The Equality of Chemical Potentials

Consider a closed system containing k nonreacting components and two phases I and II, that reaches equilibrium at constant temperature T and pressure P. We will demonstrate next that the requirement that the total Gibbs free energy of the system is minimized at equilibrium, leads to the equality of chemical potentials in the two phases for any component i. From Eq. 12.3.9  

In order that Eq. 12.4.2 be valid for all values of the Independent variables Nj(7), 2(7),., A (7), each bracketed term must be equal to zero. Hence  

We conclude, therefore, that the chemical potential of component i has the same value in both phases that are in equilibrium at constant temperature and pressure. (Should this be expected from the discussion of Section 12.3.2 ) 

This is demonstrated numerically in the next Example, while in Example 12.4 we use this equality to develop the Clapeyron equation and in Section 12.4.4 to conclude that the fiigacities of component i in the two phases are also equal to each other. 

Show that the vaporization of saturated water at 1 atm to saturated steam at the same pressure, leads to no change of the chemical pot tial the chemical potential in die two equilibrium phases is the same. 

---

The criterion for phase equilibrium is given by Eq. (8.14) to be the equality of chemical potential in the phases in question for each of the components in the mixture. In Sec. 8.8 we shall use this idea to discuss the osmotic pressure of a... 

Thermodynamics provide a straightforward method for quantifying this situation. The criterion for equilibrium is the equality of chemical potential in... 

There is a similar expression for polymer activity. However, if the fluid being sorbed by the polymer is a supercritical gas, it is most useful to use chemical potential for phase equilibrium calculations rather than activity. For example, at equilibrium between the fluid phase (gas) and polymer phase, the chemical potential of the gas in the fluid phase is equal to that in the liquid phase. An expression for the equality of chemical potentials is given by Cheng (12). 

Earlier, Gavach et al. studied the superselectivity of Nafion 125 sulfonate membranes in contact with aqueous NaCl solutions using the methods of zero-current membrane potential, electrolyte desorption kinetics into pure water, co-ion and counterion selfdiffusion fluxes, co-ion fluxes under a constant current, and membrane electrical conductance. Superselectivity refers to a condition where anion transport is very small relative to cation transport. The exclusion of the anions in these systems is much greater than that as predicted by simple Donnan equilibrium theory that involves the equality of chemical potentials of cations and anions across the membrane—electrolyte interface as well as the principle of electroneutrality. The results showed the importance of membrane swelling there is a loss of superselectivity, in that there is a decrease in the counterion/co-ion mobility, with greater swelling. 

The equality of chemical potentials in a first-order phase transition leads to two important relationships. The first is the Clapeyron equation1... 

Most of these aspects of water-sorption equilibrium correspond to the equality of chemical potentials of water in the medium and in the polymer. The consequences of this principle are illustrated by the experiment of Fig. 14.2, where an interface is created between water and a nonmiscible liquid (oil, hydrocarbon, etc.), and a polymer sample is immersed into the organic liquid. It can be observed that, despite the hydrophobic character of the surrounding medium, the sample reaches the same level of water saturation as in direct water immersion or in a saturated atmosphere. What controls the water concentration in the polymer is the ratio C/Cs of water concentrations in the organic phase, where Cs is the equilibrium concentration, which can be very low but not zero. In other words, hydrophobic surface treatments can delay the time to reach sorption equilibrium but they cannot avoid the water absorption by the substrate. 

We now consider more on the equality of chemical potentials at equilibrium. 

Partial molar Gibbs free energies (chemical potentials) can most easily be measured by taking advantage of the equality of chemical potentials in different phases at equilibrium. The chemical potential of a component of a solution is its chemical potential in the vapor that is in equilibrium with the solution. Thus, if the component has a measurable vapor pressure, it can be used to determine its chemical potential in the solution. Assuming that the vapor can be considered an ideal gas, its chemical potential is determined by its vapor pressure ... 

By successively considering pairs of phases, we may readily generalize to more than two phases the equality of chemical potentials the result for 7r phases is... 

Prove that the common tangent construction is equivalent to the equality of chemical potentials of the phases whose compositions are given by the points of tangency. 

In 1951, Robert Thomas Sanderson (1912-1989) introduced the principle of electronegativity equalization that proposes, when two or more atoms combine, the atoms adjust to the same intermediate Mulliken electronegativity (Sanderson, 1951). Density functional theory tells us that the Mulliken electronegativity is the negative of the chemical potential (Parr et al., 1978). Sanderson s principle then becomes very appealing in that it can be considered analogous to a macroscopic thermodynamic phenomenon - the equalization of chemical potential. When atoms interact, the electronegativity, or chemical potential, must equalize. 

Equation (1) is the central equation of LAST, specifying the equality of chemical potential in the bulk gas and the adsorbed phase (which is assumed to be ideal in the sense of Raoult s law). Equation (2) calculates the spreading pressure from the pure-component isotherm. The total amount adsorbed and the selectivity are given by equations (3) and (4), respectively. 

The general condition for phase equilibrium is the equality of chemical potentials. For practical reasons this is replaced by the equality of the fugacity/of each of the... 

The theory of DNA compactization proposed in Refs. 34 and 35 is based on two main assumptions, the volume approximation and the condition of electroneutrality. The total volume of a system is divided into two parts, the volume occupied by the DNA coil and the external solution. The size of the macromolecule and the composition of the solvent within these two parts are determined by the usual equilibrium conditions the equality of osmotic pressures and the equality of chemical potentials of the components. In addition it is proposed that these two parts are electrically neutral, i.e., the net charge of the DNA macromolecule is compensated by oppositely charged counterions moving freely but within the effective volume of the DNA macromolecule. 

Hardness has been calculated in various other ways. For example, a five-point finite difference formula has been used [55] to approximate (d2E/BN2). The equality of chemical potential with the total electrostatic potential at the covalent radius [56-58] has been made use of in calculating rj. The electron density required for this work [56] has been obtained from a self-consistent numerical solution of a quadratic Euler-Lagrange equation [59,60]. Orsky and Whitehead [61] have proposed another defi-... 

The equilibrium condition is the equality of chemical potentials in the gas phase and in the adsorbed layer, therefore an adsorption isotherm follows naturally... 

One can therefore use either the equality of chemical potentials or fu-gacities as the condition for equilibrium. 

This diagram is worth careful thought. It illustrates several things that are useful in understanding activities, chemical potentials, and standard states, such as the absolute nature of chemical potentials and the necessity of using differences, the equality of chemical potentials in each phase, and the arbitrary nature of the standard state. To further illustrate the la.st point, suppose we choose a new energy level for the standard state more or less at random, such that (/xa — when Xa is 0.5 is 5000 cal mol" . This implies a value of oa of 10 and this in turn defines the physical... 

For any component i of a multicomponent, multiphase system, derive (4-12), the equality of fugacity, from (4-8), the equality of chemical potential, and (4-11), the definition of fugacity. 

Here, in contrast to Eq. (12-3) in the strong-electrolyte case, the equality of chemical potentials is assumed to imply the existence of a chemical equilibrium. Since it is impossible to freeze the reaction [Eq. (12-38)] in order to study the components independently, the chemical-equilibrium assumption is nonoperational. 

As was previously shown (see Figure 2.24), in undersaturated solution the mineral is dissolved, in oversaturated it is formed and in meta-saturated these processes are as if absent. In actuality the dissolution and miner-ogenesis always run simultaneously but at different rate. Balance means that these rates are equal, which corresponds with the equality of chemical potential of reagents and process products. The equilibrium between the solution and the mineral is determined by the product values of solubility or directly the solubility. 

The condition of equilibrium of phases is the equality of chemical potentials of components in both phases ... 

The last Eq. (2.116) expresses the equality of chemical potentials in both phases. As a result only the equality is valid in (2.112) and therefore all processes in this model of a two-phase system may be considered those of zero entropy production and equilibrium (and those reversible). Namely, it is an analogical result as for model A in Sect. 2.2 because we confine to the model without memory assuming the independence of wjd) and we confine to the system the equilibrium of which (between... 

In Chapter 8 we discussed the mechanical stability of a pure fluid in terms of the behavior of a subcritical isotherm on a Pv diagram. A sample isotherm is shown at the top of Figure 9.3, computed using the van der Waals equation of state. Also in Chapter 8 we showed that pure-fluid vapor-liquid equilibrium states are found by solving the equilibrium conditions (9.1.8). The equality of chemical potentials in (9.1.8) can also be expressed as an equality of fugacities in the case of pure-fluid vapor-liquid equilibria. 

These so-cajted field or formulation variables are. with temperataure and pressure, the intensive physicochemical variables that define the thermodynamic equilibrium conditions of the system through the equality of chemical potentials. The typical expression of the chemical potential of a surfactant molecule in solution (either in oil or water) can be written as ... 

Here, we explore the reciprocal case usually treated in chemical reactivity, namely, to derive the energetic shape for chemical valence behavior starting from the equalization of chemical potentials of atoms in molecules [92-95],... 

The last equation can be solved if one takes into account the equality of chemical potentials of a component in two co-existing phases of a stratified system. 

---