Fundamental partition equation

Equation 1 is a fundamental resolution equation Q,2) which contains the measured terms a, k and n. Theoretical Plates (n) are a measure of a column s overall efficiency, the partition ration (k) is a measure of the amount of time a solute... 

Name of Ensemble Independent Variables Types of Contact with Next System Partition Function Fundamental Thermodynamic Equation... 

We now consider the connection between the preceding equations and the theory of Aharonov et al. [18] [see Eqs. (51)-(60)]. The tempting similarity between the structures of Eqs. (56) and (90), hides a fundamental difference in the roles of the vector operator A in Eq. (56) and the vector potential a in Eq. (90). The fomrer is defined, in the adiabatic partitioning scheme, as a stiictly off-diagonal operator, with elements (m A n) = (m P n), thereby ensuring that (P — A) is diagonal. By contiast, the Mead-Truhlar vector potential a arises from the influence of nonzero diagonal elements, (n P /i) on the nuclear equation for v), an aspect of the problem not addressed by Arahonov et al. [18]. Suppose, however, that Eq. (56) was contracted between (n and n) v) in order to handle the adiabatic nuclear dynamics within the Aharonov scheme. The result becomes... 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

Thus, the fraction of dose absorbed is exponentially related to the absorption number. Equation (10) shows that the absorption number (and therefore the membrane permeability) is a fundamental parameter while other parameters such as the partition coefficient and pKa are useful guides but not fundamental parameters. For highly soluble drugs with linear absorption kinetics, dose and dissolution have no effect on the fraction of dose absorbed. In the case of drugs that are absorbed by a carrier-mediated process, a mean permeability should be used [30],... 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

Because processes such as diffusion and partitioning are fundamentally the same in water and air, equations that describe uptake from air can be obtained by replacing the subscripts in equations for water with those appropriate for air... 

Equations (l)-(6) constitute a remarkable contribution by Oddershede-Sabin to the improvement of the KT in the analysis of atomic and molecular stopping. The partitioning in the stopping contributions from different atomic/molecular orbitals requires the fundamental assumption implied by equation (5) whereby... 

This is the fundamental equation for batch melting. More complex batch melting models deal with the changes of the bulk partition coefficient D. Although this formula may also be expressed as... 

The term 0 + (l-0)D can be regarded as effective partition coefficient between the latest drop of extracted melt and total residue (residual solid + residual melt). Eq. (3.5) is the fundamental equation for dynamic melting. 

Adsorption isotherms may be derived from a consideration of two-dimensional equations of state, from partition functions by statistical thermodynamics, or from kinetic arguments. Even though these methods are not fundamentally different, they differ in ease of visualization. We consider examples of each method in Sections 9.3 and 9.4. 

The slope of a plot of the partition coefficient vs. the reciprocal of the temperature (in Kelvin) is PUR. This is the fundamental equation of gas and liquid chromatography. In our laboratory, we coat a capillary column with a polyurethane of interest and measure the retention time of chemicals passing through it. The retention time is colinear with the partition coefficient. 

Expressions for the partition function can be obtained for each type of energy level in an atom or molecule. These relationships can then be used to derive equations for calculating the thermodynamic functions of an ideal gas. Table 11.4 or Table A6.1 in Appendix 6 summarize the equations for calculating the translational, rotational, and vibrational contributions to the thermodynamic functions, assuming the molecule is a rigid rotator and harmonic oscillator.yy Moments of inertia and fundamental vibrational frequencies for a number of molecules are given in Tables A6.2 to A6.4 of Appendix 6. From these values, the thermodynamic functions can be calculated with the aid of Table 11.4. 

For any real binding case in condensed phases the situation is much too complex to allow the synthetic composition of the partition function. Nevertheless, the fundamental equations yield some insight into molecular binding processes that have been ignored or underappreciated in discussions of artificial molecular recognition. 

The semigrand partition function can be written for a system of biochemical reactions at specified concentrations of coenzymes, and G" = —kT n T". The fundamental equation for G" can be used to calculate the other thermodynamic... 

All of the statistical figures of merit used for judging the quality of least-squares fits are based upon the fundamental relationship shown in Equation 5.15, which describes how the total sum of squares is partitioned into two parts (1) the sum of squares explained by the regression and (2) the residual sum of squares, where y is the mean concentration value for the calibration samples. 

Solving the equations derived for practical mass transport problems requires values for fundamental material constants which describe the solubility of the diffusant in the polymer matrix or the partition coefficient of the diffusant between two contacting media. This chapter deals with estimation methods for solubility values and partition constants. 

Here pt and pj denote the fractional populations of states i and j (p( = exp —Ei/kT /q in thermal equilibrium, where q is the partition function) pm and pn denote the corresponding fractional populations of the energy levels, and dm and dn the degeneracies (pf = pm/dm, etc.). The absorption intensity Gji9 and the Einstein coefficients and Bji9 are fundamental measures of the line strength between the individual states i and j they are related to each other by the general equations... 

Hydrophobicity (or lipophilicity) characterizes the readiness of a molecule to escape or to prefer the water environment. It plays a fundamental role in biochemical processes and influences the fate of a molecule in the environment. Thus, hydrophobic descriptors play an important role in QSAR modeling that is used in drug research and for risk characterization. The most widely used hydrophobic descriptor is the octanol-water partition coefficient (log P) proposed by Hansh [49]. P is a quotient between solubihties in octanol and water. It is defined by following equation ... 

Distribution potential established when ionic species are partitioned in equilibrium between the aqueous and organic phases, W and O, is a fundamental quantity in electrochemistry at liquid-liquid interfaces, through which the equilibrium properties of the system are determined. In any system composed of two immiscible electrolyte solutions in contact with each other, the equilibrium is characterized by the equality of the electrochemical or chemical potentials for each ionic or neutral species, respectively, commonly distributed in the two phases [4]. It follows from the former equality that the distribution potential Aq inner electrical potential of the aqueous phase, 0, with respect to the inner potential of the organic phase, 0°, is given by the Nernst equation [17,18],... 

This expression for resolution may be written in an alternative form in terms of the relative retention Fj/Kj (Section 24-1) and the partition ratio k. Thus it can be shown that (24-34) is equivalent to the following fundamental equation involving three nearly independent factors ... 

When three dimensional particles were used instead of coatings, the diffusional and structural kinetics showed marked differences in the rates of mass to area in the concentration dependent regions of partition coefficient (lower temperature and mass sorbed). Thus, the fundamental assumption of the Kiselev Yashin equation that the mass/area ratio was equal for prepeak and peak regions was not met, and agreement with static data was fortuitous at best. 

In equation 3.1 the spin terms of the negative species have been canceled out. The quantity 12.43 is obtained from fundamental constants and the translation partition function of the electron. Qim is the ratio of the remaining partition function of the anion to that of the neutral. If the partition function ratio for the anion and neutral are assumed to be the same, this term is zero. With one value of the equilibrium constant the electron affinity of the molecule can be estimated. The statistical mechanical expression for Keq refers to the absolute zero of temperature so that no temperature correction to Ea is necessary. Unfortunately, there were no values for the equilibrium constants or electron affinities. Thus, the value of Keq for one molecule, anthracene, was determined and the electron affinity of other aromatic hydrocarbons referenced to that value. If the partition function ratios are equal,... 

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