Conductive sphere

When one chooses a radical as the reference system, the stability of a carbocation or carbanion can be defined by the free-energy change for discharging the ion in vacuum, and the change can be approximately described by the classical Born equation (3) (Bom, 1920), provided that the ion is represented by a conducting sphere on which the charge is located. 

In (3) N is Avogadro s constant, r the radius of a conducting sphere, Z the charge number, and e the elementary electric charge. 

An early continuum treatment of solvation, associated with Born,17 comes out of the analysis of the electrostatic work involved in building up a charge Q on a conducting sphere of radius R in a medium with dielectric constant e. From Poisson s equation, it follows that the potential outside of the sphere is Q/eR. Thus the work of charging is the result of each additional element dq interacting with the charge q already present 87... 

This gives the electrostatic contribution to the free energy of solvation of a conducting sphere with charge Q and radius R in a medium with dielectric constant e. 

To illustrate the use of the solution (20.1) and of some of the properties of Legendre functions we shall now consider the problem in which nn insulated conducting sphere of radius a is placed with its centre ut the origin of coordinates in nn electric field whose potential is known to be... 

For conducting spheres of equal size and equal and like charge, the force of repulsion is given by... 

As previously discussed, for two equally sized, equally and oppositely charged conducting spheres, the capacitance is given by... 

It is also easily shown that the corresponding equations for a charged conducting sphere near a grounded conducting plane are given by... 

Consider a conducting sphere bearing charge q, which may be taken as an approximation to a monatomic ion. The charge on such an object spreads out uniformly on the surface, and the charge density at any point on the surface may thus be expressed as... 

Assuming that the molecule can be approximated as a perfectly conducting sphere (radius r) with volume V, the dipole moment fi induced in the sphere upon its placement in an electric field of magnitude Fis given by equation 19 ... 

The polarizability is strongly related to the molecular volume. In simple electrostatic theory the polarizability of a perfectly conducting sphere of radius a is a = a3, but in real molecules of more complex shapes the average polarizability is still proportional to the size of the molecule the larger molecules have the higher polarizabilities. 

The fusion temperature of these polymers is low enough to allow the spinning of fibres and melt pressing of films 263). They can also be blended with normal thermoplastics such as polystyrene or polyethylene oxide)2711. The conductivity shows a percolation threshold of about 16% which is expected for a random distribution of conducting spheres. 

Another situation of interest in which this equation is applicable is that of a single conductive sphere (or nanoparticle) on an electro-inactive surface (np) since it can be considered as a model system for the study of the voltammetric behavior of spherical nanoparticles adhered to a support [26, 27]. In this last case, the expression for the current-potential curve is identical to Eq. (3.101) by using... 

Henry187 derived a general electrophoretic equation for conducting and non-conducting spheres which takes the form... 

According to Booth and Henry188, the equation relating electrophoretic mobility with zeta potential for non-conducting spheres with large kq when corrected for surface conductance takes the form... 

The particle is a rigid, non-conducting sphere with its charge... 

In [43], the carbon cage was represented as a classical infinitesimally thin spherical conducting sphere. Methods of classical physics were used to determine the dipole polarizability ad co) of the sphere. Accordingly,... 

Consider a ball probe of a conductive sphere such as a stainless steel ball bearing which is connected to an electrometer. When the ball probe is placed in a flow system, the electric current induced by charge transfer between the ball and the flowing charged particles can be measured. The amount of current measured will vary with the particle velocity and particle mass flux. 

Fig. 3. Heat production is an important consideration for devices using electric fields in the liquid near cells. This figure shows the theoretical distribution of heat production in and around a spherical cell at the centre of a quadrupole electrode chamber in a solution of low electrical conductivity (top) and high conductivity (bottom). The heat production is given by gE2 where g is the conductivity of the solution or cell component and E is the (local) electric field strength. The contour interval is 7% of the maximum in each case. The cell is modelled as an electrically conductive sphere enveloped by an insulating but capacitive membrane.

Here p(A) is the Fourier transform of p(R). This Born ion is considered as a conducting sphere with its charge Q being smeared over the surface of its cavity p(R) = (Q/4Tra2)8(R - a), p(k) = Qsm(ka)/ka. Outside the cavity the electrostatic field created by this charge is fully equivalent to the field due to the point charge Q considered earlier. By this means for R > a... 

In the Debye-Huckel theory, an ion in solution is treated as a conducting sphere. The distance of closest approach of two ions is a.4 The solution beyond a... 

To estimate its scattering cross section an electron is considered as a charge e uniformly spread over a spherical surface of radius R. The energy stored in such a system, which constitutes an isolated conducting sphere, is calculated by simple electrostatics [95] as E = e2/Sitc0R and equated with the rest energy of an electron of mass me to define the classical radius of the electron ... 

Rather full calculations of /. (r) vs. r for various p values must be compared to the experimental results to determine p. Equation (6) gives a widely used expression for solvent reorganization energy that can be substituted into k expressions. It was derived by Marcus over 40 years ago and is both simple and useful [61]. It models the donor and acceptor as two conducting spheres imbedded in a dielectric continuum. 

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