Microscopic equilibrium

In a microscopic equilibrium description the pressure-dependent local solvent shell structure enters tlirough... 

Because Kcff applies to a scheme that involves more that one equilibrium (see Eq. (1.7)), it is referred to as a macroscopic equilibrium constant, to distinguish it from the microscopic equilibrium constants KA and E, which describe the individual equilibria. 

Figure 19.17 Spherical macroparticle with radius ra consisting of an aggregate of microparticles separated by micropores filled with water. A chemical with constant concentration C° diffuses into the pore volume of the macroparticle. The local dissolved pore concentration Cw is at instantaneous equilibrium with the local sorbed phase C ( K d is microscopic equilibrium coefficient). Note that the macroscopic distribution coefficient Kd is time dependent (see Eq. 19-78.)...

Acid dissociation constants do not tell us which protons dissociate in each step. Assignments for pyridoxal phosphate come from nuclear magnetic resonance spectroscopy [B. Szpoganicz and A. E. Martell, Thermodynamic and Microscopic Equilibrium Constants of Pyridoxal 5 -Phosphate, J. Am. Chem. Soc. 1984,106, 5513]. 

From Eqs. (38) and (39) together with the KH values in Table XXI the microscopic equilibrium constants K ai and Ka2 can be calculated. 

The same approach can be applied not only to the bulk equilibrium constants (K) but also to the microscopic connection processes (given the symbol k). Recall that the macroscopic equilibrium constant is simply the sum of all the microscopic equilibrium constants. For example, if an acid (H2A) has two non-equivalent ionisable protons there are two distinct but equivalent ways to remove a proton to produce HA- and hence there are two microscopic equilibrium constants kx and k2) for this deprotonation process. Thus the macroscopic acid dissociation constant, K = k1+k2. Don t get confused between microscopic equilibrium constants and rate constants, both of which have the symbol k. So, in terms of... 

As one wishes to deduce from such experiments the microscopic equilibrium properties of the material under study, it is essential to first establish a link to the equilibrium fluctuations AM(l) of the macroscopic dipole moment M(f) = —that is, the sum of the permanent molecular moments... 

It is evident that this is a way of rewriting the exact solution of Eq. (47). However, it is interesting to recover the fluctuation-dissipation prediction from a perspective that might lead to a free diffusion with no upper limit if an error is made that does not take into account the statistical properties of the fluctuation E,(f). Let us evaluate the correlation function of E,(f). Using the property of Eq. (48) and moving to the asymptotic time limit reflecting the microscopic equilibrium condition, we obtain... 

Equations (A.23) and (A.25) pertain to equilibrium conditions of homogeneous systems. Such systems have constant properties over space and time and there is no entropy production. We shall now be interested in systems, away from equilibrium where properties vary as functions of location as well as time. Tb apply the results of thermodynamics to nonequilibrium systems., the principle of local (microscopic) equilibrium is invoked. For that reason it is useful to work with the thermodynamic variables on a unit volume basis. Equation (A.25) then becomes... 

In the DSMC technique, the probability that a chemical reaction occurs is the ratio of the reaction cross section to the elastic cross section. The most commonly applied chemistry model is the Total Collision Energy (TCE) form employed by Boyd based on a general model proposed by Bird. In this model, the probability of reaction, P, is obtained by integrating the microscopic equilibrium distribution function for the total collision energy, and equating it to a chemical rate coefficient, Kf. Specifically, the mathematical form of the probability is obtained from the following integral ... 

Let us denote by Rc the rate with which an unoccupied acceptor captures a free electron from the conduction band in a unit time, and by kg the rate constant for emission of a captured electron from an occupied acceptor into the conduction band. Similarly to Eq. 15, the principle of microscopic equilibrium relating these is given by... 

Equation 24 for F( in Eq. 23 was derived above under the condition of AG < 0. When AG > 0, the backward rate constant, A bw, can be obtained, since the reverse of the forward reaction for this free energy of reaction satisfies this condition. The value A pw can be determined from A bw by the relation of microscopic equilibrium, Fw/ BW = exp(- ff AG ) in Eq. 15. The rate constant k given by Eq. 23 satisfies this relation without any modification, since = I( x). Therefore, Eq. 23 turns... 

If K is the microscopic equilibrium constant for the association of the monotopic ligand A with B, the stepwise constants are K = 3K Ki =K Ks= K/3. The statistical factors are easily understood in the first equilibrium, three equivalent binding sites A are available for binding in the second, two equivalent binding sites A are available in the forward reaction but two equivalent bonds A—B can dissociate in the reverse reaction in the third, there are three equivalent bonds A—B which can dissociate in the reverse reaction. In general, for the interaction of a symmetrical m-topic ligand with a monotopic metalloporphyrin, the stepwise constants K( are given by Eq. 2, from which Eq. 3 is easily derived [20,29-31]. 

The observed, or macroscopic, equilibrium constant, K, of a generic chemical equilibrium (Eq. [34]) can be regarded as being given by the product of an intrinsic, or microscopic, equilibrium constant, K, and a statistical factor,... 

The classical example is that of a symmetrical dicarboxylic acid. Since the two ends are chemically identical, the microscopic energy and therefore the microscopic equilibrium constant for the acid dissociation of each of the two groups is identical to, say, K. If we now look at the macroscopic level, we are not able to distinguish between the two ends of the molecules, and the observed, or macroscopic, first acid dissociation constant, is related to the probability that either site is deprotonated, thus Kai — 2. On the... 

Each stepwise binding constant Kj can be factored into the product of a statistical factor and the corresponding microscopic equilibrium constant K. The statistical factor is easily obtained considering that in the forward reaction there are n—j+1 empty sites available for the ligand, whereas in the reverse reaction there are j Hgands that may dissociate from the receptor. Accordingly, Eq. [39] holds ... 

Figure 7.1 shows the pK s of some dicarboxylic acids along with the pK s of some related monocarboxylic acids. As noted earlier, complications can arise when a compound contains groups that have dissociations with close pK values [93,173,434]. In such a case one must consider the equilibria shown in Figure 7.2 for dissociation of an original acid RH Ht [173], In this scheme K, Kg, Kc, and are so-called microscopic equilibrium constants and is the equilibrium constant for tautomerization between RH and RH. . The experimentally measured (macroscopic) dissociation constants Kj and K2 pertain to the following equilibria ... 

Equation (8.21) combines the microscopic equilibrium rate constants for two elementary processes for which the initial and final states are degenerate, g gj and gigm times, respectively, the reduced masses [x and p are different and the energy change is AEjj For non-rearrangement collisions = p,. ... 

To build a model of water sorption and swelling in PEMs, three microscopic equilibrium conditions of water must be accounted for in the PEM and the adjacent medium. The global equilibrium state corresponds to the minimum of the appropriate thermodynamic free energy, in this case the Gibbs energy. 

If there is interaction between the groups, the number of different microscopic constants is much increased. In the special case, often postulated, in which the groups interact in pairs, the number of different interaction constants is q q — l)/2. In the general case the total number of independent microscopic equilibrium constants is one less than the number of microscopically distinguishable molecular forms, that is, 2 — 1. In any case it is greater than q, the number of overall constants X which are involved in the experimentally determinable relation... 

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