22
1 A Historical Review of the Structures of Water and Ice
Fig. 1.17 The free energy of
a Zundel (H 5 O
+
2 ) cation as a
function of distance d
between oxygen atoms.
Black dots show hydrogen
atoms, open circles are
oxygen atoms. The
proton-transfer energy
barrier disappears when the
distance is 2.36 Å. Drawn
using data from [60]
+
Free energy (eV)
d (Å)
r OO =2.75 Å
r OO =2.55 Å
r OO =2.45 Å
r OO
|d|
r OO =2.95 Å
0
0.5
1.0
0
-0.2
-0.4
0.2
0.4
r OO =2.35 Å
which favors effective proton transfer. The rate of proton transfer by the Grotthuss
mechanism is limited by the rate of the adaptation of molecules around the hydronium
ion [61], and thus it is diffusion controlled. Consequently, ionic transport in water is
not just the electric field dragging solvated hydronium ions, but the rearrangement
of water molecules in the way suitable for fast proton exchange, with a speed that is
much higher than for conventional molecular diffusion [57]. However, the diffusion
coefficient D H + = 9.3·10
−9 m
2 /s [9, 62] is related to the charge of the proton, but not
the mass of the proton, the diffusion coefficient of which is D = 2.3·10
−9 m
2 /s, the
same as that for the water molecule (see Sect. 1.4). Water, therefore, has a structure
in which the diffusion of mass and the charge does not coincide. It is in this property that water differs from other dielectrics and ionic conductors. Interestingly, the
mobility of the proton charge in ice and water is of the same order of magnitude [63,
64]. This fact shows that the short-order molecular arrangement, which is the same
for both ice and water, defines the proton conduction in both ice and water.
18
1.3.3 Pondus Hydrogenii (pH)
The concentration of hydronium ions in pure water that has been established by DC
electrical conductivity measurements was used as a reference value for the concept
18 Note that the modeling of protonic transport by quantum-mechanical calculations is still difficult,
because it involves too many particles to be calculated without simplifying assumptions [57].
1 A Historical Review of the Structures of Water and Ice
Fig. 1.17 The free energy of
a Zundel (H 5 O
+
2 ) cation as a
function of distance d
between oxygen atoms.
Black dots show hydrogen
atoms, open circles are
oxygen atoms. The
proton-transfer energy
barrier disappears when the
distance is 2.36 Å. Drawn
using data from [60]
+
Free energy (eV)
d (Å)
r OO =2.75 Å
r OO =2.55 Å
r OO =2.45 Å
r OO
|d|
r OO =2.95 Å
0
0.5
1.0
0
-0.2
-0.4
0.2
0.4
r OO =2.35 Å
which favors effective proton transfer. The rate of proton transfer by the Grotthuss
mechanism is limited by the rate of the adaptation of molecules around the hydronium
ion [61], and thus it is diffusion controlled. Consequently, ionic transport in water is
not just the electric field dragging solvated hydronium ions, but the rearrangement
of water molecules in the way suitable for fast proton exchange, with a speed that is
much higher than for conventional molecular diffusion [57]. However, the diffusion
coefficient D H + = 9.3·10
−9 m
2 /s [9, 62] is related to the charge of the proton, but not
the mass of the proton, the diffusion coefficient of which is D = 2.3·10
−9 m
2 /s, the
same as that for the water molecule (see Sect. 1.4). Water, therefore, has a structure
in which the diffusion of mass and the charge does not coincide. It is in this property that water differs from other dielectrics and ionic conductors. Interestingly, the
mobility of the proton charge in ice and water is of the same order of magnitude [63,
64]. This fact shows that the short-order molecular arrangement, which is the same
for both ice and water, defines the proton conduction in both ice and water.
18
1.3.3 Pondus Hydrogenii (pH)
The concentration of hydronium ions in pure water that has been established by DC
electrical conductivity measurements was used as a reference value for the concept
18 Note that the modeling of protonic transport by quantum-mechanical calculations is still difficult,
because it involves too many particles to be calculated without simplifying assumptions [57].