Titration curves coulometric

Obtaining a coulometric titration curve is often far from straightforward, since the requirement for faradaic charge insertion makes for slow and painstaking work. 

The phase diagram of an electrode material may be determined from the slope of the coulometric titration curve. An electrode of N components shows activities which are independent of the composition as long as the maximum number of N phases are in equilibrium with each other. Relative changes in the amounts of the different phases do not change the activities of the components and therefore keep the cell voltage constant. This causes voltage plateaux for any region of the equilibrium of the maximum number of phases. 

Fig. 8.7 Coulometric titration curve for the addition of Li to Sb and Bi. The plateaux indicate two-phase regions whereas the voltage drops correspond to single phase regions.

Fig. 8.8 Gnbbs energy of formation of Lij+,Sb as a function of the composition. The curve is derived from the integration of the coulometric titration curves.

The slope of the coulometric titration curve is accordingly proportional to the Wagner factor in the case of predominant electronic conductivity. 

Fig. 8.9 Coulometric titration curve for the ternary system Li-In-Sb. Plateaux correspond to three-phase regions. The phases are indicated.

A variety of other kinetic quantities may be readily derived from this information and the coulometric titration curve. Only a few examples will be given. 

As discussed in Section 8.2 the relation between the chemical diffusion coefficient and diffusivity (sometimes also called the component diffusion coefficient) is given by the Wagner factor (which is also known in metallurgy in the special case of predominant electronic conductivity as the thermodynamic factor) W = d n ajd In where A represents the electroactive component. W may be readily derived from the slope of the coulometric titration curve since the activity of A is related to the cell voltage E (Nernst s law) and the concentration is proportional to the stoichiometry of the electrode material ... 

Q Spreadsheet Summary In the second experiment in Chapter 11 of Applications of Microsoft Excel in Analytical Chemistry , a spreadsheet is developed to plot a coulometric titration curve. The end point is located by first- and second-derivative methods. 

Construct a coulometric titration curve of 100.0 mL of a 1 M H2SO4 solution containing Fe(ll) titrated with Ce(lV) generated from 0.075 M Ce(lll). The titration is monitored by potentiometry. The initial amount of Fe(II) present is 0.05182 mmol. A constant current of 20.0 mA is used. Find the time corresponding to the equivalence point. Then, for about 10 values of time before the equivalence point, use the stoichiometry of the reaction to calculate the amount of Fe produced and the amount of Fe + remaining. Use the Nemst equation to find the system potential. Find the equivalence point potential in the usual manner for a redox titration. For about 10 times after the equivalence point, calculate the amount of Ce " produced from the electrolysis and the amount of Ce + remaining. Plot the curve of system potential versus electrolysis time. 

FIGURE 9.1. Coulometric titration curves for the systems Li-Sb and Li-Bi using a molten salt electrolyte for lithium ions. The equilibrium cell voltage is plotted with reference to elemental lithium as a function of the lithium concentration. 

The discussed techniques may be extended to the determination of phase diagrams with any number of components. A general treatment is described in the literature. The determination of thermodynamic data from the coulometric titration curves will be illustrated in the next chapter. 

The variation of the Gibbs energy of formation of a phase with an even narrower stoichiometric width may be determined by integration of the coulometric titration curve as shown in Figure 9.9 for LiAlCl4. The data are given relative to the ideal stoichiometiy. The resolution is well below 1 J/mol and is hard to be matched by other techniques. 

The diffusivity of the mobile ions in a predominantly electronic conducting sample may be calculated from the chemical diffusion coefficient and the slope of the coulometric titration curve according to the relation... 

The Wagner enhancement factor d In a ,/d In c , may be determined from the coulometric titration curve according to Equation (9.32). When using Gil l, all the necessary information is already included in the quantities AEj, AE and the stoieMometiy of the sample. Therefore, D may be directly obtained from the formnla given in Table 9.2. 

Typical examples of coulometric titration curves are reported in Fig. 11 for La2Ni04+5 compound at different temperatures [Nakamura et al, 2009b]. 

Diffusion resistance R can be generally expressed as a function of a slope of the coulometric titration curve dV/dx of the host material with molar share X of diffusing ions [34] ... 

Where C = oxygen concentration in gas space, V = volume of gas space, = molar volume of the sample, A = surface area of tne sample, = oxygen-discharge reaction rate constant, = oxygen diffusion coefficient, dV/d5 = thermodynamic factor (slope of coulometric titration curve at a selected sample non-stoichiometry 8), and = diffusion layer or film thickness. 

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