Chemical potential, real

Note that a constant of integration p has come mto the equation this is the chemical potential of the hypothetical ideal gas at a reference pressure p, usually taken to be one ahnosphere. In principle this involves a process of taking the real gas down to zero pressure and bringing it back to the reference pressure as an ideal gas. Thus, since dp = V n) dp, one may write... 

In connection with the thermodynamic state of water in SAH, it is appropriate to consider one more question, i.e., their ability to accumulate water vapor contained in the atmosphere and in the space of soil pores. It is clear that this possibility is determined by the chemical potential balance of water in the gel and in the gaseous phase. In particular, in the case of saturated water vapor, the equilibrium swelling degree of SAH in contact with vapor should be the same as that of the gel immersed in water. However, even at a relative humidity of 99%, which corresponds to pF 4.13, SAH practically do not swell (w 3-3.5 g g1). In any case, the absorbed water will be unavailable for plants. Therefore, the only real possibility for SAH to absorb water is its preliminary condensation which can be attained through the presence of temperature gradients. 

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

According to the above equations, the surface potential is the difference between the real and chemical potentials of charged particles dissolved in the liquid phase. 

By virtue of the function (3.6), concentrations, which are readify determined parameters, can be used instead of chemical potentials in the thermodynamic equations for ideal systems. The simple connection between the concentrations and chemical potentials is lost in real systems. To facilitate the changeover from ideal to nonideal systems and to avoid the use of two different sets of equations in chemical thermodynamics,... 

Using activities instead of chemical potentials has the major advantage that the equations derived for ideal systems can be retained in the same form for real systems, but with activities in the place of concentrations. For the practical application of these equations, we must know the values of activity as a function of concentration. 

In real mixtures and solutions, the chemical potential ceases to be a linear function of the logarithm of the partial pressure or mole fraction. Consequently, a different approach is usually adopted. The simple form of the equations derived for ideal systems is retained for real systems, but a different quantity a, called the activity (or fugacity for real gases), is... 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

The expression for the chemical potential of a component of a real solution can be separated into two terms ... 

Consider the transport of gaseous species A from a bulk gas to a bulk liquid, in which it has a measurable solubility, because of a difference of chemical potential of A in the two phases (higher in the gas phase). The difference may be manifested by a difference in concentration of A in the two phases. At any point in the system in which gas and liquid phases are in contact, there is an interface between the phases. The two-film model (Whitman, 1923 Lewis and Whitman, 1924) postulates the existence of a stagnant gas film on one side of the interface and a stagnant liquid film on the other, as depicted in Figure 9.4. The concentration of A in the gas phase is represented by the partial pressure, pA, and that in the liquid phase by cA. Subscript i denotes conditions at the interface and 8g and are the thicknesses of the gas and liquid films, respectively. The interface is real, but the two films are imaginary, and are represented by the dashed lines in Figure 9.4 hence, Sg and 8( are unknown. 

Figure 2.10 Schematic illustration of the pressure dependence of the chemical potential of a real gas showing deviations from ideal gas behaviour at high pressures.

In mixtures of real gases the ideal gas law does not hold. The chemical potential of A of a mixture of real gases is defined in terms of the fugacity of the gas, fA. The fugacity is, as discussed in Chapter 2, the thermodynamic term used to relate the chemical potential of the real gas to that of the (hypothetical) standard state of the gas at 1 bar where the gas is ideal ... 

There is also a topological term which is essential in order to satisfy the t Hooft anomaly conditions [32-34] at the effective Lagrangian level. It is important to note that respecting the t Hooft anomaly conditions is more than an academic exercise. In fact, it requires that the form of the Wess-Zumino term is the same in vacuum and at non-zero chemical potential. Its real importance lies in the fact that it forbids a number of otherwise allowed phases which cannot be ruled out given our rudimentary treatment of the non-perturbative physics. As an example, consider a phase with massless protons and neutrons in three-color QCD with three flavors. In this case chiral symmetry does not break. This is a reasonable realization of QCD for any chemical potential. However, it does not satisfy the t Hooft anomaly conditions and hence cannot be considered. Were it not for the t Hooft anomaly conditions, such a phase could compete with the CFL phase. 

To complete our discussion of non-relativistic superfluids let us briefly mention some of the alternatives to the LOFF and DFS phases. One possibility is that the system prefers a phase separation of the superconducting and normal phases in real space, such that the superconducting phase contains particles with the same chemical potentials, i.e. is symmetric, while the normal phase remains asymmetric [20, 21],... 

Fig. 1-6. Energy level of a charged particle i in a condensed phase e, = energy of particle i p, = electrochemical potential a, = real potential Pi = chemical potential z, = charge number of particle i VL = vacuum infinity level OPL = outer potential levd.

Fig. 2-8. Ihe electrochemical potential, p., the real potential, a, and the chemical potential, , of electrons in metals 4 = inner potential X = surface potential = outer potential MS= metal surface VL = vacuum infinity level.

The real potential and the chemical potential of electrons in metals... 

The real potential, a , of electrons in metals, as shown in Eqn. 2-4, comprises the electrostatic surface term, - ex, due to the surface dipole and the chemical potential term, M., determined by the bulk property of metal crystals. In general, the electrostatic surface term is greater the greater the valence electron density in metals whereas, the chemical potential term becomes greater the lower the valence electron density in metals. 

The unitary real potential, ay., of the surface metal ion consists of the chemical potential, Py, and the electrostatic surface term e x as shown in Eqn. 3-7 ... 

Fig. S-14. Energy change in hydration of A ions p = outer potential of aqueous solution x = surface potential Pa (.,) il A <.q)) = unitary electrochemic (chemical) potential of hydrated A ions Oa ( i) unitary real potential of hydrated A ions = Pa ( )-zex( ...

As mentioned in Chap. 1, the ion level represented by the real potential a consists of the chemical potential and the electrostatic surface energy Zi e x (a + 2i e x). Since the surface potential, x, of aqueous solutions is constant (x 0.13 V), the relative energy level of hydrated ions may be expressed in terms of the chemical potential... 

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