Equilibria statistical factors

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

Symmetry corrections for rate or equilibrium constant are usually expressed in a formally different but perfectly equivalent way by means of statistical factors (Leffler and Grunwald, 1963). If the two reagents of (33) have pA and ph equivalent reactive sites, then the rate constant for the forward reaction must be divided by a statistical factor of pA-pB- Since similar considerations apply to the back reaction as well, the equilibrium constant must be divided by a statistical factor ofpApJpc Pv> as n (35). The assign-... 

The relative magnitude of the two concentrations evidently depends on the magnitude of the equilibrium constants and of the concentrations of M and X. Two further points need to be made, (i) It is likely that the formation of 3 from 2 will be favoured, i.e., the equilibrium constant will be enhanced, by a statistical factor akin to an entropy of mixing, (ii) For the species 2 there is no front and back , the MP+BM is symmetrical, and therefore the probability of the propagation, i.e., the, is twice as great as it would be for XP+BM, even if the free energy of complexing is the same for X and M. 

An analysis of the [Co-( )-pn3]3+ system may be carried out if the statistical term is considered solely an entropy effect and the conformational term an enthalpy contribution. Also since the four tris and four mixed species are not differentiated statistically, only the equilibrium constant k = tris/mixed is considered. For (4-)-pn/(—)-pn= 1 and assuming the ligands are distributed binomially around the metal ion, the statistical factor gives fc = 0.33 (J7/=0 assumed) which leads to TAS= -0.66 kcal/mole at 25°. 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

Although a statistical factor contributes to the equilibrium constants for many types of reactions, it is in reactions of the type being considered here—the replacement of one neutral ligand by another neutral ligand— that this factor may be the principal factor in causing variation of Kn with n. Other series of reactions of this type are the formation of ammonia complexes in aqueous solution. The variation of Kn with n observed for the chromium (III)/water-methyl alcohol system is slightly smaller than observed for ammonia complexes of cobalt(II) (18) and nickel(II) (19) ... 

We obtain the same final velocity equation for steady-stale conditions, except replaces Kg. This is not surprising since the steady-state assump. tion does not change the form of the velocity equation for the uninhibited reaction while the reaction between E and 1 to yield El must be at equilibrium, (There is nowhere for El to go but back to E-H.) The velocity equation differs from the usual Michaelis-Menten equation in that the K term is multiplied by the factor [1 4- ([I]/J i)]- The above derivation confirms our original prediction that is unaffected by a competitive inhibitor, but that the apparem K value is increased. The increase in the value does. loi mean that the El complex has a lower affinity for the substrate. El has no affinity at all for the substrate, while the affinity of E (the only form that can bind substrate) is unchanged. The apparent increase in results from a distribution of available enzyme between the full affinity and "no affinity forms. The factor [14-([I]/ffi)] may be considered as an [I]-dependent statistical factor describing the distribution of enzyme between the E and El forms. Figure 4-21 shows the effect of a competitive inhibitor on the v versus [S] plot. 

S.W Benson, Statistical Factors in the Correlation of Rate Constants and Equilibrium Constants, J. Am. Chem. Soc., 1958, 80, 5151 V. Gold, Statistical Factors in the Bronsted Catalysis and Other Free Energy Correlations, Trans. Faraday Soc., 1964, 60, 738 ... 

MCT considers interacting Brownian particles, predicts a purely kinetic glass transition, and describes it using only equilibrium structural input, namely the equilibrium structure factor Sq [3,46] measuring thermal density fluctuations. MCT-ITT extends this statistical mechanics, particle based many-body approach to dispersions in steady flow assuming a linear solvent velocity profile, but neglecting the solvent otlrerwlse. 

Because the energy differences 8E between the sublevel states of each multiplet of mA and nB are very small, 8E ambient temperature and above, they are nearly equally populated under equilibrium conditions (Boltzmann s law, Equation 2.9) and the probability of the formation of any given encounter spin state will be equal to all the others as there are mn choices, it will be equal to the spin-statistical factor a = (mn) ... 

The equilibrium constant for the reaction as written is then 4ew/lcT, where 4 is a statistical factor (symmetry number of 2 for <8> <8> and OO). Using the equilibrium constant, it is then easy to find the equilibrium number of pairs of both occupied nearest neighbor sites as a function of temperature—which essentially solves the problem. 

FIGURE 78 Calculation of statistical factors by using the method of the symmetry numbers for equilibrium (83) and (84). (A) The metal ions are considered as nonsolvated species and (B) the metal ions are considered as nine-coordinate tricapped-trigonal prismatic solvates. 

If there is symmetry, statistical effects appear. Tom, Creutz, and Taube noted that in the equilibrium in Reaction 3 (where 4,4 -bipy = 4,4 -bipyridine), the mixed-valence ion is favored by a statistical factor of 4 even in the absence of other effects (31), When one compares the... 

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