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Proper evaluation of the Gibbs free energy and other properties of seawater and other aqueous solutions is essential in the analysis of desalination systems. Standard seawater has been studied extensively and property data are readily accessible. However, many aqueous solutions requiring desalination have significantly different composition from seawater and seawater data is generally not accurate for these solutions. Experimental data for a given aqueous solution may be unavailable under the conditions of interest. Therefore, there is a need to model relevant physical properties from chemical thermodynamic principles. In particular, for solutions that are not ideal, the activity and fugacity coefficients must be considered. In this paper, the effect of nonidealities in sodium chloride (NaCl) solutions is considered through a parametric study of the least work of separation for a desalination system. This study is used to determine the conditions under which the ideal solution approximation is valid and also to determine when an NaCl solution is a good approximation to standard seawater. It is found that the ideal solution approximation is reasonable within ranges of salinities and recovery ratios typical of those found in the seawater desalination industry because many of the nonidealities cancel out, but not because the solution behaves ideally. Additionally, it is found that NaCl solutions closely approximate natural seawater only at salinities typically found in seawater and not for salinities found in typical brackish waters. NOMENCLATURE Roman symbols Units A Debye-Hückle constant L1/2/mol1/2 a activity b Davies constant L/mol c molarity mol/L solution e electron charge C F Faraday constant C/mol G Gibbs free energy J/kg Ġ Gibbs free energy flow rate J/s Ḣ enthalpy flow rate J/s Ic molar ionic strength mol/L Im molal ionic strength mol/kg M molecular weight kg/mol m molality mol/kg solvent ṁ mass flow rate kg/s Na Avogadro’s number 1/mol n number of moles mol ṅ mole flow rate mol/s p pressure Pa Q̇ heat rate J/s R universal gas constant J/mol-K r recovery ratio, mass basis kg/kg r̄ recovery ratio, mole basis mol/mol S salinity (TDS) kg solute/kg solution Ṡ entropy flow rate J/s-K T temperature K Ẇ work rate (power) J/s 1 Copyright © 2012 by ASME w mass fraction kg/kg x mole fraction mol/mol z valence of ion Greek symbols Units γc molar activity coefficient γ f fugacity coefficient γm molal activity coefficient γx rational activity coefficient ε0 permittivity of free space F/m εr relative permittivity/dielectric constant μ chemical potential J/mol ν sum of stoichiometric coefficients ρ density kg/m3 φ osmotic coefficient Subscripts a ambient b brine f feed i species (solvent or solutes) j stream (b, f , or p) least reversible operation p product s solute species sep separation 0 solvent ± mean property for anion and cation + cation − anion Superscripts id ideal nid nonideal rev reversible ◦ standard state Acronyms Units DHLL Debye-Hückle Limiting Law ppt parts per thousand g solute/kg solution ppm parts per million mg solute/kg solution TDS total dissolved solids kg solute/kg solution INTRODUCTION Desalination research is being fueled by growing water demand resulting from rising population, by increasing standards of living, and by the contamination of existing water sources. Analytical studies are a critical part of desalination research, and in order to make reliable calculations, it is essential to accurately evaluate the physical properties of the particular water source that is to be treated. Seawater has been studied in depth and seawater physical properties are well documented [1–4]. However, these properties are only appropriate for water sources that have an ionic composition similar to standard seawater. For many natural and produced waters, including river water, ground water, flowback from hydraulic fracturing, and industrial waste waters, the composition may be substantially different from that of seawater. Additionally, when studying nanofiltration systems, which may have different permeabilities for different solutes, the brine and product streams can have substantially different compositions from the feed stream. Further, scale formation in desalination systems is a direct function of the solution composition. Therefore, for many desalination related calculations, it is essential to evaluate physical properties in detail. Accurate evaluation of solution properties requires treatment of the activity and fugacity coefficients in order to properly address nonidealities. There are numerous ways to evaluate the activity coefficients, including Debye-Hückle theory and empirical data. For simplicity, it is common to use the ideal solution approximation, thus entirely avoiding the problem of setting the activity coefficients [5–9]. Unfortunately, it is unclear when this approximation is justifiable. In this paper, the validity of the ideal solution approximation is analyzed through calculation of the least work of separation. Gibbs free energy for a sodium chloride (NaCl) solution is evaluated using various property models and the least work is evaluated as a function of feed salinity and recovery ratio. The NaCl solution results are also compared to the least work calculation evaluated using seawater properties because the use of aqueous NaCl solutions is common in laboratory studies of desalination systems [10–14] as well as in industry [15]. It is found that for salinities and recovery ratios typically found in desalination systems, the ideal solution approximation has lower-than-expected error due to fortuitous cancellation of terms, rather than near-ideal solution behavior. THERMODYNAMICS AND DEFINITIONS The Gibbs free energy of a mixture is

Effect of Nonideal Solution Behavior on Desalination of a Sodium Chloride (NaCl) Solution and Comparison to Seawater