Equilibrium constant from half-cell potentials

Equation 2.16 shows that potentiometry is a valuable method for the determination of equilibrium constants, ffowever, it should be borne in mind that the system should be in equilibrium. Some other conditions, which are described below, also need to be fulhlled for use of potentiometry in any application. The basic measurement system must include an indicator electrode that is capable of monitoring the activity of the species of interest, and a reference electrode that gives a constant, known half-cell potential to which the measured indicator electrode potential can be referred. The voltage resulting from the combination of these two electrodes must be measured in a manner that minimises the amount of current drawn by the measuring system. This condition includes that the impedance of the measuring device should be much higher than that of the electrode. 

This equation allows one to compute the chemical equilibrium constant from measured standard-state electrochemical cell potentials (usually referred to as standard cell potentials). Some standard half-cell potentials are given in Table 14.6-1. The standard potential of an electrochemical cell is obtained by combining the two relevant half-cell potentials. 

Calculation of the Equilibrium Constant from Standard Half-Cell Potentials... 

Using Eq. (17.50), we can calculate the equilibrium constant for any reaction from the standard cell potential which, in turn, can be obtained from the tabulated values of the standard half-cell potentials. The following method and examples illustrate a procedure that will ensure obtaining the with both a correct sign and magnitude. 

Since the values of equilibrium constants are obtained from the standard half-cell potentials, the method of obtaining the S° of a half-cell has great importance. Suppose we wish to determine the of the silver-silver ion electrode. Then we set up a cell that includes this electrode and another electrode the potential of which is known for simplicity we choose the SHE as the other electrode. Then the cell is... 

The voltage for a complete reaction is the difference between the potentials of the two half-reactions E = E+ — E, where E+ is the potential of the half-cell connected to the positive terminal of the potentiometer and E is the potential of the half-cell connected to the negative terminal. The potential of each half-reaction is given by the Nemst equation E = E° — (0.059 16/n) log Q (at 25°C), where each reaction is written as a reduction and Q is the reaction quotient. The reaction quotient has the same form as the equilibrium constant, but it is evaluated with concentrations existing at the time of interest. Electrons flow through the circuit from the electrode with the more negative potential to the electrode with the more positive potential. 

All species are aqueous unless otherwise indicated. The reference state for amalgams is an infinitely dilute solution of the element in Hg. The temperature coefficient, dE°/dT, allows us to calculate the standard potential, E°(T), at temperature T E°(T) — Ec + (dE°/dT)AT. where A T is T — 298.15 K. Note the units mVIK for dE°ldT. Once you know E° for a net cell reaction at temperature T, you can find the equilibrium constant, K, for the reaction from the formula K — lOnFE°,RTln w, where n is the number of electrons in each half-reaction, F is the Faraday constant, and R is the gas constant. 

Because we can calculate E° from standard potentials, we can now also calculate equilibrium constants for any reaction that can be expressed in terms of two half-reactions. Toolbox 12.2 summarizes the steps involved, and Example 12.7 shows the steps in action. Equation 6 also shows that the magnitude of E° for a cell reaction is an indication of the equilibrium composition. It follows from the equation that a reaction with a large positive E° has a very large K. A reaction with a large negative E° has a K much less than 1. 

The equilibrium constant of a reaction can be calculated from standard potentials by combining the equations for the half-reactions to give the reaction of interest and determining the standard potential of the corresponding cell. 

The free energies in (18) are illustrated in Fig. 10. It can be seen that GA is that part of AG ° available for driving the actual reaction. The importance of this relation is that it allows AGXX Y to be calculated from the properties of the X and Y systems. In thermodynamics, from a list of n standard electrode potentials for half cells, one can calculate j (m — 1) different equilibrium constants. Equation (18) allows one to do the same for the %n(n— 1) rate constants for the cross reactions, providing that the thermodynamics and the free energies of activation for the symmetrical reactions are known. Using the... 

In the half-cell of Eq. (5.24), the concentration of AgClj" must be small compared to that of Cl-, or a liquid-junction potential will result because the mobilities of AgClJ and Cl- are not the same. Thus, for a reference electrode of the second kind to be elfective in cells without appreciable junction potentials, the equilibrium constant for the reaction of Eq. (5.25) must be smaller than unity (preferably <0.1). In water, methanol, formamide, and V-methyl-formamide, this criterion is met, but in most organic solvents the equilibrium constant for the reaction of Eq. (5.25) ranges from 30 to 100. The silver chloride electrode is not recommended for general use in organic solvents.27... 

The third largest class of enzymes is the oxidoreductases, which transfer electrons. Oxidoreductase reactions are different from other reactions in that they can be divided into two or more half reactions. Usually there are only two half reactions, but the methane monooxygenase reaction can be divided into three "half reactions." Each chemical half reaction makes an independent contribution to the equilibrium constant E for a chemical redox reaction. For chemical reactions the standard reduction potentials ° can be determined for half reactions by using electrochemical cells, and these measurements have provided most of the information on standard chemical thermodynamic properties of ions. This research has been restricted to rather simple reactions for which electrode reactions are reversible on platinized platinum or other metal electrodes. 

When the pH is specified, each biochemical half reaction makes an independent contribution to the apparent equilibrium constant K for the reaction written in terms of reactants rather than species. The studies of electochemical cells have played an important role in the development of biochemical thermodynamics, as indicated by the outstanding studies by W. Mansfield Clarke (1). The main source of tables of ° values for biochemical half reactions has been those of Segel (2). Although standard apparent reduction potentials ° can be measured for some half reactions of biochemical interest, their direct determination is usually not feasible because of the lack of reversibility of the electrode reactions. However, standard apparent reduction potentials can be calculated from for oxidoreductase reactions. Goldberg and coworkers (3) have compiled and evaluated the experimental determinations of apparent equilibrium constants and standard transformed enthalpies of oxidoreductase reactions, and their tables have made it possible to calculate ° values for about 60 half reactions as functions of pH and ionic strength at 298.15 K (4-8). 

Half a century ago it was shown by physical chemists from the laws of thermodynamics that the equilibrium constant of the over-all cell reaction can be calculated from the potential of the cell. In fact, we can calculate from standard potentials of the couples as given in Table 32-2 values of equilibrium constants for the couples. These values are also given in the table. 

Before we discuss redox titration curves based on reduction-oxidation potentials, we need to learn how to calculate equilibrium constants for redox reactions from the half-reaction potentials. The reaction equilibrium constant is used in calculating equilibrium concentrations at the equivalence point, in order to calculate the equivalence point potential. Recall from Chapter 12 that since a cell voltage is zero at reaction equilibrium, the difference between the two half-reaction potentials is zero (or the two potentials are equal), and the Nemst equations for the halfreactions can be equated. When the equations are combined, the log term is that of the equilibrium constant expression for the reaction (see Equation 12.20), and a numerical value can be calculated for the equilibrium constant. This is a consequence of the relationship between the free energy and the equilibrium constant of a reaction. Recall from Equation 6.10 that AG° = —RT In K. Since AG° = —nFE° for the reaction, then... 

Tables of this sort are extremely useful, because they feature much chemical and electrical information condensed into quite a small space. A few electrode potentials can characterize quite a number of cells and reactions. Since the potentials are really indices of free energies, they are also ready means for evaluating equilibrium constants, complex-ation constants, and solubility products. Also, they can be taken in linear combinations to supply electrochemical information about additional half-reactions. One can tell from a glance at an ordered list of potentials whether or not a given redox process will proceed spontaneously.

Write down the expression for the copper-zinc electrochemical cell. Write the reducing reactions for the half-cells and the redox reaction for the whole cell. Assume that equilibrium has been reached and from the standard Cu /Cu and Zn" /Zn potentials calculate the equilibrium constant. 

In electrical and electrochemical processes, electrical work is defined as the product of charges moved (Q) times the potential (E) through which it is moved. If this work is done in an electrochemical cell in which the potential difference between its two half-cells is E, and the charge is that of 1 mol of reactant in which n mol of electrons are transferred, then the electrical work (w) done by the cell must be - E. In this relationship, the Faraday constant F is required to convert coulombs from moles of electrons. In an electrochemical cell at equilibrium, no current flows and the energy change occurring in a reaction is expressed in Eq. (4.1). 

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