Systems, closed equilibrium displacements

Equations (5.63) and (5.64) are actually more general than is apparent from the derivation. Consider a closed system at a given temperature and pressure with /t , moles of the components 1.2,3,... distributed among the phases A,B, C,... For the flow of mass between the phases due to an infinitesimal reversible (equilibrium) displacement we can write... 

BASIC KINETIC BEHAVIOR. Relaxation experiments are usually carried out with systems that are fairly close to equilibrium, and as such they can be treated by linear differential equations. If a perturbation leads to a larger displacement from equilibrium, the mathematics becomes intractable, and this is certainly the case if multiple step systems are substantially displaced from equilibrium (Fig. 11). 

The variations of entropy S and energy U express the content of the second law in a deep way, whether as displacements (8) from equilibrium (e.g., for a closed system),... 

Let us first define the external MEC in M, consisting of m atoms. Consider the global equilibrium of M in contact with a hypothetical electron reservoir (r) fi0 = fj1 where fi= fi, the chemical potential of r. Let z = N — N° = d/V denotes the vector of a hypothetical AIM electron-population displacements from their equilibrium values N°. Since d/V = - d/Vr, the assumed equilibrium removes the first-order contribution to the associated change due to z in the energy, = M + , of the combined (closed) system (Mir) moreover, taking into account the infinitely soft character of a macroscopic reservoir, the only contribution to the energy change in the quadratic approximation is ... 

Equilibrium Displacements in Closed Systems the Theorems of van t Hoff and Le Chatelier. 

We shall now study the modification, or the displacement of thermodynamic equilibrium, as we pass from one system to the other. First we shall compare two closed systems composed of the same initial quantities of the various components, but differing in the final equilibrium values of T, p,... 

The Gaussian approximation can therefore be seen to be equivalent to a quadratic potential or linear elastic restoring force. Deviations from the Gaussian distribution will correspondingly yield nonlinear force terms in the dynamics. The Gaussian approximation should therefore be an appropriate simplification for describing systems close to equilibrium or at most linearly displaced from the equilibrium state. 

Figure 7. Using a theoretically determined equilibrium fractionation to interpret measured isotopic fractionations in a hypothetical mineral-solution system. Three sets of data are shown. The theoretical equilibrium fractionation for this system is indicated by the gray arrow. The first set of data, indicated by circles, closely follow the calculated fractionation, suggesting a batch equilibrium fractionation mechanism. The second set of data (stars) is displaced from the theoretical curve. This may either indicate a temperature-independent kinetic fractionation superimposed on an equilibrium-like fractionation, or that the theoretical calculation is somewhat inaccurate. The third set of data (crosses) shows much greater temperature sensitivity than the equilibrium calculation this provides evidence for a dominantly non-equilibrium fractionation mechanism. For the first data set, the theoretical fractionation curve can be used to extrapolate beyond the measured temperature range. The second data set can also be extrapolated along a scaled theoretical curve (Clayton and Kieffer 1991).

These results raise the prospect of dynamics of nucleosomes in linker histone-free chromatin, that is, of a thermal fluctuation of nucleosomes between closed negative , open , and closed positive states identified in the minicircle system. If this equilibrium exists, an extra supercoiling constraint applied to the fiber should displace it in one direction or the other depending on the sign of that constraint, and this displacement should be reversible upon its removal. 

While it may not be intuitively obvious, if the displacement from equilibrium is small, the rate of return to equilibrium can always be expressed as a first-order process (e.g., see Eq. 9-13). In the event that there is more than one chemical reaction required to reequilibrate the system, each reaction has its own characteristic relaxation time. If these relaxation times are close together, it is difficult to distinguish them however, they often differ by an order of magnitude or more. Therefore, two or more relaxation times can often be evaluated for a given solution. In favorable circumstances these relaxation times can be related directly to rate constants for particular steps. For example, Eigen measured the conductivity of water following a temperature jump18 and observed the rate of combination of H+ and OH for which x at 23°C equals 37 x 10 6 s. From this, the rate constant for combination of OH and H+ (Eq. 9-52) was calculated as follows (Eq. 9-53) ... 

In a liquid binary solution, this accumulation is accompanied by the corresponding displacement of another component (solvent) from the surface region into the bulk solution. At equilibrium a certain amount of the solute will be accumulated on the surface in excess of its equilibrium concentration in the bulk solution, as shown in Figure 2-6. Excess adsorption E of a component in binary mixture is defined from a comparison of two static systems with the same liquid volume Vo and adsorbent surface area S. In the first system the adsorbent surface considered to be inert (does not exert any surface forces in the solution) and the total amount of analyte (component 2) will be no = VoCo. In the second system the adsorbent surface is active and component 2 is preferentially adsorbed thus its amount in the bulk solution is decreased. The analyte equilibrium concentration Ce can only be measured in the bulk solution, so the amount VoCe is thereby smaller than the original quantity no due to its accumulation on the surface, but it also includes the portion of the analyte in the close proximity of the surface (the portion U Ce, as shown in Figure 2-6 note that we did not define V yet and we do not need to define... 

The fact that only one spectral set is observed for both of the chelate rings in 4 - 5 in the and C NMR spectra is evidence that (a) the equilibrium between 4 and 5 is rapid relative to the NMR time scale and, (b), that the open and closed chelate moieties in 5 are equivalent, i.e. they also undergo simultaneous rapid equilibration. The equilibration of the chelate rings must follow a rapid flip-flop type mechanism, by which one NMe2 group displaces the other, as described previously for other systems [8]. 

After washing the columns in an open-loop system for 4-6 h with PBS, and then overnight in a closed-loop system with 20 mL of PBS, a further 30% of the initially bound protein was removed, until an equilibrium had been established between the bound and unbound proteins. On changing the PBS eluent to heat-defibrinated plasma (containing antithrombin III), the desorption of both 125I-thrombin and 125I-antithrombin III from the column increased dramatically. In the experiments with radiolabeled antithrombin III, the column had been exposed previously to thrombin, therefore, the displaced radiolabeled antithrombin III should more properly be described as labeled inactive complex. Because defibrinated plasma contains anti-... 

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