near wall. But
430 J. Phys. Chem., Vol. 100, No. 1, 1996
F(1)(wall) )
p′′(V) ∂ ln Q ) kT ∂V T,N
(
)
Koper and Reiss
where once again p′′ is the pressure of the system at some fluctuated volume and not necessarily the constant external pressure p to which the system is subject. Combining eqs 5.6 and 5.7, we find
dQ ) Q(V) F(1)(wall) dV ) Q(V)
p′′(V) dV kT
(5.8)
Without the middle expression, this equation is identical with eq 5.2, but with that middle term it illustrates graphically, through its singlet density, the important role of the molecule near the wall. In a recent molecular theory of vapor phase nucleation, the “molecule near the wall” played an important role.14-16 The authors called this molecule a “shell molecule” and made no reference to the constant pressure ensemble although there is some connection. In the theory, a physical cluster representing a density fluctuation was defined as a group of i + 1 molecules contained within a spherical volume V + dV whose center coincided with the center of mass of the i + 1 molecules. One of the molecules, the shell molecule, was localized within the spherical shell of volume dV bounding V. These physical clusters, referred to as i/V clusters, were used to evaluate the full partition function of the supersaturated vapor in an almost exact manner for the case of vapors that were slightly imperfect gases. In the process, the correct equilibrium distribution of clusters (density fluctuations) was obtained and made available for the application of the principle of detailed balance to the calculation of the nucleation rate. To evaluate the full partition function of the vapor, it is necessary to count configurations of molecules exhaustively and nonredundantly. The shell molecule made this possible. The i/V cluster forms an example of a small system to which, in effect, the constant pressure ensemble is applied. Progress depends on the use of a clear and fundamental formulation of the ensemble. 6. Some Examples Involving Application to Small Systems In this section we apply the theory to small systems, noting some of the differences that arise in comparison with the application to macroscopic systems and examining how the results for small systems converge on those for macroscopic ones as the system is enlarged toward the thermodynamic limit. We begin by returning to eq 5.3 and substituting eq 5.2 to obtain
∆)
∫V
{ }
p′′(V) pV Q(N,V,T) exp dV kT kT
(6.1)
In the thermodynamic limit, the maximum term in the sum is all that counts and this term is obtained by setting the derivative of the logarithm of the integrand in eq 6.1 equal to zero, yielding
p′′ ∂ ln p′′ p - )0 kT kT ∂V N,T
(
)
(6.2)
We can use
(∂ ln∂Vp′′)
)N,T
1 p′′Vκ(V)
1 f0 p′′Vκ(V)
(5.7)
(6.3)
where κ is the isothermal compressibility. Both the compressibility and the pressure are intensive properties, so that in the thermodynamic limit (where V f ∞)
(6.4)
Thus the last term in eq 6.2 may be set equal to zero, and the equation yields
p′′(V) ) p
(6.5)
We see that the maximum term corresponds to the volume for which p′′ is identical with the constant external pressure to which the system is subject. This means that in the thermodynamic limit the pressure of the system corresponding to the maximum term is the same as the maintained constant pressure, and because in the logarithmic sense, usual to statistical mechanics, the maximum term is all that matters, eq 4.12 assumes the form of the Laplace transform claimed by Sack.5 Thus the present development reduces to the expected form in the thermodynamic limit. Next, we examine eq 6.2 in the case that the system is an ideal gas and study the situation for systems of all sizes. For an ideal gas we have p′′ ) NkT/V. Substitution of this relation into eq 6.2 yields
p
p′′ )
1-
(6.6)
1 N
This equation shows that, for an ideal gas, even a system so small that it contains only 100 molecules has a p′′ that differs from p by only about 1%. Thus, in this case, the representation for a system in the thermodynamic limit remains applicable for systems that are almost of molecular size. Nevertheless, when the system is small enough, the large system representation is no longer applicable. One might wonder how the system can have an intrinsic pressure different from that of its environment. The answer lies in the fact that (as mentioned earlier) p′′ is not a “pressure” in the conventional sense. Intensive variables such as pressure and temperature are usually defined as quantities determined by external barostats and thermostats with which they are in contact. However p′′ is really a derivative of ln Q and is therefore determined by the properties of the system itself. Whether such intrinsic intensive variables should be regarded as pressures or temperatures, etc., and especially questions concerning their fluctuation, are the subjects of a lively dialogue in the literature. A conclusion related to the above example can be drawn by using the probability specified by eq 4.11 to calculate the relative volume fluctuation for an isothermal, isobaric ideal gas. In this case Q is obtained from eq 5.1 by replacing Z with VN and using the ideal gas equation of state for p and p′′. The result for the relative fluctuation is
(
)
〈V2〉 - 〈V〉2 〈V〉
2
1/2
)
1 xN
(6.7)
Now, as we indicated in section 1, it frequently happens that the representation of the constant pressure ensemble partition function as a sum over an unspecified set of discrete volumes is converted to an integral (as in the denominator of eq 4.11) by simply multiplying the typical term in the sum by dV and replacing the sum by an integral. This uncritical procedure leads to an expression like that in the denominator of eq 4.11, except that the factor p′′/kT in the integrand is omitted. The result is a partition function that is not dimensionless, when it should be, but has the dimension of volume. Even highly respected authors7 can be guilty of this error. However, if the representa-
Constant Pressure Ensemble
J. Phys. Chem., Vol. 100, No. 1, 1996 431
tion for a system in the thermodynamic limit is used, the factor becomes the constant p/kT which will cancel out of the ratio in eq 4.11. If we now use the resulting expression (regardless of the size of the system) to calculate the relative volume fluctuation for an ideal gas, we find
(
)
〈V2〉 - 〈V〉2 〈V〉
2
1/2
1 )
xN + 1
(6.8)
Even when N is as small as 10, the result in eq 6.8 differs from that in eq 6.7 by only 5%. Thus for the case of an ideal gas, the relative fluctuation can be calculated fairly accurately for a small system by using the representation for a macroscopic one. However, the ideal gas is a special case. In order to emphasize this point we return to eqs 6.2 and 6.3, substituting the latter into the former. The result is a quadratic equation for p′′ whose solution is
p′′ )
(
p p2 kT + 2 4 κV
)
1/2
(6.9)
The quantity whose square root appears in this equation will be negative when kT/κV exceeds p. Thus the smallest value of p that can yield a real value of p′′ is (4kT/κV)1/2. What is the meaning of this lower limit on p? To answer this question, we note that as the external pressure is reduced and the square root quantity becomes smaller, so does p′′ in eq 6.9. A reduction of p′′ implies an increase of V, i.e. an increase in the most probable volume of the system. Put in another way, the volume to which the maximum in the integrand corresponds increases. As p is further reduced, the square root term in eq 6.9 will vanish and, at this point, p′′ ) p/2. At lower values of p there is no real root, which means that there is no longer a most probable (equilibium) volume. The implication is that the maintained constant external pressure p can no longer “hold” the system, which will then expand freely until some other dramatic event occurs. To examine a quantitative example, we study a relatively incompressible liquid like water, whose density F ) N/V is roughly constant and whose κ is very small and also relatively constant. The zero of the square root quantity, the condition that corresponds to the lower limit on p, occurs when p ) (4kT/κV)1/2 ) (4kTF/κN)1/2. From this expression we see that if both κ and N are small, (4kT/κV)1/2 will be large and the lower limit on p will be large; i.e., a large pressure will be required in order to “hold” N molecules at the temperature T. We can evaluate this lower limit for water when N is given. Alternatively, we can specify a value for p and ask for the smallest value of N that can be held at that pressure. If we choose T ) 300 K and p ) 105 Pa (1 atm) and use the values measured for the parameters for bulk water at 300 K, namely F ) 3.3 × 1028 m-3, κ ) 4.5 × 10-10 m2/Pa, and k ) 1.38 × 10-23 J/K, we can insert these into the following equation,
N)
4kTF κp2
(6.10)
to yield
N ) 1.21 × 108
(6.11)
Thus, at 300 K, a pressure of 1 atm will not be able to contain a drop of water having less than 1.21 × 108 molecules. Of course we must remember that the parameters for bulk water are almost certainly not applicable to such a small system, so that the result in eq 6.11 must be regarded as highly approximate. However, the physical picture may still be described
approximately in the following way. We can regard the system as a small drop of water and choose the external pressure to be equal to or greater than its vapor pressure. If p exceeds the vapor pressure, the vapor will have been compressed away and only the liquid will survive. If p is reduced to the vapor pressure, the system, drop plus vapor, will undergo an expansion as the vapor begins to coexist with the liquid. The value of p at which this happens represents the lower limit prescribed by the vanishing of the square root term in eq 6.9. However, as soon as vapor appears, the system, which includes the vapor, becomes compressible so that the value of κ appearing in eqs 6.9 and 6.10 becomes large and a real root of the quadratic equation reappears. This is the “dramatic event” referred to above, and the system is once again stabilized. Note that in the calculation corresponding to eq 6.10, p/2 is 0.5 atm. This is not too far from the vapor pressure of water at 300 K. If, for the small drop, a more accurate value of F/κ had been used, the value of p at which the real root disappeared would probably have been closer to the actual vapor pressure of water, since κ for a small drop should be larger than that for bulk water. The importance of including the correct volume scale, when dealing with a small system, can be emphasized by defining the partition function without it, as in the erroneous definition of ref 7. In that case the partition function is defined as the integral of eq 6.1 with the factor p′′/kT in the integrand omitted. Now, setting the volume derivative of the integrand to zero in order to determine the most probable V yields the result
(∂ ∂Vln Q)
) T,N
p p′′ ) or p′′ ) p kT kT
(6.12)
This relation does not depend on the size of the system and, in contrast to the case in which the scale is included, indicates that there is always a solution for p′, namely the constant pressure to which the system is subject. This result stands in dramatic contrast to the result obtained from the correct expression (eq 6.1) for the partition function and emphasizes the importance of the volume scale especially for the treatment of small systems. 7. Summary In this paper we have further addressed the problem of the constant pressure ensemble and the volume scale that must be introduced in order to represent the corresponding partition function as a dimensionless integral. The volume scale problem is one example of the difficulties that arise when one is forced to derive fundamental statistical mechanical relations by working in configuration space rather than in the full classical phase space of the system. We have shown that when the system is small the volume scale differs from that derived by previous authors for macroscopic systems. In particular, the constant pressure partition function can no longer be obtained from the canonical ensemble partition function by Laplace transformation of the latter. Our analysis indicates that, if one wishes to keep the unifying idea of the Laplace transformation, in the case of partition functions describing type 1 fluctuations, it is the derivative of the partition function that needs to be transformed whereas, in the case of type 2 partition functions, the partition function itself is transformed. Although this latter fact is hinted at by Sack,5 no clear indication is given which quantity is to be transformed in which case. We have referred to the fact that the constant pressure ensemble has found only modest application in statistical thermodynamics and that, probably for this reason, several respected workers have made serious logical errors in using it
432 J. Phys. Chem., Vol. 100, No. 1, 1996 although (in the thermodynamic limit) these errors have little practical significance. However, the situation is different with respect to small systems. There, the errors can be serious! This situation has become more significant in view of the fact that the constant pressure ensemble has begun to find application to small systems that arise only in theoretical studies. In this respect, the necessity of the so-called “shell molecule” has been rationalized for the constant pressure ensemble as well as for certain types of cluster. We have also drawn a connection between the constant pressure ensemble and the original Einstein fluctuation theory in which two kinds of fluctuation were considered. These fluctuations, type 1 and type 2, are discussed at some length in the current paper. Acknowledgment. We thank D. Bedeaux for “trivial” discussions. Part of this work has been performed under the auspices of the EC Network Thermodynamics of Complex Systems (Contract Number CHRX-CT92-0007), and we also acknowledge support from NSF Grant Number CHE93-1959. References and Notes (1) Guggenheim, E. A. J. Chem. Phys. 1939, 7, 103.
Koper and Reiss (2) Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956; p 63. (3) Munster, A. Z. Phys. 1953, 136, 179; Statistiche Thermodynamik; Springer: Berlin, 1956; Chapter 7; Mol. Phys. 1959, 2, 1. (4) Byers, Brown, W. Mol. Phys. 1958, 1, 68. (5) Sack, R. A. Mol. Phys. 1959, 2, 8. (6) Tolman, R. C. The Principles of Statistical Mechanics; Dover: New York, 1979; Chapters I-IV, XIII, and XIV. (7) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: New York, 1986; p 25. (8) Ellerby, H. M. Phys. ReV. E 1994, 49, 4287. (9) Kegel, W. K.; Groenwold, J.; Reiss, H. Submitted to Langmuir. (10) Einstein, A. Ann. Phys. 1910, 338, 1275. (11) Tolman, R. C. The Principles of Statistical Mechanics; Dover: New York, 1979; pp 637-641. (12) Reiss, H.; Ellerby, H. M.; Manzanares, J. A. Chem. Phys. 1993, 99, 9930. (13) Hill, T. L. The Principles of Statistical Mechanics; Mcgraw-Hill: New York, 1956; Appendices II-IV, pp 66-68. (14) Ellerby, H. M.; Weakleim, C. L.; Reiss, H. J. Chem. Phys. 1991, 95, 9209. (15) Weakleim, C. L.; Reiss, H. J. Chem. Phys. 1993, 99, 5374. (16) Weakleim, C. L.; Reiss, H. J. Chem. Phys. 1994, 101, 2389.
JP951819F
