From Pure Water to Concentrated Salt Solutions
Ionic Strength and Ionic Activity
Ionic Strength and Ionic Activity
Up to now we have considered ions dissolved in pure water, but seawater contains quantities of ions. The parameter used to characterize aqueous solutions with different amounts of oppositely charged electric charges (ions) is ionic strength, I. It is defined as half the summation of the concentrations multiplied by the respective squared ionic charge (z).
I = 0.5*∑c_{i}*z_{i}^{2}
The sum encompasses all ions present in the medium so that, for a NaCl solution, we have
I = 0.5*([Na^{+}]*1 + [Cl^{-}]*1)
Although NaCl is the salt most responsible for the salinity of water, the properties of seawater and a pure NaCl solution with the same concentration are different. For the standard seawater composition used here the ionic strength is approximately 0.7, which corresponds to a salinity of around 35 (grams of salts per kg of water)
Cl = 0.54586 ' Cl- Mol/kg(solution)
Na = 0.46906 ' Na+ Mol/kg(solution)
Mg = 0.05282 ' Mg++ Mol/kg(solution)
Ca = 0.01028 ' Ca++ Mol/kg(solution)
SO4 = 0.02824 ' SO4-- Mol/kg(solution)
K = 0.01021 ' K+ Mol/kg(solution)
Br = 0.00084 ' Br- Mol/kg(solution)
Sr = 0.00009 ' Sr++ Mol/kg(solution)
F = 0.00007 ' F- Mol/kg(solution)
B = 0.00042 ' B(OH)3 + B(OH)4- Mol/kg(solution)
The ionic strength of seawater may be calculated from salinity (DOE, 1994)
I = 19.924/(1000 – 1.005*S)
The behaviour of an ion dissolved in water depends on the electrical interaction with the other ions present in solution. Therefore the chemical 'activity' of an ion dissolved in fresh water and in seawater is quite different.
The activity of a chemical species, denoted by {A}, is strictly related to its concentration by the activity coefficient γ(A) :
{A} = γ(A)*[A]
For infinite dilution, the activity coefficient is 1, but it decreases as the solution becomes more concentrated. If you consider a simple electrolyte, deviation from ideal behaviour can be described through the effect of (relatively) long-range electrostatic interactions. For those interactions, approximations can be derived to describe the dependence of activity coefficients γ (i) on ionic strength I.
Seawater has however a higher ionic strength which is, in turn, due to the presence of different electrolyte charges: the combination of these two facts leads to the formation of 'ion pairing’ and complex formation in the electrolyte mixture.
As an example, bivalent charged carbonate ions may associate to positively charged Mg^{++} or Na^{+} ions forming aliovalent ion pairs such as NaCO3^{-} or MgCO3^{0}
According to Skirrow (1975), the most important ion pairing equilibria in seawater are:
Ca^{++} + CO3^{- -} CaCO3^{0}
Mg^{++} + CO3^{- -} MgCO3^{0}
Na^{+} + CO3^{- -} NaCO3^{-}
Ca^{++} + HCO3^{- -} CaHCO3^{+}
Mg^{++} + HCO3^{- -} MgHCO3^{+}
Na^{++} + HCO3^{- -} NaHCO3^{+}
The electrostatic interaction of the CO3 ^{- -} ions with opposite charges in solution isn't the only factor that decreases the activity of the ion. The ion pairing greatly impairs the same activity, as the carbonate ion in seawater is not 'free', being combined in neutral or lower charged (aliovalent) species in solution.
Considering the effect of ion pairing on activities, it is useful to talk about 'free' and 'total'
activity coefficients. If no ion pairing occurs, like in dilute solutions, free and total activity coefficients coincide. In seawater however, the total activity coefficient can be dramatically lower than the free one, as many of the bivalent carbonate ions form ion pairs.
If no ion-pairing occurs, the free activity coefficient γf_{ }of an ion in simple electrolyte solutions varies with ionic strength I according to the Debye-Hückel limiting law
log( γf ) = - A z^{2 }√I valid for I< 0.005
or to the Davies equation
log( γf ) = - A z^{2 }(√I / (√I + 1) – 0.2*I) valid for I<0.5
with
A = 1.82*10^{6}(εT)^{-1.5},
where^{ }ε
≈ 79 is the dielectric constant of water, and T is the absolute
temperature in K. At 25°, A is about 0.5 for water. Z indicates
the charge of the ion and I the ionic strength of the solution.
As the ionic strength of seawater is approximately 0.7, which is only slightly higher than the limit of the Davies equation, it should be used in a reasonable way. However, this equation and the Debye-Hückel limiting law no longer apply since they only hold for dilute solutions and simple electrolytes (as opposed to concentrated solutions and electrolyte mixtures of unlike charges) (Zeebe 2001).
The problem will be tackled in this handbook by using empirical formulas for the equilibrium constants, which employ concentrations, and without the need to calculate activities. The same approach will be used for the temperature, pressure and salinity dependence of these constants. The fitting of experimental data has been carried out by DOE 1994, Millero 1995, Weiss 1974 et al. A comprehensive review of such data can be found in Zeebe 2001, Appendix A.
Before going any further, three topics must be clarified, pH, salinity and density.
In chemical oceanography, three main different pH scales are currently used; free, total and seawater. This point is not to be neglected; when dealing with acidity constants of hydrogen ion transfer reactions (as is the case of H2CO3) the use of a consistent pH scale is mandatory. The values of different pH scales in seawater differ by up to 0.12 units (Zeebe 2001).
The pH value is in theory defined as the negative logarithm of the activity of hydrogen ions
pH = -log(10) a(H+)
Unfortunately, individual ion activities cannot be determined experimentally. Indeed, the concentration of one single ion cannot be varied independently, because electroneutrality is required. Therefore the 'free' pH scale for seawater has been proposed:
pH_{F} = -log(10)[H^{+}]_{F}
where [H^{+}]_{F} stands for the free hydrogen ion concentration, including hydrated forms, like H3O^{+} and H9O4^{+} (Dickson 1984)
Indeed, free protons do not exist in any significant amount in aqueous solutions. Rather, the proton is bonded to a water molecule thus forming H3O^{+}. This in turn is hydrogen bonded to three other water molecules to form an H9O4^{+ }ion^{ }(Dickson 1984, p.2299). To be noted that, as usual in ocean chemistry, the concentrations in square brackets are expressed in mol/Kg (water) and not in mol/L as is more usual in general chemistry.
In 1973, Hanson defined a total scale for pH so as to include the effect of sulphate ions in its definition.
pH_{T} = -log(10)[H^{+}]_{T} where [H^{+}]_{T} = [H^{+}]_{F} + [HSO4^{ - }]
The bisulphate ion (HSO4^{ -}) is a rather weak acid (Ka1 ≈ 2.1∙10^{-2}) so it is not completely dissociated in H^{+} and SO4^{ – }ions. Once the Ka is known, a relationship between the two scales can be inferred. This in turn requires an accurate value of Ka in seawater, which would be difficult to obtain. But by using Hanson’s total pH scale in seawater, the calculation of Ka for bisulphate ion can be avoided.
This total scale will be used in the empirical expressions for the various acidity constants, including the ionic water product. This choice seems to conflict with the usual free pH scale as used in general chemistry, but is necessary due the use of the total scale in the experimental determination of the various constants. Therefore, in the code shown below, this total scale will be employed.
The third scale is the so-called seawater scale, which only slightly differs from the preceding one. The need to introduce this scale is due to the presence of fluoride ions (F ^{-) }in seawater. Consequently, we have to account for the protonation of F ^{–} ions according to the equilibrium:
HF H^{+} + F ^{- } with a Ka2 ≈ 3.5∙10^{-4}
Indeed, hydrofluoric acid is a weak acid. In standard seawater however, the concentration of fluoride ions is 7.0∙10^{-5 }Mol/Kg of water, about 400 times lower than the concentration of sulphate ions, 2.8∙10^{-2 }Mol/Kg, therefore the seawater scale differs by no more than 0.01 pH units from the total scale. In the following table the transformation between the three pH scales are shown, both in terms of concentrations and pH units.
[H^{+}]_{T} = [H^{+}]_{F} ∙(1 + [SO4^{ - - }]/Ka1 ) |
[H^{+}]_{SW} = [H^{+}]_{F} ∙(1 + [SO4^{ - - }]/Ka1 + [F^{ - }]/Ka2 ) |
[H^{+}]_{SW} = [H^{+}]_{T} ∙(1 + [F^{ - }]/Ka2 ) |
pH_{T} = pH_{F} - log_{10}(1 + [SO4^{ - - }]/Ka1 ) |
pH_{SW} = pH_{F} - log_{10}(1 + [SO4^{ - - }]/Ka1 + [F^{ - }]/Ka2 ) |
pH_{SW} = pH_{T} - log_{10}(1 + [F^{ - }]/Ka2 ) |
If
Ka1≈ 2.1∙10^{-2}
and [SO4^{ -
- }] =
2.8∙10^{-2 } then the
difference between pHT
and pHF
scale would be ≈ 0.37 pH unit, but this would be valid only in
pure water.
Under the same conditions (pure water) then the difference between pHSW and pHT scale would be ≈ 0.37 pH unit.
Seawater composition varies widely, although the relative ratios of dissolved species are nearly constant. So in standard simulations only salinity can vary, while the relative composition of seawater remains constant. Salinity is expressed in grams of dissolved species per kg of solution. The values are taken from DOE 1994 (with borates).
[Cl-]
= 0.54586 * S/35 mol/kg(solution) |
For those who enjoy modifying code, rewriting that part of the code with other compositions might work, if the composition is not too far from electrical neutrality. Indeed, the neutrality is always assured by the addition of hydrogen (H^{+}) or hydroxyl (OH^{-}) ions, which in turn changes the pH. But, if the initial salt composition is very unbalanced, the resulting pH may be outside 0-14 limits, causing the program to crash.
The standard seawater composition as listed may be supersaturated with respect to calcite or aragonite formation. Calcite is the less soluble form of calcium carbonate, so theoretically it should be the first to precipitate. Coral reef is however made up of aragonite, fact that should be considered.
The calculation of the density of seawater requires a number of steps (Millero and Poisson, 1981; Gill, 1982; Zeebe, 2001). In the following, the density of seawater (roSTP) is expressed in kg m^{-3}, or g dm^{-3}) and is calculated in function of temperature (in °C), pressure p(in bars) and salinity.
First the density of pure water (roPw) is calculated as a function of temperature with a fifth degree polynomial. (see code below).
As a second step, the density of seawater at 1 bar (i.e. P = 0) is calculated as a function of salinity starting from roPw and employing a second order mixed polynomial, whose first two coefficients depend on temperature according to a fourth and second degree polynomial.
As a third and final step, the density of seawater according to pressure P (roSTP) is given using the secant bulk modulus Ksb3, which in turn derives from Ksb2 and Ksb1, as shown in the code below.
Care must be taken for two reasons. First, in ocean chemistry, the unit measure for pressure is the ‘bar’, which is similar but not equal to atmosphere: 1.000 atm = 1.01325 bar. For those who like S.I. Units, 1 atm = 101325 Pa (pascal) and 1 bar = 0.1 MPa (megapascal). Second, the algorithms employed only assume the effect of water column pressure, so on the sea surface itself the pressure is assumed to be zero. This is unrealistic, as it neglects air pressure being nearly equal to 1 atm at sea level, but should be used as the empiric algorithms assume so. The code is listed in the appendix (code006.bas).
In ocean chemistry, the use of a non-standard concentration scale is widely diffused, the so-called gravimetric unit. It is expressed by the moles (mol) of a solute per kg of solution. It differs therefore from molarity (used in most chemistry, being the number of moles per litre of solution) and from molality (number of moles per kg of solvent, here water). If not otherwise stated, the gravimetric scale will be used here.
Knowing the composition of seawater, changing from the gravimetric scale to molality involves the following steps:
1- Calculation of the total mass of the substances in 1 kg of solution, on the basis of their atomic weights and their concentration.
2- Calculation of the mass of water in 1 kg of solution, the difference being 1.000 - Σmass of solutes.
3- The gravimetric concentration of each ion or substance is divided by the above mass of water to give the corresponding molality.
Knowing the density of seawater, and changing from the gravimetric scale to molarity involves the following steps:
1- Calculation of the density of the seawater solution, according to the above procedure and expressing the result in kg/liter.
2- The gravimetric concentration of each ion or substance is multiplied by the above density to give the corresponding molarity.