4.5 The Microscopic Origin of the Electrodynamic Properties of Water and Ice
163
ature. As one can see, the model (4.24) describes the terahertz part of the spectrum
in the 1–10 THz interval, with the parameters given in the figure caption. The mass
m of a charged particle is the mass of the ion (H 3 O
+ or OH
− ), the concentration of
charges n ± = 1 mol/l, and the residence time of the charge in the potential τ r = 1.6
ps, which coincides with the lifetime t s of the charge in the oscillatory state (see
Table 4.2).
The general terms of the simple model of forced oscillations with friction reproduce the experimentally observed spectrum of water in the terahertz region with
the concentration of vibrating species equal to 1 mol/l, as found in the ionic model
above. The intense IR spectrum of water is associated with a high concentration of
short-lived ionic species, which oscillate in the potential of the surroundings, and
also show the translational conductivity as the time of observation increases. The
observed deviation of the model curve from the calculated one in the area of the
secondary relaxations (see yellow area in Fig. 4.14) is associated with the contribution from the orientation of permanent molecular dipoles, which were not taken
into account in the model (4.21). However, the contribution of this part to the static
dielectric constant of water (0) is only a few percent of the contribution from the
main dielectric relaxation, associated with the polarization caused by the relative
displacement of short-lived ionic species.
4.5.5 Autoionization and pH
The ionic model of water allows us to look at the concept of pH from a new angle.
The conductometric method of to determine the dissociation constant K w of water
uses data on the proton DC conductivity of pure water, σ dc , and aqueous electrolyte
solutions (see Sects. 1.3.1 and 1.3.3). The dynamics of ions, both of pure water and of
dilute electrolytes, are considered independent (no electrostatic interaction between
ions is assumed), and the conductivity created by them is supposed to be additive.
According to (1.11), the concentration of hydronium ions in pure water is
H 3 O
+
=
K w = α [H 2 O] = 10
pH
,
(4.26)
where α is a dissociation constant. Despite the simplicity of finding the ion concentration, it has an unusual temperature dependence, [H 3 O
+ ](T), which is unexplained
in the Bernal–Fowler model of water.
Figure 4.15 shows the concentration [H 3 O
+ ] of ionic species of pure water as
a function of reciprocal temperature (the Arrhenius scale), calculated from K w (T)
shown in Fig. 1.18 by (4.26). Interestingly, the low-temperature activation energy
E a = k B T ·ln([H 3 O
+ ]/[H 3 O
+ ] 0 ) of the ionic species (the barrier to the generation
of the charge carriers) is only about 0.4 eV, which is an order of magnitude lower
than the 5 eV that is expected for the autoprotolysis of water [62], but close to the
activation energy of the static DC conductivity of pure water (see Table 2.3). It is
163
ature. As one can see, the model (4.24) describes the terahertz part of the spectrum
in the 1–10 THz interval, with the parameters given in the figure caption. The mass
m of a charged particle is the mass of the ion (H 3 O
+ or OH
− ), the concentration of
charges n ± = 1 mol/l, and the residence time of the charge in the potential τ r = 1.6
ps, which coincides with the lifetime t s of the charge in the oscillatory state (see
Table 4.2).
The general terms of the simple model of forced oscillations with friction reproduce the experimentally observed spectrum of water in the terahertz region with
the concentration of vibrating species equal to 1 mol/l, as found in the ionic model
above. The intense IR spectrum of water is associated with a high concentration of
short-lived ionic species, which oscillate in the potential of the surroundings, and
also show the translational conductivity as the time of observation increases. The
observed deviation of the model curve from the calculated one in the area of the
secondary relaxations (see yellow area in Fig. 4.14) is associated with the contribution from the orientation of permanent molecular dipoles, which were not taken
into account in the model (4.21). However, the contribution of this part to the static
dielectric constant of water (0) is only a few percent of the contribution from the
main dielectric relaxation, associated with the polarization caused by the relative
displacement of short-lived ionic species.
4.5.5 Autoionization and pH
The ionic model of water allows us to look at the concept of pH from a new angle.
The conductometric method of to determine the dissociation constant K w of water
uses data on the proton DC conductivity of pure water, σ dc , and aqueous electrolyte
solutions (see Sects. 1.3.1 and 1.3.3). The dynamics of ions, both of pure water and of
dilute electrolytes, are considered independent (no electrostatic interaction between
ions is assumed), and the conductivity created by them is supposed to be additive.
According to (1.11), the concentration of hydronium ions in pure water is
H 3 O
+
=
K w = α [H 2 O] = 10
pH
,
(4.26)
where α is a dissociation constant. Despite the simplicity of finding the ion concentration, it has an unusual temperature dependence, [H 3 O
+ ](T), which is unexplained
in the Bernal–Fowler model of water.
Figure 4.15 shows the concentration [H 3 O
+ ] of ionic species of pure water as
a function of reciprocal temperature (the Arrhenius scale), calculated from K w (T)
shown in Fig. 1.18 by (4.26). Interestingly, the low-temperature activation energy
E a = k B T ·ln([H 3 O
+ ]/[H 3 O
+ ] 0 ) of the ionic species (the barrier to the generation
of the charge carriers) is only about 0.4 eV, which is an order of magnitude lower
than the 5 eV that is expected for the autoprotolysis of water [62], but close to the
activation energy of the static DC conductivity of pure water (see Table 2.3). It is