Activity coefficients (x- vs m-based)

Updated October 08, 2024 17:45

Objective:

This article highlights the critical role of activity coefficients in accurately modeling electrolyte solutions, while clarifying the distinctions between mole fraction-based and molality-based activity coefficients.

Advantages of the m-based approach

Traditionally, researchers in aqueous electrolyte thermodynamics used an activity model based on a molar (moles/L) or molal (moles/kg H2O) representation. Early in OLI's history, the molal concentration scale was adopted since the activity coefficients are independent of changes in density.

This approach has some other advantages. In the molal-based approach (hereafter referred to as m-based), when the concentration of the solutes approaches 0 moles/kg H2O (infinite dilution), the activity coefficients approach 1.0 (unity). The slope of the activity coefficient as a function of molal concentration follows the Debye-Hückle limiting law.

Reference:

LibreTexts. The Debye-Hückel Theory. Physical Chemistry (LibreTexts). https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/25%3A_Solutions_II_-_Nonvolatile_Solutes/25.06%3A_The_Debye-Huckel_Theory (accessed Sept 13, 2024).

Shortcomings of the m-based approach

This works well at low concentrations, but many extensions must be added to account for more concentrated solutions. At extremely high concentrations, where water is more of a solute than a solvent, many of the m-based activity models fail to work reliably.

Introduction to the x-based approach

In 2002, OLI published its Mixed-Solvent Electrolyte (MSE) model (Wang, P., Anderko, A., & Young, R. D. (2002). A speciation-based model for mixed-solvent electrolyte systems. Fluid Phase Equilibria, 203, 141-176.) This model uses the mole fraction-based concentration scale (x-based).

In the x-based concentration scale, the activity coefficient approaches 1.0 (unity) when the mole fraction of the solute approaches 1.0. This concentration scale and the MSE model allow for simulations at extremely high concentrations.

Below, we we show an example of how to compare the two different types of activity coefficients.

Comparing Activity Coefficients: Example

A user of the OLI software submitted a question.

User Question:

I am comparing the activity coefficient of the nitrate ion (NO₃^1-). I looked up the value in OLI Studio for the AQ and MSE thermodynamic frameworks. They differ by over 50%, with the MSE-derived values being much larger. Why?

Response: From Peiming Wang, PhD. OLI Systems (retired)

I agree with the concept of obtaining different activity coefficient values from the two different models (i.e., AQ vs. MSE) under the same total salt concentration, temperature, and pressure conditions if speciation is different (e.g., with or without the ion-pair, such as NaNO3(aq) in the NaNO3 solution). This is because activity coefficients are a function of each and all individual species (ions or neutral), and the speciation in the two models is different, resulting in different equilibrium concentrations for individual species and, thus, different activity coefficients.

Below, I would like to provide further reasoning based on thermodynamic fundamentals.

Part 1.

In an equilibrium solution at a given temperature and pressure, the Gibbs-Duhem equation must hold, i.e.,

Where a_i is the activity of species i: a_i=m_i∙γ_i (m is molality and γ is activity coefficient) and a₀=a_w (water activity), and n is the number of species in the solution. Equation (1) can be integrated (where n₀ is set to be 55.508 for 1 kg H₂O):

To simplify the explanation, only the binary solution (e.g., NaNO3+H2O) is considered here. For an electrolyte fully dissociated in the aqueous solution (e.g., NaNO₃ in the MSE model), Na+ and NO3- molalities are the same as the total NaNO₃ concentration.

Suppose ion-pairing is considered (e.g., NaNO3 in the AQ model). In that case, the Na+ and NO3- concentrations will be different from (i.e., smaller than) the total NaNO3 concentration due to the formation of ion-pair, NaNO3(aq). In this case, the molality (m) and activity coefficient (γ) of NaNO_3(aq) will also need to be included in Eq. (1) and (2).

The value of the activity coefficient for each species will be determined based on concentrations of all species present (i.e., Na+ and NO3- in the MSE mode, and Na+, NO3-, NaNO3(aq) in the AQ model, in addition to other less critical species such as H+/H3O+ and OH-).

At the same time, for a given solution (i.e., fixed T, P, and total salt concentration), water activity, a_w, which is an experimentally measurable property, must be the same (i.e., a value that matches the experimental results under the given condition) regardless of the model (e.g., AQ vs. MSE) used in the calculation.

Part 2

The explanation may be further extended to the solid saturation, i.e., a system in which the solid is in equilibrium with the solution (e.g., at solid-liquid equilibrium). For the solid dissolution reaction:

The equilibrium condition is

For the NaNO3 + H2O system, the concentrations (e.g., molality) of the dissociated cation (M=Na+) and anion (X=NO3-) in the MSE model are the same as the total NaNO3 concentration at saturation (as NaNO3 is fully dissociated). These ionic concentrations (e.g., mNa and mNO3), however, are different from (i.e., smaller than) the total NaNO3 concentration at the saturation in the AQ model due to the formation of NaNO3(aq) ion-pair:

At the same time, concentrations and activity coefficients of both cation (Na+) and anion (NO3-) must satisfy the equilibrium condition defined by Eq. (4), in which Ksp is the thermodynamic equilibrium constant that both models should fulfill.

Thus, the two models can't have the same activity coefficients of NO3- and/or Na+ at the saturated solution due to their different ionic concentrations, which must satisfy Eq. (4).

It should be noted, however, that the equilibrium constant Ksp from the two models may have some (usually small) deviation depending on the data used in parameterizing the model. These parameters include standard-state Gibbs free energy of formation leading to the Ksp values.

Conclusion:

When systems are multi-component, the variation of the activity coefficients becomes more complicated. It is unsurprising that if speciation is different in the two models, the activity coefficient for the same ion can be distinct under the same inflow conditions at a given T and P.