The drift speed

When the potential difference between two electrodes a distance l apart is ∆φ, the ions in the solution between them experience a uniform electric field of magnitude

E=

In such a field, an ion of charge ze experiences a force of magnitude

F=zeE=

(In this chapter we disregard the sign of the charge number and so avoid notational complications.) A cation responds to the application of the field by accelerating towards the negative electrode and an anion responds by accelerating towards the positive electrode. However, this acceleration is short-lived. As the ion moves through the solvent it experiences a frictional retarding force, F_fric, proportional to its speed. If we assume that the Stokes formula (eqn 19.12) for a sphere of radius a and speed s applies even on a microscopic scale (and independent evidence from magnetic resonance suggests that it often gives at least the right order of magnitude), then we can write this retarding force as

Ffric = fs f= 6πηa

The two forces act in opposite directions, and the ions quickly reach a terminal speed, the drift speed, when the accelerating force is balanced by the viscous drag. The net force is zero when

s =

It follows that the drift speed of an ion is proportional to the strength of the applied field. We write

s =uE

where u is called the mobility of the ion (Table 21.6). Comparison of eqns 21.41 and 21.42 and use of eqn 21.40 shows that

u==

Illustration 21.4 Calculating an ionic mobility

For an order of magnitude estimate we can take z = 1 and athe radius of an ion such as Cs+ (which might be typical of a smaller ion plus its hydration sphere), which is 170 pm. For the viscosity, we use η = 1.0 cP (1.0 × 10−3 kg m−1 s−1, Table 21.4). Then u ≈ 5 × 10−8 m2 V−1 s−1. This value means that, when there is a potential difference of 1 V across a solution of length 1 cm (so E = 100 V m−1), the drift speed is typically about 5 µm s−1. That speed might seem slow, but not when expressed on a molecular scale, for it corresponds to an ion passing about 104 solvent molecules per second.

Fig. 21.16 The mechanism of conduction by hydrogen ions in water as proposed by N. Agmon (Chem. Phys. Letts. 244, 456 (1995)). Proton transfer between neighbouring molecules occurs when one molecule rotates into such a position that an O-H···O hydrogen bond can flip into being an O···H-O hydrogen bond. See text for a description of the steps.

Because the drift speed governs the rate at which charge is transported, we might expect the conductivity to decrease with increasing solution viscosity and ion size. Experiments confirm these predictions for bulky ions (such as R4N+ and RCO2 −) but not for small ions. For example, the molar conductivities of the alkali metal ions increase from Li+ to Cs+ (Table 21.6) even though the ionic radii increase. The paradox is resolved when we realize that the radius a in the Stokes formula is the hydrodynamic radius (or ‘Stokes radius’) of the ion, its effective radius in the solution taking into account all the H2O molecules it carries in its hydration sphere. Small ions give rise to stronger electric fields than large ones (the electric field at the surface of a sphere of radius r is proportional to ze/r2 and it follows that the smaller the radius the stronger the field), so small ions are more extensively solvated than big ions. Thus, an ion of small ionic radius may have a large hydrodynamic radius because it drags many solvent molecules through the solution as it migrates. The hydrating H2O molecules are often very labile, however, and NMR and isotope studies have shown that the exchange between the coordination sphere of the ion and the bulk solvent is very rapid. The proton, although it is very small, has a very high molar conductivity (Table 21.6)! Proton and 17O-NMR show that the times characteristic of protons hopping from one molecule to the next are about 1.5 ps, which is comparable to the time that inelastic neutron scattering shows it takes a water molecule to reorientate through about 1 rad (1 to 2 ps). According to the Grotthuss mechanism, there is an effective motion of a proton that involves the rearrangement of bonds in a group of water molecules. However, the actual mechanism is still highly contentious. Attention now focuses on the H9O4 + unit, in which the nearly trigonal planar H3O+ ion is linked to three strongly solvating H2O molecules. This cluster of atoms is itself hydrated, but the hydrogen bonds in the secondary sphere are weaker than in the primary sphere. It is envisaged that the rate-determining step is the cleavage of one of the weaker hydrogen bonds of this secondary sphere (Fig. 21.16a). After this bond cleavage has taken place, and the released molecule has rotated through a few degrees (a process that takes about 1 ps), there is a rapid adjustment of bond lengths and angles in the remaining cluster, to form an H₅O₂ + cation of structure H₂O ···H+ ···OH2 (Fig. 21.16b). Shortly after this reorganization has occurred, a new H⁹O⁴⁺ cluster forms as other molecules rotate into a position where they can become members of a second ary hydration sphere, but now the positive charge is located one molecule to the right of its initial location (Fig. 21.16c). According to this model, there is no coordinated motion of a proton along a chain of molecules, simply a very rapid hopping between neighbouring sites, with a low activation energy. The model is consistent with the observation that the molar conductivity of protons increases as the pressure is raised, for increasing pressure ruptures the hydrogen bonds in water. The mobility of NH⁴⁺ is also anomalous and presumably occurs by an analogous mechanism.

اخر الاخبار

اخبار العتبة العباسية المقدسة

قسم صناعة الشبابيك يشرع بإدامة شباكين قديمين لمرقد الإمامين العسكريين (عليهما السلام)

فرقة العباس (عليه السلام) تختتم خدماتها المقدمة للزائرين في سامراء

موكب العتبتين المقدستين يحيي ذكرى استشهاد الإمام الهادي (عليه السلام) في سامراء

العتبة العسكرية المقدسة تكرّم وفد العتبة العباسية المقدسة لمشاركته في تقديم الخدمات للزائرين

المزيد

الآخبار الصحية

كنز غذائي على طبقك.. ماذا تفعل السبانخ بجسمك؟

ما لا يتوقعه أحد.. "سر غذائي" في الجبن الدسم

لتحسين الهضم والنوم.. هذا أفضل وقت لتناول العشاء

10 طرق بسيطة وغير مألوفة لحرق المزيد من السعرات يوميا

المزيد

الآخبار التكنلوجية

سابقة بعالم الطيران.. طائرة تقرر الهبوط بنفسها بعد ظرف طارئ

"الإنفلونزا الخارقة" تكتسح دول العالم.. وتثير قلقا كبيرا

مرحلة ما قبل الأعراض.. دواء جديد يستهدف "شرارة" الزهايمر

إزالة ثاني أكسيد الكربون من الغلاف الجوي.. ما دور المحيطات والبحار؟

المزيد

مواضيع ذات صلة

Monitoring the progress of a reaction

The measurement of diffusion coefficients

Solutions of the diffusion equation

Diffusion with convection

Conductivities and ion–ion interactions

Strong electrolytes

Conductance and conductivity

Superconductors

Induced magnetic moments

Magnetic susceptibility

Nonlinear optical phenomena

Light emission by solid-state lasers and light-emitting diodes