Consistency with Classical Thermodynamics

Integration of (28-34) allows one to express entropy in terms of Pi for a canonical ensemble of molecules that obey Boltzmann statistics. The result is exact to within an integration constant that comprises the third law of thermodynamics at absolute zero, where Pi is unity for the state of lowest energy and zero for all higher-energy states. Hence, 

This relation is equivalent to Boltzmann s equation (i.e., S = A lnf2) for the microcanonical ensemble, where all quantum states are equally probable and is the thermodynamic multiplicity of states. The correspondence between S and Z is obtained by invoking the Boltzmann distribution for P, in equation (28-35) using results from (28-31)  

On the second line in equation (28-36), the first term in brackets is unity via (28-23) and the second term is the internal energy U via (28-25). The generalized correspondence between S and Z is 

Equation (28-41) from classical thermodynamics indicates, via the chain rule, that 

If one employs results from Sections 28-3 and 28-4, and calculates (dS/dU)v N from statistical thermodynamics when p constant, a relation is obtained between p and T. The independent variables are U, V, and N, and 

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The parameter as defined in the ergodic problem via equation (28-20), has dimensions of reciprocal energy and is given by l/kT. This claim will be justified, and consistency with classical thermodynamics will be demonstrated in Section 28-5. 

Thus far we have explored the field of classical thermodynamics. As mentioned previously, this field describes large systems consisting of billions of molecules. The understanding that we gain from thermodynamics allows us to predict whether or not a reaction will occur, the amount of heat that will be generated, the equilibrium position of the reaction, and ways to drive a reaction to produce higher yields. This otherwise powerful tool does not allow us to accurately describe events at a molecular scale. It is at the molecular scale that we can explore mechanisms and reaction rates. Events at the molecular scale are defined by what occurs at the atomic and subatomic scale. What we need is a way to connect these different scales into a cohesive picture so that we can describe everything about a system. The field that connects the atomic and molecular descriptions of matter with thermodynamics is known as statistical thermodynamics. 

The strategy in a molecular dynamics simulation is conceptually fairly simple. The first step is to consider a set of molecules. Then it is necessary to choose initial positions of all atoms, such that they do not physically overlap, and that all bonds between the atoms have a reasonable length. Subsequently, it is necessary to specify the initial velocities of all the atoms. The velocities must preferably be consistent with the temperature in the system. Finally, and most importantly, it is necessary to define the force-field parameters. In effect the force field defines the potential energy of each atom. This value is a complicated sum of many contributions that can be computed when the distances of a given atom to all other atoms in the system are known. In the simulation, the spatial evolution as well as the velocity evolution of all molecules is found by solving the classical Newton equations of mechanics. The basic outcome of the simulation comprises the coordinates and velocities of all atoms as a function of the time. Thus, structural information, such as lipid conformations or membrane thickness, is readily available. Thermodynamic information is more expensive to obtain, but in principle this can be extracted from a long simulation trajectory. 

Thermodynamic systems are parts of the real world isolated for thermodynamic study. The parts of the real world which are to be isolated here are either natural water systems or certain regions within these systems, depending upon the physical and chemical complexity of the actual situation. The primary objects of classical thermodynamics are two particular kinds of isolated systems adiabatic systems, which cannot exchange either matter or thermal energy with their environment, and closed systems, which cannot exchange matter with their environment. (The closed system may, of course, consist of internal phases which are each open with respect to the transport of matter inside the closed system.) Of these, the closed system, under isothermal and iso-baric conditions, is the one particularly applicable for constructing equilibrium models of actual natural water systems. 

However, natural systems consist of flows caused by unbalanced driving forces, and hence the description of such systems requires a larger number of properties in space and time. Such systems are away from the equilibrium state, and are called nonequilibrium systems, they can exchange energy and matter with the environment, and have finite driving forces (Figure 2.1). The formalism of nonequilibrium thermodynamics can describe such systems in a qualitative and quantitative manner by replacing the inequalities of classical thermodynamics with equalities. 

The relationships of Equations 5 and 2 are unquestionably valid for unlimited surface coverage on ideal external open (flat, planar, accessible) surfaces ranging from nil at E to infinity at E=0. All of the inherent assumptions (tabulated above) are equally valid as models for physical adsorption in internal constricted regions. These are classically denoted as ultramicropores ( 2 nm), micropores(<2 nm), mesopores (2<1000nm) and macropores (very large and difficult to define with adsorption isotherm). In these instances there are finite concentration limits corresponding to the volume (space, void) size domain(s). Although caution is needed to deduce models from thermodynamic data, we can expect to observe linear relationships over the respective domains. The results will be consistent with, albeit not absolute proof of the models. 

Throughout, emphasis has been placed on the logical structure of the theory and on the need to correlate every analysis with experimental operating conditions and constraints. This is coupled with an attempt to remove the mystery that seems so often to surround the basic concepts in thermodynamics. Repeatedly, the attention of the reader is directed to the tremendous power inherent in the systematic development of the subject matter. Only the classical aspects of the problem are taken up no attempt has been made to introduce the statistical approach, since the subject matter of classical thermodynamics is self-consistent and complete, and rests on an independent basis. 

In section 3 the contribution to the surface chemical potentials arising from the field of the potential drop Aif was derived from the combination of classical thermodynamics with the electrostatic theory of dielectrics. This contribution may be alternatively calculated as follows. Suppose that in general an adsorbed layer with M lattice sites, area A and thickness 1 consists of N neutral molecules of the i-th species, i 1, 2,. .., N. If... 

Comparison of the molecular dynamics calculations with the predictions of classical thermodynamics indicates that the Laplace formula is accurate for droplet diameters of 20 tT j (about 3400 molecules) or larger and predicts a Ap value within 3% of the molecular dynamic.s calculations for droplet diameters of 15 o-y (about 1400 molecules). Interestingly, vapor pres-sures calculated from the molecular dynamics simulations suggested that the Kelvin equation is not consistent with the Laplace formula for small droplets. Possible explanations are the additional assumptions on which the Kelvin relation is ba.sed including ideal vapor, incompressible liquid, and bulk-like liquid phase in the droplet. 

However, as expressions for both partition functions have been derived quantum mechanically, they do not permit us per se to calculate equilibriiun properties of thermal sj stems because quantum (as well as classic) mechanics as such does not have any concept of temperature. However, an inspection of Eqs. (2.37) and (2.38) reveals that botli partition functions still contain one or more yet-to-be-determined Lagiangian multipliers. These need to be calculated in a way that the resulting expressions become consistent with thermodynamics as it was introduced in Chapter 1. 

Nine years after Shannon s paper, Edwin T. Jaynes published a synthesis of the work of Cox and Shannon (11). In this paper Jaynes presented the "Maximum Entropy Principle" as a principle in general statistical inference, applicable in a wide variety of fields. The principle is simple. If you know something but don t know everything, encode what you know using probabilities as defined by Cox. Assign the probabilities to maximize the entropy, defined by Shannon, consistent with what you know. This is the principle of "minimum prejudice." Jaynes applied the principle in communication theory and statistical physics. It was easy to extend the theory to include classical thermodynamics and supply the equations complementary to the Rothstein paper(12). 

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