Electroanalytical Chemistry. Gary A. Mabbott

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СКАЧАТЬ under the influence of an electric field, they also migrate in the direction of the oppositely charged electrode. The velocity of an ion caused by an electric field is sometimes called the drift velocity or the migration velocity. It is proportional to the strength of the electric field, ε, driving the current.

      (1.29)

      where the electric field, ε, has the units of V/cm. It is the voltage difference between the electrodes divided by the distance between them. v is the drift velocity of the ion in cm/s and the proportionality constant, u, is the ion mobility. The units for the ion mobility are cm²/(s V). The reason that the mobilities vary among ions is the fact that collisions with solvent molecules and other particles cause drag. Drag is related to the size of the ion. In this context, the size of the ion includes the sheath of solvent molecules that the ion drags with it, its solvent sphere. The bigger the solvated ion, the greater the viscous drag force opposing the ion's movement. All ions are slowed down by the viscosity of the solution. Because the viscosity decreases with temperature, the conductance increases with temperature as indicated in Eq. (1.28). The ion mobility is also proportional to the charge on the ion. Also, the bigger the charge, the greater the tug that the electric field exerts on the ion. Each ion carries a fraction of the total current in proportion to its mobility and its contribution to the total number of charges in solution. An important consequence of the variation in ion mobilities is the fact that the current is shared unevenly among the ions.

      There are two practical applications of conductance or conductivity measurements that are of interest to analytical chemists. The first application is a semiquantitative estimate of ion concentration. One can calibrate conductivity measurements for an accurate quantitative determination of a specific salt, if that is the only source of ions in the sample. However, such a situation constitutes a special case. In the area of environmental science and agriculture, conductivity measurements are often made as a general indicator of water purity. Conductivity is also a parameter that is monitored at the outlet of a water purifying system commonly used for analytical and biochemistry laboratories. The quality of the water is often described in terms of the specific resistance. Theoretically, the specific resistance of water with no ions other than those from the dissociation of water is 18.3 MΩ.

There is one common application where conductivity provides good data for quantitative determinations of specific ions. Conductivity detectors are very popular for ion chromatography. In this case, the conductivity cell has been miniaturized so that it can operate on volumes on the microliter scale. Ions eluting from the separation column flow through the electrochemical device and cause a surge in conductance in proportion to their concentration. A full discussion of how a conductivity detector works in liquid chromatography can be found elsewhere [14].

1.6 Mass Transport by Convection and Diffusion

      The movement of ions and molecules in solution is important in many different aspects of electrochemical analysis. The term “mass transport” is often used to mean that reactant material is being driven by some force to the surface of an electrode. The rate at which reactant material is brought to the electrode surface influences the sensitivity of methods in many cases. The two most common mass transport mechanisms are convection and diffusion. In the first case, the bulk solution is mechanically stirred or pushed past an electrode such as in a flowing stream. The term “hydrodynamic system” is also used to mean a flowing or stirred solution that continuously brings material to the electrode.

The other mechanism for mass transport that is exploited in electroanalysis is called diffusion. Diffusion moves material by the force of a concentration gradient. This mechanism is subtler and deserves some discussion here. Imagine two solutions separated by a square window, 1 cm on each edge (see Figure 1.12). Molecules move very rapidly at room temperature, but they are frequently colliding with each other and the solvent. Consequently, the path of any individual molecule changes direction many times per second. The molecule appears to be moving randomly. How fast it moves depends on its solvated radius. Imagine also that one can count the molecules that pass through the window in each direction. The net excess going one direction or the other per second is called the flux for that molecule. A flux has the same dimensions as the product of a concentration and a velocity. The normal units are mol/(cm² s) (equivalent to mol/cm³ × cm/s). When the concentration for some molecule, M, on both sides of the window is equal, the number going from left to right through the window matches the number going from right to left each second. Consequently, the flux is zero.

Figure 1.12 The definition of flux is the net number of moles of molecules per second crossing a plane of solution with an area of 1 cm².

Now, imagine starting the experiment over with a concentration of M at a value of C_M on the left side of the window and a concentration of 0 on the right side of the window. Because there are no molecules on the right side initially, none move through the window from right to left. However, many are going from left to right initially. Consequently, the flux is not zero initially. For ease of discussion, let the direction of left to right represent movement along the x‐axis in the positive direction. Intuitively, the flux is never going to go from right to left as long as the C_M on the left is greater than it is on the right. It will never be greater on the right. At best, the concentration on the right will reach a value that equals that on the left, but only at equilibrium. Because there are more molecules to consider on the left side, the probability is always greater for a molecule to move from left to right through the window than in the other direction until equilibrium is reached. It also seems intuitive that the bigger the discrepancy in the concentrations on the two sides, the greater the excess in the number of molecules going in one direction. The following equation is a more elegant statement of these ideas. It is known as Fick's first law of diffusion.

      (1.30)

      where J_M is the flux of molecule, M, in mol/(cm² s), C_M is the concentration of M. The proportionality constant, D_M, is called the diffusion coefficient in cm²/s. The gradient in concentration is the driving force for moving molecules across the plane perpendicular to the direction of motion. By convention a decreasing concentration in the x‐direction is represented by a negative gradient, that is dC/dx < 0 in that case. The negative sign in front of the diffusion coefficient arises in order to make the flux positive for a concentration that decreases in the direction of increasing x. (It is just a convention.) The diffusion coefficient is related to the ion mobility described earlier by the Einstein–Smoluchowski equation [15]:

      (1.31)

As with ion mobility, the diffusion coefficient decreases with the solvated radius of an ion. The diffusion coefficients for a few ions in water are given in Table 1.2. Note that the diffusion coefficients for OH⁻ and H⁺ are rather large despite the fact that their hydrated radii are also large. That is the case because these ions do not move through water as an individual particle, but rather by exchange of hydrogen ions with molecules of water in their solvation sphere [5].

TABLE 1.2 Diffusion СКАЧАТЬ