Aqueous molar volumes and density of solutions

The molar volume of a salt can be calculated from the measured densities of a salt solution and of pure water:
            
where V_m is the molar volume of the salt (cm³/mol), m is the molality (mol/kg H₂O), MW is the molecular weight of the salt (g/mol), and ρ and ρ₀ are the densities of the solution and of pure water at the same pressure and temperature, respectively (g/cm³).
Figure 1 shows the molar volumes of HCl, NaCl and CaCl₂ (symbols from measured densities, lines from models) as a function of the square root of the ionic strength (I, mol/kg H₂O) at 25°C (PHREEQC input file Vm_salt.phr)

Figure 1.

The molar volumes increase almost linearly with the square root of the ionic strength. The increase can be linked with the pressure derivative of the excess free energy in the solution, ΔV_m = ∂G_E / ∂P = -RT × ∂(ln γ) / ∂P. It was calculated from Debye-Hückel theory by Redlich and Rosenfeld in 1931:

            
where V_m⁰ is the molar volume at infinite dilution, A_v is the Debye-Hückel limiting slope (cm³/mol / √(mol/kg H₂O)), ν_i is the stoichiometric coefficient of element i in the salt, and z_i is the charge number.
The Debye-Hückel limiting slope is :
            
where R is the gas-constant (8.2 cm³ atm/mol), T is the absolute temperature (K), DH_A is the Debye-Hückel A parameter (0.51 (mol/√(mol/kg H₂O at 25°C), ε_r is the relative dielectric constant of pure water (-), P is pressure (atm), and κ₀ is the compressibility of pure water (1/atm).
The Redlich-Rosenfeld equation has been validated often, and is useful for extrapolating to infinite dilution where differences in ρ cannot be measured with sufficient precision.

Figure 1 shows that, at infinite dilution, the molar volume of NaCl is smaller than of HCl. This is, because the density of a NaCl solution increases more than is given by replacement of 1 H⁺ (1 g/mol) by 1 Na⁺ (23 g/mol). Thus, the molar volume of Na⁺ is smaller than of H⁺ since the molar volume of Cl^- is the same in both solutions. Also at I = 0, the volume of CaCl₂ is only slightly larger than of NaCl, and thus, Ca²⁺ must have even smaller volume than Na⁺.
Aqueous volumes of solute species can be defined by convention relative to the volume of H⁺ = 0 cm³/mol. The molar volume of HCl, calculated with eqn (1), is then equal to the conventional molar volume of Cl^-. If the conventional volume of Cl^- is subtracted from the molar volume of NaCl, the conventional molar volume of Na⁺ is obtained, etc. The additivity of molar volumes applies, anyhow, at infinite dilution, but is valid also in concentrated solutions as will be demonstrated further below.

The conventional molar volumes at infinite dilution are calculated with PHREEQC input file Vm0_tc.phr and shown in figure 2.

Figure 2.

The conventional volumes of Na⁺, Mg²⁺ and Ca²⁺ are smaller than of H⁺, and therefore negative. As was noted with Figure 1, this is because the density of a NaCl solution increases more than follows from replacing H⁺ by Na⁺. The density difference is wholly attributed to Na⁺, but physically, it is due to compaction of water molecules around the Na⁺ ion by electrostatic attraction of the dipoles on H₂O to the ion. The divalent cations Mg²⁺ and Ca²⁺ need more water molecules for charge compensation than Na⁺, and their conventional volumes are more negative. Also, HCO₃^- has the largest volume of all the ions shown in Figure 2, but the divalent anion CO₃^2- has a much smaller (even negative) molar volume. The pressure effect on the solubility of carbonates is therefore strongly pH-dependent.

It can be concluded that the dielectric properties of water determine the conventional aqueous molar volumes of the ions. The volume change that takes place when an ion is transferred from vacuum into water can be calculated from the pressure derivative of the solvation energy, which is given (approximately) by the Born equation:
            
where ΔG_s is the solvation energy (cal/mol), N_Avogadro is Avogadro's number (6.022e23/mol), q_e is the electron charge (1.602e-19 C), ε₀ is the dielectric permittivity of vacuum (3.704e-11 C²/cal/m), and r_i is the ion radius (m).
The variation of the pressure derivative of the solvation energy with temperature and pressure allows estimating how the aqueous volume will change with T and P. The (T, P) dependent pressure derivative is shown in Figure 3 (PHREEQC input file eps_r.phr).

Figure 3.

The pressure derivative of the Born function is a parabola of similar shape as the molar volumes shown in Figure 2 and is useful for extrapolating and fitting the molar volumes at infinite dilution as a function of temperature and pressure. It is part of the equation used by SUPCRT for calculating V_m⁰.

Figures 1-3 show that the aqueous molar volumes are a complex function of of temperature, pressure, and solution composition, which PHREEQC calculates with the SUPCRT-modified-Redlich-Rosenfeld (SmoRR) equation:

            
and
            
The coefficients a_1...4, W, å and b_1...4 are entered with '-Vm a₁ a₂ a₃ a₄ W å b₁ b₂ b₃ b₄' when SOLUTE_SPECIES are defined. For example,
      SOLUTION_SPECIES
      Ca+2 = Ca+2
       -gamma 5.0 0.1650
       -dw 0.793e-9
       -Vm -0.3456 -7.252 6.149 -2.479 1.239 5 1.60 -57.1 -6.12e-3 1 # supcrt modified

Input files that compare measured and calculated (T, P, I)-dependent molar volumes are in the directory
c:\phreeqc\high_P_T\Vm_sol.
The files are installed by phreeqc3.Installer.exe’.

With the molar volumes defined, the density of a solution can be calculated from eqn (1)’:
            

Figure 4 compares measured and calculated densities (PHREEQC input files rho_NaKMgCaCl.phr and rho_MgNaClSO4.phr), showing how well the additivity of molar volumes applies even at (very) high concentrations.

Figure 4.

Another example is given in density of evaporating seawater .

See Appelo, Parkhurst and Post (2014), GCA 125, 49-67, and Appelo (2015), AG 55, 62-71 for detailed information on PHREEQC`s molar-volume calculations and pressure effects on solubilities. (Ask Tony Appelo for reprints)

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