Equilibrium uniqueness

The early Universe can be reasonably described as a dilute gas of particles and radiation in thermal equilibrium, uniquely characterised by its instantaneous density and temperature. The expansion of space causes further dilution and coohng of this gas. 

The values of the enthalpy obtained do not depend on the model chosen for the dimer, as shown by Lussan (19) in a comprehensive summary of the possible theoretical treatments of the NMR data. Lussan demonstrates the form of the equilibrium constant calculation in the case of (1) monomer-dimer (open or cyclic), (2) monomer-cyclic trimer, and (3) monomer-higher acyclic multimers in the two cases of (3a) all K s equal and (3b) ki for monomer-dimer equilibrium unique, k s for higher multimers all equal. He then takes the experimental curves for a number of alcohols in carbon tetrachloride and achieves a reasonable fit to the data up to 0.6 mole fraction by using one or the other of the theoretical relationships. In some cases two sets of theoretical points are plotted on the same graph as the experimental data both are a good fit in the low concentration region, up to 0.1 mole fraction. Above this concentration one or the other of the theoretical curves is much closer to the experimental curve. Lussan implies that hypothesis 3b may be a more accurate fit to the data in the more concentrated solutions. Methanol, ethanol, 2-methyl-2-propanol (tert-butyl alcohol) and 2,2,4-trimethyl-3-pentanol follow the curve for equilibrium 3a, while 2,2,4,4-tetramethyl-3-pentanol fits the monomer-dimer data. Lussan points out that the behavior of the latter alcohol fits in with that of two similar heavily substituted tertiary alcohols which have been found by infrared methods to form only dimers. 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

For a closed chemical system witli a mass action rate law satisfying detailed balance tliese kinetic equations have a unique stable (tliennodynamic) equilibrium, In general, however, we shall be concerned witli... 

The situation in singlet A electronic states of triatomic molecules with linear equilibrium geometry is presented in Figme 2. This vibronic structure can be interpreted in a completely analogous way as above for n species. Note that in A electronic states there is a single unique level for K =, but for each other K 0 series there are two levels with a unique character. 

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

Since the concentrations of Na+, A-, HA, H3O+, and OH- are unknown, five equations are needed to uniquely define the solution s composition. Two of these equations are given by the equilibrium constant expressions... 

Weltin, E. Are the Equilibrium Goncentrations for a Ghemical Reaction Always Uniquely Determined by the Initial Goncentrations /. Chem. Educ. 1990, 67, 548. 

In accordance with (3.53) the functional II/(x) + Ilg( ) is coercive and weakly lower semicontinuous on the space H, consequently, the problem (3.48) (or the problem (3.54)) has a solution. The solution is unique. Note that the equilibrium equations... 

Separation operations achieve their objective by the creation of two or more coexisting zones which differ in temperature, pressure, composition, and/or phase state. Each molecular species in the mixture to be separated reacts in a unique way to differing environments offered by these zones. Consequently, as the system moves toward equilibrium, each species establishes a different concentration in each zone, and this results in a separation between the species. 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

With the microfocus instrument it is possible to combine the weak Raman scattering of liquid water with a water-immersion lens on the microscope and to determine spectra on precipitates in equilibrium with the mother liquor. Unique among characterization tools, Raman spectroscopy will give structural information on solids that are otherwise unstable when removed from their solutions. 

In contrast to statics, the relaxational kinetics of living polymers and of giant wormlike micelles is unique (and different in both cases). It is entirely determined by the processes of scission/recombination and results in a nonlinear approach to equilibrium. A comparison of simulational results and laboratory observations in this respect is still missing and would be highly desirable. 

The overall direction of the reaction will be determined by the relative concentrations of ATP, ADP, Cr, and CrP and the equilibrium constant for the reaction. The enzyme can be considered to have two sites for substrate (or product) binding an adenine nucleotide site, where ATP or ADP binds, and a creatine site, where Cr or CrP is bound. In such a mechanism, ATP and ADP compete for binding at their unique site, while Cr and CrP compete at the specific Cr-, CrP-binding site. Note that no modified enzyme form (E ), such as an E-PO4 intermediate, appears here. The reaction is characterized by rapid and reversible binary ES complex formation, followed by addition of the remaining substrate, and the rate-determining reaction taking place within the ternary complex. 

Methane is unique among hydrocarbons in being thermodynamically stable with respect to its elements. It follows that pyrolytic reactions to convert it to other hydrocarbons are energetically unfavourable and will be strongly equilibrium-limited. This is in marked contrast to the boranes where mild thermolysis of B2H6 or B4H10, for example, readily yields mixtures of the higher boranes (p. 164). Vast natural reserves of CH4 gas exist but much is wasted... 

N2O4 has been extensively studied as a nonaqueous solvent system and it is uniquely useful for preparing anhydrous metal nitrates and nitrato complexes (p. 468). Much of the chemistry can be rationalized in terms of a selfionization equilibrium similar to that observed for... 

A two-step sequence of nitrile oxide-olehn cycloaddition and reduction of the resulting A -isoxazolines offers a unique and attractive alternative to the classical aldol reaction and its many variants (2J). The procedure bypasses traditional problems, including enolate equilibrium and cross condensation (20). 

The vapor pressure (P ) of a pure liquid at a given temperature (T) is the pressure exerted by its vapor in equilibrium with the liquid phase in a closed system. All liquids and solids exhibit unique vapor pressure-temperature curves. For instance, in Figure 2-79, lines BA and AC represent the equilibrium vapor pressure curves of the solid and liquid phases, respectively. 

For high temperatures, the spin-glass system behaves essentially the way conventional Ising-spin systems behave namely, a variety of different configurations are accessible, each with some finite probability. It is only at low enough tempera tures that a unique spin-glass phase - characterized chiefly by the appearance of a continuum of equilibrium states - first appears. 

The (vapor + liquid) equilibrium line for a substance ends abruptly at a point called the critical point. The critical point is a unique feature of (vapor + liquid) equilibrium where a number of interesting phenomena occur, and it deserves a detailed description. The temperature, pressure, and volume at this point are referred to as the critical temperature, Tc. critical pressure, pc, and critical volume, Vc, respectively. For COi, the critical point is point a in Figure 8.1. As we will see shortly, properties of the critical state make it difficult to study experimentally. 

The critical point is unique for (vapor + liquid) equilibrium. That is, no equivalent point has been found for (vapor + solid) or (liquid + solid) equilibria. There is no reason to suspect that any amount of pressure would eventually cause a solid and liquid (or a solid and gas) to have the same //m, Sm, and t/m. with an infinite o and at that point. mC02 was chosen for Figure 8.1 because of the very high vapor pressure at the (vapor + liquid + solid) triple point. In fact, it probably has the highest triple point pressure of any known substance. As a result, one can show on an undistorted graph both the triple point and the critical point. For most substances, the triple point is at so low a pressure that it becomes buried in the temperature axis on a graph with a pressure axis scaled to include the critical point. 

Presto, a third-order rate law This multiplication should not be taken as representing a chemical event or as carrying such implications it is only a valid mathematical manipulation. Other similar transformations can be given,2 as when one multiplies by another factor of unity derived from the acid ionization equilibrium of HOC1. (The reader may show that this gives a second-order rate law.) These considerations illustrate that it is the rate law and not the reaction itself that has associated with it a unique order. 

One can define a special equilibrium constant K (special in that it lacks the partition function for the unique vibration), giving for the rate constant... 

The investigation of plutonium chemistry in aqueous solutions provides unique challenges due in large part to the fact that plutonium exhibits an unusually broad range of oxidation states -from 3 to 7-and in many systems several of these oxidation states can coexist in equilibrium. Following the normal pattern for polyvalent cations, lower oxidation states of plutonium are stabilized by more acidic conditions while higher oxidation states become more stable as the basicity increases. 

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