25.5: Electrolytes Solutions are Nonideal at Low Concentrations
A solution of a strong electrolyte such as NaCl in water is perhaps one of the most obvious systems to consider but unfortunately it is also one of the more difficult ones. The reason is that the salt produces two charged solutes Na^{+} and Cl^{+} (both in hydrated form) in solution. First of all we need to consider the dissociation process and its stoichiometry as we are bringing more than one solute species into solution.
Secondly, we need to consider electrostatic interactions between solutes. The charges introduce a strong interaction that only falls off with r^{1} as opposed to r^{6} or so if only neutral species are present. This causes very serious divergence from ideality even at very low concentrations. Consider a salt going into solution:
\[C_{ν_+}A_{ν_} \rightarrow ν_+C^{z+} + ν_A^{z} \label{Eq1}\]
where: \(ν_+\) and \(ν_\) are the stoichiometric coefficients and \(z_+\) and \(z_\) the formal charges of the cations and anions resp. As we shall see the stoichiometric coefficients involved in the dissociation process are important for a proper description of the thermodynamics of strong electrolytes. Charge neutrality demands:
\[ν_+z_+ + ν_ z_ = 0 \label{Eq2}\]
Thermodynamic potentials versus the dissociation
For the salt we can write:
\[μ_2 =μ_2^o + RT \ln a_2 \label{Eq3}\]
However we need to take into account the dissociation, to do so we write:
\[μ_2 = ν_+μ_+ + ν_μ_ \label{Eq4}\]
Obviously this implies:
\[μ_2^o = ν_+μ_+^o + ν_μ_^o \label{Eq5}\]
where
\[μ_+ =μ_^o + RT\ln a_+ \label{Eq6}\]
\[μ_ =μ_^o + RT\ln a_ \label{Eq7}\]
Usually Henry's law is taken as standard state for both type of ions. Admittedly, these equations are rather formal. We cannot measure the activities of the ions separately because it is impossible to add one without adding the other, nevertheless we can derive a useful formalism with them that takes into account the dissociation process.
If we substitute the last two equations in the ones above we get:
\[ν_+\ln a_+ + ν_ \ln a_=\ln a_2 \label{Eq8}\]
Taking the exponent of either side of Equation \(\ref{Eq8}\), we get:
\[a_2 =a_+^{ν_+}a_^{ ν} \label{Eq9}\]
Notice that the stoichiometric coefficients (Equation \(\ref{Eq1}\)) are exponents in Equation \(\ref{Eq9}\). We now introduce the sum of the stoichiometric coefficients:
\[ν_+ + ν_ = ν \label{Eq10}\]
and define the mean ionic activity \(a_±\) as:
\[a_±^ ν ≡ a_2 =a_+^{ν+}a_^{ ν}\]
Note
The mean ionic activity \(a__{±\)} and the activity of the salt are closely related but the relationship involves exponents due to stoichiometric coefficients involved in the dissociation process. For example:
 For Na_{1}Cl_{1}: ν= 1+1 = 2: a_{±} ^{ 2 }= a_{NaCl}
 For Al_{2}(SO_{4})_{3}: ν= 2+3 = 5: a_{±} ^{ 5 }= a_{Al2(SO4)3}
Activity coefficients
All this remains a formality unless we find a way to relate it back to the concentration of the salt. Usually molality is used as a convenient concentration measure rather than molarity, because we are dealing with pretty strong deviations from ideal behavior and that implies that volume may not be an additive quantity. Molality does not involve volume in contrast to molarity.
Working with molalities we can define activity coefficients for both ions, even though we have no hope to determine them separately
\[a_+ =γ_+ m_+ \label{Eq11}\]
\[a_ =γ_ m_ \label{Eq12}\]
Stoichiometry dictates the molalities of the individual ions must be related to the molality of the salt \(m\) by:
\[m_=ν_m \label{Eq13}\]
\[m_+=ν_+m \label{Eq14}\]
Note
We cannot measure the activities of the ions separately because it is impossible to add one without adding the other
Analogous to the mean ionic activity we can define a mean ionic molality as:
\[m_{\pm}^ν ≡ m_+^{ν+}m_^{ ν} \label{Eq15}\]
We can do the same for the mean ionic activity coefficient
\[γ_{\pm}^ν = γ_+^{ν+}γ_^{ν} \label{Eq16}\]
Using this definitions we can rewrite
\[a_2=a_{\pm}^ν=a_+^{ν+}a_^{ν} \label{Eq17}\]
as
\[a_2=a_{\pm}^ ν =γ_{\pm}^ ν m_{\pm}^ ν \label{Eq18}\]
Note
Note that when preparing a salt solution of molality \(m\), we should substitute
\[m_=ν_m \nonumber\]
\[m_+=ν_+m \nonumber \]
into
\[m_±^ν ≡ m_+^{ν+}m_^{ ν} \nonumber \]
Example 25.5.1: Aluminum Sulfate
For Al_{2}(SO_{4})_{3} we get:
 ν= 2+3 = 5
 a_{±} ^{ 5 }= a_{Al2(SO4)3}
 m_{}=3m
 m_{+}=2m
So:

 m_{±} ^{ ν }=m_{+}^{ν+}m_{}^{ ν}=(2m)^{2}(3m)^{3}=108m^{5}
 a_{Al2(SO4)3}= a_{±} ^{ 5 } =108m^{5}γ_{±}^{5}
As you can see the stoichiometry enter both into the exponents and into the calculation of the molality. Notice that the activity of the salt now goes as the fifth power of its overall molality (on top of the dependency of γ_{±} of exp(√m) as shown below).
Measuring mean ionic activity coefficients
In contrast to the individual coefficients, the mean ionic activity coefficient \(γ_{\pm}\) is a quantity that can be determined. In fact we can use the same GibbsDuhem trick we did for the sucrose problem to do so. We simply measure the water vapor pressure above the salt solution and use
\[\ln γ_{\pm} = φ 1 + \int_{m'=0}^m [ φ 1 ]m' \,dm'\]
The fact that the salt itself has a negligible vapor pressure does not matter. Particularly for ions with high charges the deviations from ideality are very strong even at tiny concentrations. Admittedly doing these vapor pressure measurements in pretty tedious, there are some other procedures, e.g. involving electrochemical potentials. However, they too are tedious.