Radiophysical properties of solutions
Aquacomplexes polarized due to deformation of the solvation shell are located in an external electric field. As a result, there is a separation of aquacomplexes experiencing more and less displacement. The polarization charges of the separated aquacomplexes differ at least in absolute value. Thus, separation of charges and electrostatic forces arise in the volume of the solution. The latter should lead to the excitation of intrinsic electrostatic oscillations into the neutral volume on average (in sufficiently large volumes or for sufficiently large periods of time) of the solution.
As a result of the separation of polarized aquacomplexes, a polarization charge ppol arises in solution.
Given the fact that the dielectric susceptibility of the solution is determined by the ratio
where a is the polarizability coefficient of the solvated ion (cluster); n is the average number of clusters per unit volume of the solution, and n = 2n , where n is the number of dissociated salt molecules
per unit volume of the solution, the polarization charge of one cluster ( P i'
p = —17- can be determined by the expression v A. "
In accordance with the law of conservation of the charge
where j is the current density. Since the current is carried by the polarized aquacomplexes then
where v is the velocity of aquacomplexes carrying current. The equation of motion of a polarized aquacomplex having mass m is written in the form
Differentiating equations (3.59) and (3.60) with respect to time and substitution by adding (3.60) to (3.59), we obtain
Substituting in (3.62) the expression for 8/8t obtained from (3.49),and considering that div£ = 4nq, we obtain
The resulting equation describes a simple harmonic oscillation with a circular frequency
If the solution is placed bdtwe'dh two flat electrodes, the distance between which is small, then the process can be considered in one?dimensional geometry. Let E denote the absolute value of the electric field acting on the solution. Then
where Д is the thickness of the solution layer between the potential and grounded flat electrodes.
Thus, the density of the volume polarization charge in the solution will oscillate with a circular frequency co0 (v0 = со0/2тг) after the action of an external electric field on the solution. In this case, the oscillation frequency is determined not by the amplitude of the field strength, but by the divergence (divergence) of the field strength in the solution volume.
The movements of the water molecule that is part of the solvate shell under the action of an external electric field, as well as the dielectric characteristics of the water structured around the ion, should have features compared to the case of unstructured water. Consequently, the electrophysical properties of the solvation shell formed by water molecules should differ from the properties of ordinary water, since, being part of the shell, water molecules are ‘fixed’ by the electric field of the ion.
One can describe the polarization properties of a water molecule ‘fixed’ by an ion.
For this, it is convenient to use a simple model of the molecule - a solid rod, the ends of which carry ‘positive’ and ‘negative’ charges, respectively.
A water molecule ‘attached’ to the cation by a hydrogen atom
We assume that the molecule is attached to a positively charged ion (cation) and, to simplify the analysis, is attached to a homogeneous positively charged surface (Fig. 3.1). In the absence of an external electric field, the molecule assumes an equilibrium position, which is the angle 0f between the direction of the constant dipole moment of the molecule p0 and the direction of the external constant electric field E. When exposed to an external field, the dipole rotates through an angle 0 relative to the equilibrium provisions. An additional dipole moment appears, the projection of which onto the field direction is determined by the relation
As a result of the deviation of the molecule from the equilibrium state, a quasielastic force Fret arises, which tends to return the molecule back. The molecule will be in equilibrium when the torque of the external electric field is equal to the rotating moment of quasi-elastic force:
The dipole moment arising under the action of an electric field is equal to the difference between the total moment after rotation and the dipole moment before rotation in the absence of an electric field:
If the external electric field is variable, then the molecule will perform forced oscillations described by a linear differential equation of the second order (3.69), in which 0 is taken as a generalized coordinate (see Fig. 3.1):
where I is the moment of inertia relative to the fixing point of the axis of rotation of the molecule, p is the coefficient characterizing the internal friction (fluid viscosity), c is the displacement of the dipole under the action of the driving force, and / is the length of the dipole.
The expressions for the driving force (3.70) and displacement (3.71) are determined according to Fig. 3.1 and the sine theorem:
Using the Steiner theorem , it is possible to calculate the moment of inertia of a water molecule ‘fixed’ to the cation by an oxygen atom that is part of the molecule. Its value will be / = 3.0639
Fig. 3.1. The motion scheme of a dipole of a water molecule fixed on a positively charged surface.
• 10~47 kg • m2. The frequency of natural vibrations of the water molecule is coQ = 1.1851 • 1011 s'1.
Substituting (3.70) and (3.71) into the equation of forced vibrations (3.69), taking into account the small angle 0, we can obtain the equation of forced vibrations of a water molecule ‘fixed’ to a solvated cation (positively charged ion) by an electronegative oxygen atom under the action of an external periodic electric field
The solution to equation (3.72) is
For the case of dipole oscillations in an electric field, when the angle of the deviation of the dipole from the equilibrium state is small, the expressions for polarizability take the form
Dependence graphs of real (a) and imaginary (Z>) parts of polarizabilities versus frequency are shown in Fig. 3.2.
The real part of this function has a narrow resonance peak. The position of the maximum of the imaginary part of polarizability is close to co0. The peak width and amplitude depend on the attenuation coefficient b = l/2x. At low values of b, the frequency ю tends to the value of the natural frequency of the system co0.
The broadening of the spectral line due to an increase in the attenuation coefficient b gradually turns the resonance spectrum into a relaxation one (Fig. 3.3).
Polarization processes in the case of a ‘fixed’ molecule for elastic electron and ion polarization are similar to the case of a free molecule. For these types of polarization, the ‘fixing’ method of the water molecule (oxygen atom or hydrogen atoms) to the ion does not matter.
Intermolecular vibrations include librational and translational vibrations of molecules, which correspond to frequencies 
со = 1.2905 • 1014 S"1, co_, = 4.0035 • 1014 s"1, соПЙ = 0.3636 • 1014 s-‘.
The corresponding polarizabilities are equal to, respectively
ct' = 3.7521 • 10-41 Ф -m a'= 0.3899 • 1Г41 Ф -m2, a'= 4.7268
- 4 ’5 ’6
- • 10-40 Ф -m2
The dielectric permittivity for this frequency range has the value
At the resonant frequency of vibrations of a free water molecule corresponding to the radio frequency region coQ7 = 4.5274 • 1012 s_1, the dielectric constant
The high-frequency dielectric permittivity of a water molecule ‘attached’ to an oxygen cation corresponds to the value
Fig. 3.2. Dependence of the real (a) and imaginary (b) parts of the polarizability on the frequency (Hz).
Fig. 3.3. The dependence of the real (a) and imaginary (b) parts of the polarizability on the frequency (Hz) for different attenuation coefficients b.
At the natural frequency co0 = 1.1851 • 1011 s the dielectric permittivity is
A water molecule ‘attached’ to the anion by a hydrogen atom
Consider the case of a negatively charged ion (anion). In our simplified model, a water molecule will be fixed on a negatively charged surface by hydrogen atoms (Fig. 3.4). Polarization processes in the case of ‘fixing’ a water molecule by hydrogen atoms for elastic electronic and elastic ionic polarization are similar to the case of its fixing by an oxygen atom. A significant difference in the values of polarizability and permittivity is observed for elastic dipole polarization.
The moment of inertia of a water molecule fixed on the surface by hydrogen atoms is also calculated by the Steiner theorem and is I = 11.0385 • 10-47 kg • m2.
Using the relations (3.74) and (3.75), it is possible to construct the dependence of polarizabilities on the frequency upon attachment by hydrogen atoms and compare it with the dependence for the case of attachment by an oxygen atom (Fig. 3.5).
Consider the behaviour of a molecule at the same frequencies as when fixed by an oxygen atom: co01, co02, co03. The corresponding polarizabilities: a9' = 1.0414 • 10~41 Ф m2, a'0= 0.1082-10"41 Ф т2, a'j= 1.311910-41 Ф-т2. The dielectric permittivity of water, fixed by hydrogen atoms, in the region of elastic dipole polarization will be
In the region of radio frequencies, the dielectric permittivity with allowance for con, will be s' = s'+As=3.13+21.29 = 24.42, where
As=2n0o.'12/3s0 = 21.29.
At the natural oscillation frequency co0 = 1.1851 1011 s_1, the dielectric permittivity will be
Fig. 3.4. The motion of a dipole of a water molecule attached to the surface by hydrogen atoms.
The analysis indicates that the dielectric permittivity of the water structured around the solvated ion, and therefore its electrophysical and radiophysical properties, are significantly different for the cases of cations and anions. In the case of the solvated cation, the water in the solvate shell at the frequency of natural vibrations has an almost 3 times higher dielectric permittivity (11.2 * 104) than in the case of the solvated anion (3.1 x 104). At the resonant frequency of vibrations of a free molecule of water (4.53 THz), the dielectric constant of water structured around the cation is 76, and for water structured around the anion, it is 21.3. As a result, the high-frequency (in the region of radio frequencies) dielectric permittivity of water structured around the cation is about 81, and around the anion it is about 24.4.
The effect of an electromagnetic wave on the solvated cations and anions in salt solutions in polar dielectrics, all other things being equal, should lead to the excitation of the solvation shells of cations and anions of different intensities, which in principle allows this effect to be used to separate cations and anions when an electromagnetic wave acts on a solution.
Fig. 3.5. Dependence of the real (a) and imaginary (b) parts of the polarizability on the frequency (Hz): 1 - when fixed by hydrogen atoms, 2 - fixed by an oxygen atom.