Mole number fluctuations

In summary, the general condition for the stability of the equilibrium state to thermal, volume and mole number fluctuations can be expressed by combining (12.2.8), (12.3.4) and (12.4.9)... 

Consider a one-phase binary mixture of components 1 and 2 confined to an isolated vessel, and imagine dividing the fluid into parts A and B. But unlike the pure case, region B is open to A, so that a fluctuation occurring in part B disturbs not only its internal energy and volume V, but also the mole numbers Nf and N . Consequently, the concentration in B fluctuates by transfers of material to and from part A. In addition to the constraints on U, V, and S given by (8.1.4)-(8.1.6), the total amoxmts of each component are conserved. 

To do so, use a system like that in Figure 8.3, but now consider the small region B to be of fixed volume and temperature. However, region B is open to the larger region, so the mole numbers N and N2) fluctuate in both regions. 

Suppose that the configurative point liM been moved from outside the binodal into the region between the binodal (2 2/) and spinodal x2api) with the concentration X2 = X2,o (the point D). The system is composed of r i moles of component 1 and of i2 moles of component 2, so that nj + n2 = n and X2.0 = n2/n. Further, assume that a fluctuation of composition spontaneously occurred in a certain local region and the concentration of component 2 became Xj there (Figure 1.7 and 1.8). Then, this concentration in the rest of the system is Xj. In terms of mole numbers, there are n moles in the local region (n = n l -I- n 2 n 2 = Xjw ) and n" moles in the rest of the system (n" = (n — n )x2). As a result of such a fluctuation, the Gibbs potential varies by... 

The random motion of molecules causes all thermodynamic quantities such as temperature, concentration and partial molar volume to fluctuate. In addition, due to its interaction with the exterior, the state of a system is subject to constant perturbations. The state of equilibrium must remain stable in the face of all fluctuations and perturbations. In this chapter we shall develop a theory of stability for isolated systems in which the total energy U, volume V and mole numbers Nk are constant. The stability of the equilibrium state leads us to conclude that certain physical quantities, such as heat capacities, have a definite sign. This will be an introduction to the theory of stability as was developed by Gibbs. Chapter 13 contains some elementary applications of this stability theory. In Chapter 14, we shall present a more general theory of stability and fluctuations based on the entropy production associated with a fluctuation. The more general theory is applicable to a wide range of systems, including nonequilibrium systems. 

We shall look at the stability conditions associated with fluctuations in diflferent quantities such as temperature, volume and mole numbers in an isolated system in which U, V and Nk are constant. 

Fluctuations in the mole numbers of the various components of a system occur due to chemical reactions and due to transport, such as diffusion. We shall consider each case separately. 

The fluctuations in mole numbers considered so far were only due to chemical reactions. The fluctuation in mole number can also occur due to exchange of matter between a part of a system and the rest (Fig. 12.3). As in the case of exchange of energy, we consider the total change in entropy of the two parts of the system ... 

Figure 12.3 Fluctuations in mole number can occur due to chemical reactions and exchange of molecules between the two systems. The state of equilibrium is stable if the entropy change associated with fluctuations is negative...

Let us now look at the phase separation in binary mixtures from the viewpoint of stability. The separation of phases occurs when the system becomes unstable with respect to diffusion of the two components, i.e. if the separation of the two components produces an increase in entropy then the fluctuations in the mole number due to diffusion in a given volume grow, resulting in the separation of the two components. As we have seen in section 12.4, the condition for stability against diffusion of the components is... 

The above examples apply to isolated systems in which volumes, mole numbers, and the equilibrium values of E (hence, of S) are constant. In many cases, one wishes to consider systems that interact with the surroundings, by exchanging heat or work or matter—the enlarged unit being isolated. As usual, we assume the reservoir to be so immense that fluctuations in its intensive variables—7b, Pq, and fio in the present case—remain absolutely miniscule compared to those anticipated for the system i.e., 7b, Po, and fio are presumed to be fixed. Further, the total energy, volume, and mole numbers E, Vt, and tit for the isolated compound unit at equilibrium are also assumed to be constant. Any changes in entropy are then subject to the requirement (unsubscripted variables refer to the system)... 

At the same time, as the concentration decreases the exchange of water molecules by cooperative processes becomes easier and so significant fluctuations in the coordination numbers are observed, which are estimated to be about 1. At even lower concentrations, ion-water correlation patterns will become obscured and then undetectable. As an extrapolation, dynamic processes might contribute more and more to the description of the solution. The strong interaction between Li+ and OH2 (dH. i = 34 kcal/mole in the vapor phase 130>) may cause the cation-water complexes to remain quite well-defined tetrahydrates, on the average, despite all dynamic effects. 

The turbulent transport equations are obtained in the traditional treatment of turbulence by time averaging the unsteady-state transport equations, after substituting the concentration and velocity components by the sum of their mean values and the corresponding time-dependent fluctuations. The following expression is thus obtained for the (time) average of the number of moles N that are transferred per unit time through unit area, in the direction y normal to the wall ... 

Suppose you open a 22.4 liter box to the atmosphere at 0°C. According to the ideal gas law, one mole of gas occupies 22.4 liters under those conditions, so there should be one mole of gas 6.02 x 1023 molecules) in the box. Based on the arguments in this chapter, if you tried this experiment many times, and counted the actual number of gas molecules you captured each time, how much would you expect your answer to fluctuate ... 

Chemical reactions may cause fluctuations in the number of moles of various components of a system. Equation (12.6) provides expansion in terms of affinity A (A = -XiWi). At equilibrium, SS = 0 and. = 0. So, the condition... 

This is the stability condition of an equilibrium state when the numbers of moles fluctuate. For an isothermal binary mixture, Eq. (12.12) becomes... 

The general condition for the stability of equilibrium state with respect to thermal, volume, and fluctuations in the numbers of moles is obtained by combining Eqs. (12.7), (12.9), and (12.12)... 

In eqs 2 and 3, V is the molar volume, z is the compressibility factor, n is the total number of moles in the system, and ni is the number of moles of component i. Equations 2 and 3 show that the fugacity coefficient their derivatives with respect to the number of moles of solute are known. While near the critical point the fluctuations are important and an EOS involving them should be used, we neglect for the time being their effect. 

Provided that there is a change in the number of moles upon reaction and the stoichiometry of the process is known, pressure measurements may be used to determine the order of the reaction according to equation (A). Thus Letort found that the order for the decomposition of AcH was with respect to initial concentration and 2 with respect to time (see p. 2). Such direct conclusions cannot usually be drawn from pressure measurements with oxidation reactions. However, direct information may be obtained from a very neat differential system devised by Du-gleux and Frehling (Fig. 9). Vj and V2 are two RV s, of different size connected to the inside and outside of the Bourdon gauge J. Rj allows simultaneous introduction of mixtures into Vj and V2 Any fluctuation in temperature of the furnace is thus compensated for. Rapid reactions and the direct effect of promoters and inhibitors on an oxidation may be studied. This apparatus may well be useful with other systems. 

A system is that part of the universe of immediate interest in a particular experiment or study. The system always contains a certain amount of matter and is described by specific parameters that are controlled in the experiment. For example, the gas confined in a closed box may constitute the system, characterized by the number of moles of the gas and the fixed volume of the box. But in other experiments, it would be more appropriate to consider the gas molecules in a particular cubic centimeter of space in the middle of a room to be the system. In the first case, the boundaries are physical walls, but in the second case, the boundaries are conceptual. We explain later that the two kinds of boundaries are treated the same way mathematically. In the second example, the system is characterized by its volume, which is definite, and by the number of moles of gas within it, which may fluctuate as the system exchanges molecules with the surrounding regions. 

The basic calculation involves a determination of the maximum error in moles adsorbed due to uncontrolled fluctuations in system pressure, temperature, and volume and a comparison of the weight of adsorbent or catalyst necessary to adsorb the number of moles of gas which would exactly compensate for the maximum error fluctuation (also expressed in moles). 

A macroscopic, deterministic chemical reacting system consists of a number of different species, each with a given concentration (molecules or moles per unit volume). The word macroscopic implies that the concentrations are of the order of Avogadro s number (about 6.02 x 10 ) per liter. The concentrations are constant at a given instant, that is, thermal fluctuations away from the average concentration are negligibly small (more in section 2.3). The kinetics in many cases, but far from all, obeys mass action rate expressions of the type... 

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