The Hiickel equation

Let us now examine how we can obtain an estimate of /q from the measured electromobility of a colloidal particle. It turns out that we can obtain simple, analytic equations only for the cases of very large and very small particles. Thus, if a is the radius of an assumed spherical colloidal particle, then we can obtain direct relationships between electromobility and the surface potential, if either Kit 100 or Kd 0.1, where K is the Debye length of the electrolyte solution. Let us first look at the case of small spheres (where Kd 0.1), which leads to the Hiickel equation. 

The spherically symmetric potential around a charged sphere is described by the Poisson-Boltzmann equation  

The constant A must equal zero for the potential / to fall to zero at a large distance away from the charge and the constant B can be obtained using the second boundary condition, that / = /o at r = where a is the radius of the charged particle and /o the electrostatic potential on the particle surface. Thus we obtain the result that 

The relationship between the total charge q on the particle and the snrface potential is obtained nsing the fact that the total charge in the electrical donble-layer aronnd the particle mnst be eqnal to and of opposite sign to the particle charge, that is  

In this result, the condition of small particles means that the actual size of the particles (which is often difficult to obtain) is not required. For reasons to be discussed later, we will call the potential obtained by this method the zeta potential (Q rather than the surface potential. In the following section we consider the alternative case of large colloidal particles, which leads to the Smoluchowski equation. 

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The SHMO theory was originally developed to describe planar hydrocarbons with conjugated n bonds. Each center is sp2 hybridized and has one unhybridized p orbital perpendicular to the trigonal sp2 hybrid orbitals. The sp2 hybrid orbitals form a rigid unpolarizable framework of equal C—C bonds. Hydrogen atoms are part of the framework and are not counted. The Hiickel equations (3.3) described in the first part of Chapter 3 apply, namely,... 

The possible usefulness of this relationship — which is known as the Hiickel equation — should not be overlooked. Throughout Chapter 11 we were concerned with the potential surrounding a charged particle. Equation (11.1) provides a way of evaluating the potential at the surface, i/ o, in terms of the concentration of potential-determining ions. Owing to ion adsorption in the Stern layer, this may not be the appropriate value to use for the potential at the inner limit of the diffuse double layer. Although f is not necessarily identical to i/ o, it is nevertheless a quantity of considerable interest. 

It should also be noted that in the limit of kRs - 0, Equation (47a) reduces to the Hiickel equation, and in the limit of kRs - oo, it reduces to the Helmholtz-Smoluchowski equation. Thus the general theory confirms the idea introduced in connection with the discussion of Figure 12.1, that the amount of distortion of the field surrounding the particles will be totally different in the case of large and small particles. The two values of C in Equation (40) are a direct consequence of this difference. Figure 12.5a shows how the constant C varies with kRs (shown on a logarithmic scale) according to Henry s equation. 

What is the Helmholtz-Smoluchowski equation How is it different from the Hiickel equation ... 

With Eq. (21) in mind, compare the Hiickel equation (9) with Eq. (16) defining the ectrum of topological graph of molecule ... 

The Hiickel equation (41.13), appropriately adjusted to give 7m, has been frequently employed for the analytical representation of activity coefficient values as a function of the ionic strength of the solution, and various forms of the Debye-HOckel and Br nsted equations have been used for the purpose of extrapolating experimental results. Some instances of such applications have been given earlier ( 39h, 39i), and another is described in the next section. 

The Hiickel Equation The Htickel equation [17] applies for the case kR < 1,... 

Equation (7.2.2) with the term in d/Ap neglected is known as the Hiickel equation. It is not generally applicable in aqueous media, although it may be... 

In these expressions Vi, % xe molecular orbitals of the usual LCAO type. Pariser and Parr used for their calculations molecular orbitals obtained by solving the Hiickel equations which are not, in general, consistent with Roothaan s equations. Two points should be noted at this stage. First of all it is assumed that doubly excited configurations such as unimportant in the wave... 

Keep in mind that Eqs. (8-14) are only valid for small Kr, when the electrophoresis retardation (electric-field-induced movement of ions in the electric double layer, which is opposite to the direction of particle movement) is unimportant [41J. This limitation is inherent to the Hiickel equation. Practically, a colloidal suspension always contains charged particles dispersed in a medium with surfactants (or electrolytes) of both polarities. In this case the Poisson s equation must be used for deriving the surface charge density and Zeta potential relationship. Under the Dcbyc-Hiickel approximation, i.c., the small value of potential, zey/ kgT, where v is the potential and z is the valency of ion, a simple relationship between the surface charge density and Zeta potential can be easily obtained [7], The Poisson s equation simply says that the potential flux per unit volume of a potential field is equal to the charge density in that area divided by the dielectric constant of the medium. It can be mathematically expressed as ... 

The Smoluchowski equation applies when the double layer is thin enough or R is large enough such that the motion of the diffuse part of the double layer can be considered to be nniform and parallel to a flat surface. The flow is taken to be laminar i.e. infinitesimal layers of liquid flow past each other. Within each layer, the electrical and viscous forces are balanced. By balancing these forces and nsing Poisson s equation (Eq. 3.6) with suitable boundary conditions, it can be shown that the mobility has a form similar to the Hiickel equation, althongh the numerical prefactor is different ... 

The Henry equation is a generalization of the Hiickel equation for spherical particles with arbitrary double-layer thickness. It is assumed that the charge density is unaffected by the applied field. The result for the electrophoretic mobility of non-conducting particles is... 

For very small particles in dilute solution, 1 /k is so large that kR 0 and in this limit f icR) = 1 and the Hiickel equation is recovered. In the limit of large particles, /(kR) = 1.5 and the Smoluchowski equation is obtained. The Henry equation allows for values of / kR) between these limits. 

The Hiickel equation requires that atR monovalent electrolyte in water at 25°C required to satisfy /cR = 0.1, given that R = 20 nm. 

Eq. 6.3 with f(Kr) taking the value of either 3/2 for large partilces when Kr > 200, called the Smoluchowski equation, which was derived even earlier than Henry s function [20], or 1 for small particles when Kr <0.01, called the Hiickel equation, is still the most popular way to convert a mobility distribution to a zeta potential distribution. Electrophoretic mobility can also be presented by particle s surface charge density s [21]... 

This is seen to differ from the Hiickel equation (Eq. 5.60) only by a constant in the denominator. Henry (1931) resolved the two versions of the electrophoretic mobility relation by accounting for the radius of the particle and the thickness of the double layer, in terms of 1/k, as ... 

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