Debye–Hückel Contributions to the Gibbs Energy Interaction Parameters for Ternary Electrolyte + Non-Electrolyte + Water Systems

The Debye–Hückel and non-Debye–Hückel contributions to the Gibbs energy interaction parameters are investigated for electrolyte (E) + non-electrolyte (N) + water (W) systems. A method is proposed for calculating the interaction parameters, C _n ^DH,C _n ^N, and C _n ^T, which represent the Debye–Hückel, non-Debye–Hückel, and total contributions, respectively. Four ternary E + N + W systems are chosen and the interaction parameters are computed with different forms of the Debye–Hückel equation. Results show that: (1) the Gibbs energy interaction parameters between E and N can be divided into two parts: the Debye–Hückel contribution and the non-Debye–Hückel contribution, C _n ^T=C _n ^DH+C _n ^N; (2) the signs and magnitudes of the Debye–Hückel contribution to the interaction parameters, C _n ^DH, depend mainly on the change in the dielectric constant of the solvent due to the addition of the non-electrolyte into the solvent; and (3) when the addition of the non-electrolyte only affects slightly the dielectric constant of the solvent, C ₁^DH (indicating the Debye–Hückel contribution to the interaction parameter for E + N) has a very small value and consequently can be neglected. In general, C ₁^DH is large, even larger than C ₁^N. Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users.     

Extended Hückel tight-binding approach to electronic excitations

In this work, we propose the application of a self-consistent extended Huckel tight-binding (EHTB) method in the computation of the absorption optical spectrum of molecules within the linear response time dependent density functional formalism. The EHTB approach is presented as an approximation to the Kohn-Sham energy functional. The method is applied to the computation of excitation energies and oscillator strengths of benzene, pyridine, naphthalene, diazines, and the fullerenes: C(60)(I(h)), C(70)(D(5h)), and C(80)(D(2)). The very good agreement with experimental data is very encouraging and suggests the possibility of using the EHTB as a computational efficient and reliable tool to study optical properties of a wide variety of molecular systems.     

Analytical derivation of the Hückel “4n + 2 rule”

The “4n + 2 rule” is derived analytically at the level of the simple Hückel theory for neutral even-membered chains, their double ions, as well as cations and anions of the odd-membered chains, by determining the first order energetic effect of the ring closure. The topological background of the “4n + 2 rule” is also briefly discussed.     

Dynamical Approach to Multi-Equilibria Problems Considering the Debye–Hückel Theory of Electrolyte Solutions: Concentration Quotients as a Function of Ionic Strength

We recently described a dynamical approach to the equilibrium problem that involves the formulation of the kinetic rate equations for each species. The equilibrium concentrations are determined by evolving the initial concentrations via this dynamical system to their steady state values. This dynamical approach is particularly attractive because it can be extended easily to very large multi-equilibria systems and the effects of ionic strength also are easily included. Here we describe mathematical methods for the determination of steady state concentrations of all species with the consideration of their activities using several approximations of Debye–Hückel theory of electrolyte solutions. We describe the equations for a system that consists of a triprotic acid H₃A and its conjugate bases. With these equations, two types of multi-equilibria systems were studied and compared to experimental data. The first system is exemplified by case studies of solutions of acetate-buffered acetic acid and the second system is exemplified by the hydroxide titration of citric acid. The discussion focuses on the effect of ionic strength on pH and on the amplification of acidity by ionic strength. Ionic strength effects are shown to cause significant deviations from the widely used Henderson–Hasselbalch equation.     

Tamm states in zigzag carbon nanotubes: Analytic estimates in the Hückel approximation

The frontier molecular orbitals of [C _n ] _q hydrocarbons, fragments of (n, 0) nanotubes, were considered. Asymptotic estimates for two different sets of frontier molecular orbitals were obtained in the Hückel approximation. The problem was shown to be equivalent to the search for model polyene levels with alternating resonance parameters (β and β′ = γβ). The γ parameter was determined by the symmetry type of molecular orbitals and took on values from 0 to 2. The problem was reduced to an analysis of a regular linear polyene with a modified terminal atom by taking alternant symmetry into account. The states of the system were classified according to quasi-momentum Θ values, which could be complex. Solutions with a complex Θ value were localized at tube ends and could be interpreted as Tamm states. The criterion of the appearance of Tamm solutions for the problem under consideration was formulated, γ < 1 − 1/(q + 2). The conclusion was drawn that these states for an arbitrary [C _n ] _q fragment lied close to the Fermi level, and their energies were described by the asymptotic equation α ± βρ^{(q + 1)}. Delocalized levels were always situated farther from the Fermi level than Tamm states.     

Estimation of activity coefficients of ionic species of aqueous strong electrolytes within the extended Debye–Hückel concentration range

The concentration curve of mean activity coefficient to the required power was fitted by a product function. The product function can be factorized in factor functions which represent the concentration dependence of the single-ion species (J Solid State Electrochem, in press, 1). With a simplified procedure of this method, it is possible to split the mean activity coefficients into the individual parts for the ionic species within the extended Debye–Hückel concentration range. This method is applicable to all strong electrolytes because it is not necessary to have further data or additional assumptions.     

Correlation of CO2 solubility in N-methyldiethanolamine + piperazine aqueous solutions using extended Debye–Hückel model

Solubility data of CO₂ in aqueous N-methyldiethanolamine (MDEA) solutions of concentration (2.52, 3.36, and 4.28) kmol/m³ were obtained at temperatures (313, 323, and 343) K and partial pressures ranging from about (30 to 5000) kPa. A thermodynamic model based on extended Debye–Hückel theory was applied to predict and correlate of CO₂ solubility in various aqueous amine solutions. The effect of piperazine (PZ) concentration on CO₂ loading in MDEA solutions was determined at PZ concentration (0.36, 0.86, and 1.36) kmol/m³. Using experimental data in various temperatures the interaction parameters of activity coefficient model for these systems were determined. The results show the model consistency with experimental and literature data and PZ is beneficial to the CO₂ loading. The comparison of results of this study with previous data work shows the wide range of CO₂ loading considered in this work and the better agreement of model with experimental data. The average absolute relative deviation percent (δ_AAD) for all data points were 8.11%.     

Three-Dimensional Hückel Theory for closo-Carboranes

We have recently developed a 3-dimensional Hückel method for cluster compounds. The method uses a set of approximations for Coulomb, resonance, and overlap integrals very similar to those employed in the familiar 2-dimensional Hückel theory for the pi electrons of planar conjugated hydrocarbons. The method can be adapted to heteroatomic clusters by introducing heteroatomic Coulomb integrals, alpha(Y) = alpha(X) + hbeta, whereh is a parameter for heteroatom Y. In this paper, we use the 3-dimensional Hückel method to study the properties of the closo-carboranes, C(2)B(n)()(-)(2)H(n)(). We calibrate the method by choosing a value of the heteroatomic parameter h that distinguishes positional isomers by energy and gives them relative energies in rough agreement with those established by observation and ab initio calculations. We obtain modest improvement in matching ab initio relative energies of isomers by means of a three-parameter, first-order perturbation treatment. We use the calibrated method to evaluate various mechanisms proposed for the isomerizations of C(2)B(4)H(6), C(2)B(5)H(7), and C(2)B(6)H(8), all of which have been observed to undergo intramolecular isomerizations. Rearrangements of C(2)B(6)H(8) have been satisfactorily explained by a single-DSD (diamond-square-diamond) process. Those for C(2)B(5)H(7) require at least two DSD processes, concerted, consecutive, or overlapping. Several different mechanisms have been proposed for the rearrangement of C(2)B(4)H(6). In evaluating intermediate and transition state structures, the 3-dimensional Hückel method gives higher energies to those structures with a larger number of nontriangular faces, a plausible conclusion except that occasionally it is wrong. In comparison with ab initio results, the 3-dimensional Hückel method fails to give low energies for classical structures.     

What I like about Hückel theory

We start with some biographical notes on Erich Hückel, in the context of which we also mention the merits of Otto Schmidt, the inventor of the free-electron model. The basic assumptions behind the HMO (Hückel Molecular Orbital) model are discussed, and those aspects of this model are reviewed that make it still a powerful tool in Theoretical Chemistry. We ask whether HMO should be regarded as semiempirical or parameter-free. We present closed solutions for special classes of molecules, review the important concept of alternant hydrocarbons and point out how useful perturbation theory within the HMO model is. We then come to bond alternation and the question whether the pi or the sigma bonds are responsible for bond delocalization in benzene and related molecules. M?bius hydrocarbons and diamagnetic ring currents are other topics. We come to optimistic conclusions as to the further role of the HMO model, not as an approximation for the solution of the Schr?dinger equation, but as a way towards the understanding of some aspects of the Chemical Bond.     

Hückel能量的定域分割及应用——Ⅰ.广义Hückel规则及芳香性问题

在矩和分子片断计数基础上,将Hückel总能量向矩和分子片断展开。利用已知分子的E_π计算值,在最小二乘方意义下确定展开式的系数,使分子总能量具有片断加和性意义。结果表明,各种片断能量贡献绝对值随片断增大而减小,正负号交替变化:即偶片断起稳定作用,奇片断起活化作用,称之为广义Hückel规则。以此为基础,讨论了芳香性问题,规定了参考结构条件,统一了REPE和TRE两种芳香性判据的分歧。同时也指出了REPE方案中参数化的不唯一性,讨论了它们之间的关系。     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

Px - Hückel Systems as Complex Ligands

Abstract

Recent developments in the chemistry of [Cp?Mo-(μ,n⁶?P₆)MoCp?], [Cp?Cr(μ,n⁵?P₅)CrCp?] and [Cp?₂Co₂(P₄)] are discussed.     

Mbius环与Hückel环

Mbius环是以1863年提出“Mbius带”的德国天文学家 A.F.Mbius的名字命名的,它有好些引人入胜的拓补结构特性。Huckel环则是以本世纪三十年代对有机化学做     

矩形原子簇的Hückel本征值问题

利用一维绞型链Hückel矩阵的交换性, 统一处理了二维和三维矩形原子簇包含第二和第三近邻作用的Hückel本征值问题。通过对能级分布的讨论, 在HMO水平上阐明了碱金属体心比面心稳定的原因。此外, 还估计了微晶的大小。     

改进Hückel模型变分法与同系线性规律

一、引言同系线性规律可以表示为:P=a+bF(N)其中P是同系化合物的物理化学性质,a、b 是常数,F(N)是同系线性函数,N=n+t,n 是同系序数,t 是端基当量。人们在探求 F(N)方面做了较多的工作,最近的有以下几个:     

Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.     

Hückel理论中前线分子轨道能量的计算

本文应用变分原理,从邻接矩阵的逆矩阵出发,给出了Huckel矩阵最小本征值(绝对值)的计算方法.本中说明了利用图的直观特征计算逆矩阵的简便原则和选择试探函数的依据。 在上述方法的基础上,给出了一些共轭分子同系列最小本征值的近似计算公式,它们具有封闭的拓扑参量形式,适用于同系列的任一分子.用这些公式对大约80个分子作了数值计算,绝对误差平均小于0.003(单位β)。     

The Debye-Hückel approximation: its use in describing electroosmotic flow in micro- and nanochannels

In this work we consider the electroosmotic flow in a rectangular channel. We consider a mixture of water or other neutral solvent and a salt compound, such as sodium chloride, and other buffers for which the ionic species are entirely dissociated. Results are produced for the case where the channel height is much greater than the width of the electric double layer (EDL) (microchannel) and for the case where the channel height is of the order or slightly greater than the width of the EDL (nanochannel). At small cation, anion concentration differences the Debye-Hückel approximation is appropriate; at larger concentration differences, the Gouy-Chapman picture of the electric double emerges naturally. In the symmetric case, the velocity field and the potential are identical. We specifically focus in this paper on the limits of the Debye-Hückel approximation for a simplified version of a phosphate-buffered saline (PBS) mixture. The fluid is assumed to behave as a continuum and the volume flow rate is observed to vary linearly with channel height for electrically driven flow in contrast to pressure-driven flow which varies as height cubed. This means that very large pressure drops are required to drive flows in small channels. However, useful volume flow rates may be obtained at a very low driving voltage. In the course of the solution, we establish the relationship between the wall mole fractions of the electrolytes and the zeta potential. Multivalent electrolyte mixtures are also considered.