Molecular equilibrium system, macroscopic properties

As this system nears equilibrium, the rate of the forward reaction decreases and the rate of the reverse reaction increases. At equilibrium, the macroscopic properties of this system are constant. Changes at the molecular level take place at equal rates. 

In equilibrium statistical mechanics, one is concerned with the thennodynamic and other macroscopic properties of matter. The aim is to derive these properties from the laws of molecular dynamics and thus create a link between microscopic molecular motion and thennodynamic behaviour. A typical macroscopic system is composed of a large number A of molecules occupying a volume V which is large compared to that occupied by a molecule ... 

In a non-equilibrium gas system there are gradients in one or more of the macroscopic properties. In a mono-atomic gas the gradients of density, fluid velocity, and temperature induce molecular transport of mass, momentum, and kinetic energy through the gas. The mathematical theory of transport processes enables the quantification of these macroscopic fluxes in terms of the distribution function on the microscopic level. It appears that the mechanism of transport of each of these molecular properties is derived by the same mathematical procedure, hence they are collectively represented by the generalized property (/ ... 

A system in equilibrium is fully relaxed its state does not change in time. Of course, fluctuations occur on a molecular level but these are random and do not result in changes in the macroscopic properties of the system. For instance, for the liquid-gas equilibrium of a component i, molecules of i are continuously transferred between the two phases but there is no net transport of i from the one phase to the other. 

The classical models of adsorption processes like Langmuir, BET, DR or Kelvin treatments and their numerous variations and extensions, contain several uncontrolled approximations. However, the classical theories are convenient and their usage is very widespread. On the other hand, the aforementioned classical theories do not start from a well - defined molecular model, and the result is that the link between the molecular behaviour and the macroscopic properties of the systems studied are blurred. The more developed and notable descriptions of the condensed systems include lattice models [408] which are solved by means of the mean - field or other non-classical techniques [409]. The virial formalism of low -pressure adsorption discussed above, integral equation method and perturbation theory are also useful approaches. However, the state of the art technique is the density functional theory (DFT) introduced by Evans [410] and Tarazona [411]. The DFT method enables calculating the equilibrium density profile, p (r), of the fluid which is in contact with the solid phase. The main idea of the DFT approach is that the free energy of inhomogeneous fluid which is a function of p (r), can be... 

What is called equilibrium is the state of a closed system in which its various macroscopic properties do not change with time. However, as soon as we wish to consider the interpretation of these properties at the molecular level, it must be borne in mind that the equilibrium state is of a dynamic rather than a static character, as a result of the motion and collisions of the molecules. Macroscopic properties such as pressure, temperature and entropy have a meaning therefore only as a result of averaging over a great many molecules. For example, the pressure exerted on a manometer is the mean change in the momentum of the molecules per unit time- and surface area. The number of molecules striking the surface changes from moment... 

From this discussion, it is clear that molecular simulation can provide essentially exact results for some statistical mechanics problems that would otherwise be insolvable or only tractable by approximation (Stote et al. 1999). By comparison of the simulation results with laboratory experimental results, the underlying model used in simulation can be validated. Eventually, if the model is valid, further information and insights are expected from the simulation of new systems. Thus, computer simulation works as a bridge connecting theoretical model prediction with experimental results. Some macroscopic properties of experimental interest such as equilibrium states and structural and dynamic properties can be rationalized using microscopic details of a system such as the masses of the atoms, the atomic interactions, and molecular geometries as provided from molecular simulation (Allen and Tildesley 2007). 

Thermodynamics consists of a collection of mathematical equations (and also some inequalities) which inter-relate the equilibrium properties of macroscopic systems. Every quantity which occurs in a thermodynamic equation is independently measurable. What does such an equation tell one about one s system Or, in other words, what can we learn from thermodynamic equations about the microscopic or molecular explanation of macroscopic changes Nothing whatever. What is a thermodynamic theory (The phrase is used in the titles of many papers published in reputable chemical journals.) There is no such thing. What then is the use of thermodynamic equations to the chemist They are indeed useful, but only by virtue of their use for the calculation of some desired quantity which has not been measured, or which is difficult to measure, from others which have been measured, or which are easier to measure. 

The transport phenomena discussed in Chapters 2 and 3 are the simplest examples of kinetic phenomena, if not the most familiar. For the cases considered, a single macroscopic property (number, charge, momentum, or energy density) was displaced from its equilibrium value. This perturbation caused a flux in the opposite direction, proportional to the displacement. The proportionality constant is the transport coefficient. For simple gaseous systems the relations between displacement and flux and the transport coefficient were determined using a hard-sphere model to describe molecular interaction. 

A system of many molecules has both macroscopic and microscopic (molecular) properties. The state of the system involving macroscopic properties is called the macroscopic state or macrostate. Specification of this state for a system at equilibrium requires only a few variables. The microscopic state or microstate of a macroscopic system requires information about every atom or molecule in the system, a very large amount of information. 

In a series of impressive publications. Maxwell [95-98] provided most of the fundamental concepts constituting the statistical theory recognizing that the molecular motion has a random character. When the molecular motion is random, the absolute molecular velocity cannot be described deterministically in accordance with a physical law so a probabilistic (stochastic) model is required. Therefore, the conceptual ideas of kinetic theory rely on the assumption that the mean flow, transport and thermodynamic properties of a collection of gas molecules can be obtained from the knowledge of their masses, number density, and a probabilistic velocity distribution function. The gas is thus described in terms of the distribution function which contains information of the spatial distributions of molecules, as well as about the molecular velocity distribution, in the system under consideration. An important introductory result was the Maxwellian velocity distribution function heuristically derived for a gas at equilibrium. It is emphasized that a gas at thermodynamic equilibrium contains no macroscopic gradients, so that the fluid properties like velocity, temperature and density are uniform in space and time. When the gas is out of equilibrium non-uniform spatial distributions of the macroscopic quantities occur, thus additional phenomena arise as a result of the molecular motion. The random movement of molecules from one region to another tend to transport with them the macroscopic properties of the region from which they depart. Therefore, at their destination the molecules find themselves out of equilibrium with the properties of the region in which they arrive. At the continuous macroscopic level the net effect... 

The equations of motion of a molecular system formally represent a coupled set of nonlinear differential equations. (The nonlinearity comes from the complicated distance-dependence of the pair-potentials.) It is a property of nonlinear differential equations that they are extremely sensitive to small differences in their initial conditions. In nature, these small differences are most generally created by the perturbations of the surroundings while in the computer simulations they are produced by the finite accuracy of the numerical computation. The sensitivity is manifested in the fast increase of these initial differences nearby trajectories separate exponentially until the system boundaries force them to turn back. This mechanism quickly mixes the trajectories and after a short initial period the behavior of the system forgets its past. This obviously happens for equilibrium systems when their macroscopic properties relax to fixed average values. It also occurs for NESS systems because after short transients their distribution function also becomes stationary. ... 

Chemical equilibrium, at a given temperature, is characterised by constancy of macroscopic properties — observable properties such as colour, pressure, concentration, density, etc. — in a closed system (a system containing a constant amount of matter) at a uniform temperature. Microscopic processes (changes at the molecular level), however, continue but in a balance that yields no macroscopic... 

The simulation of the macroscopic properties and of the molecular organisation obtained for a system of N model molecules at a certain temperature and pressure (T, P) typically proceeds through one of the two current mainstream methods of computational statistical mechanics molecular namics or Monte Carlo [1,2]. MD sets up and solves step by step the equations of motion for all the particles in the system and calculates properties as time averages from the trajectories obtained. MC calculates instead average properties fi-om equilibrium configurations of the system obtained with an algorithm designed... 

The characterization of quasi-equilibrium macromolecular order and conformation in these systems, as well as the dynamics of cooperative chain orientation in samples subjected to external forces are of paramount importance for the understanding of their macroscopic properties. In this work we pursue previous efforts along these lines and present the main results of a proton and deuterium nuclear magnetic resonance (NMR) study of molecular order and chain conformation, and their temperature... 

Statistical thermodynamics already provide an excellent framework to describe and model equilibrium properties of molecular systems. Molecular interactions, summarized for instance in terms of a potential of mean force, determine correlation functions and all thermodynamic properties. The (pair) correlation function represents the material structure which can be determined by scattering experiments via the scattering function. AU macroscopic properties of pure and mixed fluid systems can be derived by weU-estabhshed multiphase thermodynamics. In contrast, a similar framework for particulate building blocks only partly exists and needs to be developed much further. Besides equibbrium properties, nonequilibrium effects are particularly important in most particulate systems and need to be included in a comprehensive and complete picture. We will come back to these aspects in Section 4. 

The classical picture of molecular interactions and their effect on macroscopic properties is well developed in statistical thermodynamics of phase equilibria. Briefly, molecular interactions determine the potential of mean force which is directly related to the pair correlation function from which all macroscopic (thermodynamic) properties such as virial coefficients, osmotic pressure, and so on can be derived as a function of temperature and pressure. The question arises if this well-developed framework of equilibrium thermodynamics can be expanded to particulate systems in equilibrium. Equihbrium systems can be best studied for coUoids in solution on which we wiU focus our discussion in the following. 

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