Nernst equation, derivation

The shift in the voltammogram for a metal ion in the presence of a ligand may be used to determine both the metal-ligand complex s stoichiometry and its formation constant. To derive a relationship between the relevant variables we begin with two equations the Nernst equation for the reduction of O... 

Traditionally concepts of ion selective permeation of biological membranes have centered on differences in the effective radii of hydrated nuclei. An example of that perspective derives from consideration of the resting membrane potential, E, which in the squid axon is approximated by the Nernst equation... 

Analogously to eqn. 3.72 for stationary electrodes and a reversible redox couple of soluble ox and red, Kies derived for chronopotentiometry at a dme via insertion in the Nernst equation... 

Comparison with Eq. (2.10) shows that the measured potential is simply the difference between the equilibrium potentials of the two redox couples, each measured with respect to its own reference electrode. Admittedly, this is an obvious result, but it is useful to derive it from first principles. The corresponding Nernst equation is ... 

This derivation is based on the Nernst equation written in terms of ionic activities, but pH is usually discussed in terms of concentration. 

Both half- and overall reaction tendencies change with temperature, pressure (if gases are involved), and concentrations of the ions involved. Thus far, we have only been concerned with standard conditions. Standard conditions, as stated previously, are 25°C, 1 atm pressure, and 1 M ion concentrations. An equation has been derived to calculate the cell potential when conditions other than standard conditions are present. This equation is called the Nernst equation and is used to calculate the true E (cell potential)... 

Figure 5.13 pH dependence of Eswitch arid at 45°C as determined from the derivatives of data such as those shown in Fig. 5.1 I.The solid line indicates the best fit to the Eswitch data.Also shown is the potential of the 2H /H2 potential at 4S°C and I bar hydrogen as calculated from the Nernst equation.

The Nernst equation defines the equilibrium potential of an electrode. A simplified thermodynamic derivation of this equation is given in the Sections 5.3 to 5.5. Here we will give the kinetic derivation of this equation. 

The standard electrode potential 2 ° edta for half-reaction 15.37 is then derived from the Nernst equation for reaction 15.38 ... 

The temperature was 25°C (the standard condition) and atmospheric pressure was 751.0 Torr. Because the vapor pressure of water is 23.8 Torr at 25°C, PHi n the cell was 751.0 - 23.8 = 727.2 Torr. The Nernst equation for the cell, including activity coefficients, is derived as follows ... 

In the case of gases, properties may be tabulated til terms of their existence at 0°C and 760 mm pressure, To determine the volume of a gas at some different temperature and pressure, corrections derived from known relationships (Charles , Amonton s. Gay-Lussac s, and other laws) must be applied as appropriate. In tile case of pH values given at some measured value (standard for comparison), the same situation applies. Commonly, lists of pH values are based upon measurements taken at 25°C. The pH of pure water at 22°C is 7.00 at 25,JC, 6.998 and at 100°C. 6.13. Modern pH instruments compensate for temperature differences through application of the Nernst equation. 

In such a case the one-electron reduction potential of A(Ered) may be shifted to the positive direction with an increase in the concentration of M according to Eq. 4, which is derived from the Nernst equation of the one-electron reduction potential in the presence of M [38] ... 

Dividing by — nF, we obtain the Nernst equation, named after Walther Nernst (1864-1941), the German chemist who first derived it ... 

Equation (5.9) is the general Nernst equation giving the concentration dependence of the equilibrium cell voltage. It will be used in the next section of this chapter to derive the equilibrium electrode potential for metal/metal-ion and redox electrodes. 

The potential of this electrode is defined (Section 5.2) as the voltage of the cell Pt H2(l atm) H+(<2 = 1) MZ+ M, where the left-hand electrode, Et = 0, is the normal hydrogen reference electrode (described in Section 5.6). We will derive the Nernst equation on the basis of the electrochemical kinetics in Chapter 6. Here we will use a simplified approach and consider that Eq. (5.9) can be used to determine the potential E of the M/Mz+ electrode as a function of the activity of the products and reactants in the equilibrium equation (5.10). Since in reaction (5.10) there are two reactants, Mz+ and e, and only one product of reaction, M, Eq. (5.9) yields... 

Yonezawa et al. (1994,1995) derived a model on the basis of the Nernst equation to describe the rate of dissolution of a monodisperse system that can account for various initial amounts of solute, as long as it is less than that needed to saturate the solutionz-iaraequation was derived from the Hixson-Crowell treatment under nonsink conditions. The general form afUniS equation can be expressed as follows ... 

This is the Nernst equation defined from the electrode kinetics considerations. Later, we derive the same relationship on purely thermodynamic grounds. 

It is important to note that the electrode potential is related to activity and not to concentration. This is because the partitioning equilibria are governed by the chemical (or electrochemical) potentials, which must be expressed in activities. The multiplier in front of the logarithmic term is known as the Nernst slope . At 25°C it has a value of 59.16mV/z/. Why did we switch from n to z when deriving the Nernst equation in thermodynamic terms Symbol n is typically used for the number of electrons, that is, for redox reactions, whereas symbol z describes the number of charges per ion. Symbol z is more appropriate when we talk about transfer of any charged species, especially ions across the interface, such as in ion-selective potentiometric sensors. For example, consider the redox reaction Fe3+ + e = Fe2+ at the Pt electrode. Here, the n = 1. However, if the ferric ion is transferred to the ion-selective membrane, z = 3 for the ferrous ion, z = 2. 

The complete current-voltage characteristics of the sensor can be derived from the similar consideration that was used for derivation of the i-E curve for liquid electrolytes. Because the potentials at each electrode are reversible, their difference can be expressed by the Nernst equation for the concentration of oxygen at the anode Co(0) and at the cathode Co (A). The current flowing through the layer generates a voltage drop iRb, where Rb is the bulk resistance of the ZrC>2 layer. 

Derive the Nernst equation from the Butler-Volmer equation. 

The knowledge of the surface potential for the dispersed systems, such as metal oxide-aqueous electrolyte solution, is based on the model calculations or approximations derived from zeta potential measurements. The direct measurement of this potential with application of field-effect transistor (MOSFET) was proposed by Schenk [108]. These measurements showed that potential is changing far less, then the potential calculated from the Nernst equation with changes of the pH by unit. On the other hand, the pHpzc value obtained for this system, happened to be unexpectedly high for Si02. These experiments ought to be treated cautiously, as the flat structure of the transistor surface differs much from the structure of the surface of dispersed particle. The next problem may be caused by possible contaminants and the surface property changes made by their presence. 

As electrochemical reactions are, at their heart, chemical reactions, their thermodynamics depend on the concentrations of the species involved, as well as the temperature. The Nernst equation describes this dependence. Derivations of the Nernst equation are available in many standard texts (4-6). For our purposes, it will be simply stated that for a reaction described by... 

Nernst equation — A fundamental equation in -> electrochemistry derived by - Nernst at the end of the nineteenth century assuming an osmotic equilibrium between the metal and solution phases (- Nernst equilibrium). This equation describes the dependence of the equilibrium electrode - potential on the composition of the contacting phases. The Nernst equation can be derived from the - potential of the cell reaction (Ecen = AG/nF) where AG is the - Gibbs energy change of the - cell reaction, n is the charge number of the electrochemical cell reaction, and F is the - Faraday constant. 

Nernst equilibrium — It was - Nernst who first treated the thermodynamical - equilibrium for an -> electrode [i], and derived the - Nernst equation. Although the model used by Nernst was not appropriate (see below) the Nernst equation - albeit in a modified form and with a different interpretation - is still one of the fundamental equations of electrochemistry. In honor of Nernst when equilibrium is established at an electrode, i.e., between the two contacting phases of the electrode or at least at the interface (interfacial region), it is called Nernst equilibrium. In certain cases (see - reversibility) the Nernst equation can be applied also when current flows. If this situation prevails we speak of reversible or... 

Peters equation — Obsolete term for the - Nernst equation in the special case that the oxidized and reduced forms of a redox pair are both dissolved in a solution and a reversible potential is established at an inert metal electrode. Initially Nernst derived his equation for the system metal/metal ions, and it was Peters in the laboratory of -> Ostwald, F. W. who published the equation for the above described case [i]. The equation is also sometimes referred to as Peters-Nernst equation [ii]. 

This relationship can be derived from the Nernst Equation... 

A difference in chemical potential of species j across a membrane causes the ratio of the passive flux densities to differ from 1 (Fig. 3-12), a conclusion that follows directly from Equation 3.26. When fi° is equal to fip the influx balances the efflux, so no net passive flux density of species j occurs across the membrane (Jj = JJ1 - J°ut by Eq. 3.16). This condition (fi° = fij) is also described by Equation 3.5, which was used to derive the Nernst equation (Eq. 3.6). In fact, the electrical potential difference across a membrane when... 

Hung derived a general expression for calculating the distribution potential from the initial concentrations of ionic species, their standard ion transfer potentials, and the volumes of the two phases [19]. When all ionic species in W and O are completely dissociated, and the condition of electroneutrality holds in both phases, the combination of Nernst equations for all ionic species with the conservation of mass leads to... 

The chloride ion concentration at any point on the titration curve can be calculated from equation 20-5, which can be derived from the Nernst equation. 

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