Molecular Thermodynamics of Electrolyte Solutions

The following sections are included:

• Preface
• Foreword to Second Edition
• Contents

The following sections are included:

• Prolegomena
• Concentration Units

In solution thermodynamics, the primary quantity par excellence is the activity coefficient, which gives the non-ideal behavior of the solute species. Since electrolyte solutions obey the same thermodynamics, we shall give a brief review of solution thermodynamics…

In order to understand the interactions of charged species, some knowledge of the basic laws of electrostatics is required. We introduce below elementary principles of electrostatics: Coulomb’s law, Poisson’s law and Gauss’ law. Reader interested in more details should consult a book on electrostatics. The foundation for all three laws rests on the first law: Coulomb’s law, which we have alluded to in Eq. (1.1.6). The other two laws can be derived from (1.1.6).

The activity coefficients of electrolyte solutions show a peculiar $\left(-\sqrt{I}\right)$ (minus square root of I, I is the ionic strength) dependence at small salt concentrations. This dependence is rarely seen in the activity coefficients of neutral species. The origin of this negative square root dependence was first solved in the work of Debye and Hückel (1923). They made several ingenious simplifications based on the electrostatics and Boltzmann distribution. The result is the Debye-Hückel limiting law that firmly establishes this $\left(-\sqrt{I}\right)$ dependence…

Figure 4.2.1 shows that the Debye-Hückel ln γ_± is way off compared to the data (far too negative) and is inaccurate for practical calculations. Many attempts have been made to improve the accuracy. The methods of Güntelberg, Guggenheim, Davies, Bromley, and Meissner are such modifications. We cite only one of the more accurate ones that is much used in industry: the correlation due to Pitzer. We remark that Pitzer’s formulation is very accurate and correlated for many salts of interest (see Appendix I). However, it is semi-empirical with many fitted parameters. More rigorous approaches will be introduced in Chapter 7.

The molecular basis of thermodynamic properties is found in statistical mechanics. A substance is composed of “molecules” and falls within the study of statistical mechanics. Recognizing that this book is not to teach statistical mechanics, we summarize the basics of this important discipline of science below. Our purpose is to apply the principles of statistical mechanics to electrolyte solutions; to understand their behavior; to improve the results; and to advance, when possible, the state of art.

We have seen that the Debye-Hückel results are inaccurate in describing concentrated salt solutions. Pitzer’s equation with its virial coefficients does a remarkable job in giving accurate answers. However, Pitzer’s formulation is largely empirical (fitting constants to salt data) and difficult to generalize to multi-solvent electrolytes. Since we have introduced the statistical mechanical approach, and we know what approximations were made in the Debye-Hückel theory, we can relax the conditions in DH to see if we can make improvements. One of the severest restrictions in DH is the point ion assumption: that ions, no matter how small (such as Li⁺ ion with Pauling radius = 0.6 Å) or how big (as Br^– ion with Pauling radius = 1.95 Å), were assumed to be geometrical points, having no volume of exclusion (excluding other objects from invading the ion core). If we give the ions their proper volumes in the Ornstein-Zernike equations, what would happen to the physical properties of the ions? Would the result be better? This problem was solved in 1970s in the mean spherical approximation, (MSA or mean spherical model). During the 1980s and 1990s, this approach has been shown to give more accurate results than the Debye-Hückel theory, and have a theoretical background that Pitzer’s approach lacks. We shall adopt this approach as the basis for a number of industrial applications (i.e. acid gas treating and absorption refrigeration with electrolyte refrigerants).

Starting with Debye and Hückel in 1923, the electrolyte solutions have been studied with the solvent molecules rendered implicit and replaced by their permittivity (i.e. as a dielectric continuum with permittivity ε_m). This is known as the McMillan-Mayer picture or the McMillan-Mayer scale. This assumption greatly reduced the difficulty of treatment. However, when comparing with experimental data, the solvent has to be restored, namely account taken for the presence of the solvent molecules. All the formulas will have to be modified accordingly. To understand the difference in properties, we explain by a set of experiments from isobaric to isochoric to isoplethic…

The following sections are included:

• Introduction
• The Setchenov Principle
• The Furter Correlation
• The Taylor Expansion of the Activity Coefficients
• The Gibbs-Duhem Relation for Multi-Solvent Systems
• The Kirkwood-Buff Solution Theory for Multi-Solvent Systems

The following sections are included:

• Introduction
• The Ornstein-Zernike Integral Equations and their Closures
• The Numerical Solution Methods
  • Successive substitutions — Picard’s method
  • Successive substitutions — in Fourier space
  • Renormalization/Optimization of the total/direct correlations
• The Hypernetted Chain and the Zero-Separation Closures
• The Behavior of the Bridge Functions for Molten Salts
• Characterization of the Bridge Function
  • Development of a theory for the bridge functions
  • Renormalized bridge function
• Isothermal Compressibility and Moment Rules
  • Isothermal compressibility
  • The zeroth moment condition: The electroneutrality
  • The second moment condition

The following sections are included:

• Introduction
• The Poisson-Boltzmann Equation
• The Gouy-Chapman Theory
  • The Linear Gouy-Chapman equation
  • The sum rules for electrolytes at EDL
    • The electric field at the wall
    • The Contact value theorem
    • Grahame’s equation
  • Linear Gouy-Chapman equation — with boundary conditions
  • Nonlinear Gouy-Chapman equation
• The Stern Layer
• The Zeta Potential, ζ
• Beyond the Poisson-Boltzmann Theory
  • Ornstein-Zernike based integral equations
  • The BBGKY hierarchy
  • The Wertheim-Lovett-Mou-Buff equation
  • The Kirkwood hierarchy
• The DLVO theory
  • The potential of mean force (PMF)
    • The closure relation
    • The zero-separation theorem
  • The DLVO interaction potentials
• The Dual-Yukawa Interaction Potentials
• The Density Functional Theory for Inhomogeneous Systems
  • The Classical Density Functional Theory
  • Expansion of the Singlet DCF and the Bridge Functional
  • The Third-Order Ornstein-Zernike Relation
  • Constructing the Bridge Function with Aid of OZ3
  • Adsorption of the Yukawa Fluid on a Hard Wall — Version 1
  • Electric Double Layer: Absorption of Ions on a Hard Wall — Version 2

The following sections are included:

• Introduction
• The Absorption Refrigeration Cycle
• The Energy Balances in the Absorption Cycle
  • The individual equipment energy balance
  • The coefficient of performance, COP
• The Thermodynamic Formulas for Enthalpy Calculation
• The Efficiencies and Enthalpies in Absorption Cycle
  • Enthalpies and vapor pressures of electrolyte solutions
  • The coefficients of performance (COP)

The following sections are included:

• Introduction
• Overview of Acid Gas Treating
  • The absorption process
  • Amine solutions and the chemical reactions
• The Thermodynamic Framework
  • Hard sphere mixtures
  • The Born contribution
  • The activity coefficients of ions
  • The activity coefficients of neutral species — UNIFAC
• Practical Calculations for Acid Gas Vapor Pressures
• Results for Amine Based Acid Gas Treating
  • Acid gas loading curves
  • The speciation in amine solutions
  • The heat of absorption of acid gases
  • Hydrocarbon solubility in amine solutions
• Remarks on Other Acid Gas Treating Chemicals

The following sections are included:

• Appendix I
• Appendix II
• Appendix III
• Appendix IV
• Appendix V
• Bibliography
• Index