Chemical potential numerical values

Seeing the difficulty of working numerically with the chemical potential, whose value goes to minus infinity when the concentration goes to zero, G. N. Lewis invented a new property, which he named the fugacity/ -, defined by... 

Numerical solution of Eq. (51) was carried out for a nonlocal effective Hamiltonian as well as for the approximated local Hamiltonian obtained by applying a gradient expansion. It was demonstrated that the nonlocal effective Hamiltonian represents quite well the lateral variation of the film density distribution. The results obtained showed also that the film behavior on the inhomogeneous substrate depends crucially on the temperature regime. Note that the film exhibits different wetting temperatures on both parts of the surface. For chemical potential below the bulk coexistence value the film thickness on both parts of the surface tends to appropriate assymptotic values at x cx) and obeys the power law x. Such a behavior of the film thickness is a consequence of van der Waals tails. The above result is valid when both parts of the surface exhibit either continuous (critical) or first-order wetting. 

The curves were determined from Eqs. 24"—26" in order to apply these in a numerical calculation one first has to know the values of the following functions at — 3°C Afz, the difference in chemical potential between the "empty Structure II lattice and ice Cpg. the Langmuir constant for propane in the larger cavities of Structure II (Cpi = 0 for geometrical reasons) Cmi> Cm2> the Langmuir constants for methane in the two types of cavities of Structure II. 

The concept of potential energy in mechanics is one example of a scalar field, defined by a simple number that represents a single function of space and time. Other examples include the displacement of a string or a membrane from equilibrium the density, pressure and temperature of a fluid electromagnetic, electrochemical, gravitational and chemical potentials. All of these fields have the property of invariance under a transformation of space coordinates. The numerical value of the field at a point is the same, no matter how or in what form the coordinates of the point are expressed. 

To understand potentiometric methods, those that measure electrical potentials and determine analyte concentrations from these potentials, it is necessary that numerical values for these tendencies be known under conventional standard modes and conditions. What are these modes and conditions First, all halfreactions must be written as either reductions or oxidations. Scientists have decided to write them as reductions. Second, the tendencies for half-reactions to proceed depend on the temperature, the concentrations of the chemical species involved, and, if gases are involved, the pressure in the half-cell. Scientists have defined standard conditions to be a temperature of 25°C, a concentration of exactly 1 M for all dissolved chemical species involved, and a pressure of exactly 1 atm. Third, because every cell consists of two half-cells, it is not possible to measure the value directly. However, if we were to assign the tendency of a certain half-reaction to be zero, then the tendencies of all other half-reactions can be determined relative to this reference half-reaction. 

Here

work function, is the chemical potential of electrons in the metal, and Sxois the change of the metal surface potential upon contact with the solution. Hence, the modification of electronic distribution in the metal is due to the adsorbed solvent molecules, which change the surface potential of the metal, dxo- A similar concept was developed in numerous works of Trasatti (e.g.. Ref. 30). The value of Sxo at [Pg.7]

The numerical value of an electrode potential depends on the nature of the particular chemicals, the temperature, and on the concentrations of the various members of the couple. For the purposes of reference, half-cell potentials are taken at the standard states of all chemicals. Standard state is defined as 1 atm pressure of each gas (the difference between 1 bar and 1 atm is insignificant for the purposes of this chapter), the pure substance of each liquid or solid, and 1 molar concentrations for every nongaseous solute appearing in the balanced half-cell reaction. Reference potentials determined with these parameters are called standard electrode potentials and, since they are represented as reduction reactions (Table 19-1), they are more often than not referred to as standard reduction potentials (E°). E° is also used to represent the standard potential, calculated from the standard reduction potentials, for the whole cell. Some values in Table 19-1 may not be in complete agreement with some sources, but are used for the calculations in this book. 

The voltage delivered by a battery is determined by the innate tendencies of the active materials to lose electrons and by the concentrations of these materials. In Table 11.2, the listed numerical values representing the various electrode potentials were obtained experimentally under comparable conditions of concentration and temperature. The relative position of any electrode included in that list would be changed if the experimental measurements were made using a sufficiently different concentration of ions surrounding the electrode. The influence of a change in concentration on these equilibria is wholly in accord with facts presented earlier in connection with the study of reversible reactions and chemical equilibria. 

In electrochemistry we make it a rule that the standard chemical potential ju. of hydrogen ions is set zero as the level of reference for the chemical potentials of all other hydrated ions. The standard chemical potentials of various hydrated ions tabulated in electrochemical handbooks are thus relative to the standard chemical potential of hydrogen ions at unit activity in aqueous solutions. Table 9.3 shows the numerical values of the standard chemical potential, the standard partial molar enthalpy h°, and the standard partial molar entropy. 5 ,° for a few of hydrated ions. 

In most cases the necessary material constant can be determined by direct measurement. In practice however, because of the time and cost required to measure the numerous types and combination possibilities of plastics and contacting media, only a limited selection of such experimental constants is available. Consequently in practice one cannot avoid using estimated values. Such estimations are possible within a degree of accuracy adequate for practical purposes, when the chemical structure of the migrating substance, the polymer and the contacting media are known. Thermodynamic terms are used to characterize the equilibrium distribution of a diffusant substance between plastic (P) and contacting media (e.g. a liquid L). The most important of these terms is the chemical potential i. 

Equation 6.36 for the adiabatic potential is exact within the framework of the mean field description. However, the structure of the electric part P1 is too complex to disclose its analytic properties. Here we examine the adiabatic potential numerically following the Carlson theory of elliptic integrals [15-21], To proceed with numerical computation, it is necessary to enter a set of parameters designed to describe an experimental situation. It will not surprise the reader who has made it this far that we use values of the chemically fixed parameters specified by the n-butylammonium vermiculite gels [22], namely m+ = 74 mp and m = 36 mp. The average density n0 of the small ions is given by... 

Since one expects, for bound molecules, that the chemical potential p will be negative, the fact that J(Vp)2jp dr is a positive quantity shows from equation(120) that deviations from relation (84) of the simplest density treatment can, in principle, be of either sign, depending on the relative magnitudes of the chemical potential and density gradient corrections. We shall discuss numerical values for the deviation from equation (84) in Section 16 below, in the light of equation (120). But before doing this it is of interest to re-examine the theoretical basis of Walsh s rules. 

The formulation provided here has the great advantage that the standard chemical potential is now independent of the mode of specifying the composition of solutions. The standard state in each instance refers to that of pure i at temperature T and unit pressure. The price paid for this simplification is that the interrelations between ai(T,P,qi) and qt are now slightly more complex than those involving ai(T,P,qi) and qA when q - c,m. The at introduced in Eq. (3.6.3) will be termed the relative activity. Note that whereas ai(T,l,c ) and ai(T,l,m ) differ from unity, a1(T,l,q ) - 1 for all qi. Equations (3.6.3) also show explicitly that ai(T,P,xi) -ai(T,P,ci) = ai(T,P,mi) all have the same numerical values, independent of the composition units, whereas the corresponding... 

The numerical value of kt in (2-3) depends on how activity is defined and on the units in which concentration is expressed (molarity, mole fraction, partial pressure). Measurement of the absolute activity, or chemical potential, of an Individual ion is one of the classical unsolved problems. Since we cannot measure absolute ion activity, we are then necessarily interested in the next best—comparative changes in activities with changing conditions. To obtain comparative values numerically, we measure activity with respect to an arbitrarily chosen standard state under a given set of conditions of temperature and pressure, where the substance is assigned unit activity. The value of ki in (2-3) thus depends on the arbitrary standard state chosen accordingly, the value of the equilibrium constant also depends on the choice of standard states. 

The right-hand side is formally identical with Eq. (2.10.2b). However, there is a difference in the numerical values of in the two cases, because the reference and standard values of the chemical potential of species i are slightly different (see also Query 1.10.1). Moreover, the left-hand side depends only on temperature, with the pressure set equal to 1 bar. One also recovers Eqs. (2.10.3), (2.10.4), except that now P = 1 bar. These changes demonstrate again that equilibrium constants cannot be uniquely specified. 

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