Activity Theory

Thermodynamic activity is a fundamental concept for understanding how chemical species behave in mixtures. In aerosol science, activity determines the effective concentration of a species, accounting for molecular interactions that cause deviations from ideal behavior. This page covers the theoretical basis for activity calculations in particula.

Introduction

When organic compounds mix with water in atmospheric aerosols, they rarely behave ideally. The interactions between water molecules and organic molecules create non-ideal behavior that affects:

• Vapor pressure: How readily compounds evaporate from particles
• Water uptake: How much water aerosols absorb at a given relative humidity
• Phase behavior: Whether the mixture remains homogeneous or separates into phases

Understanding and predicting these behaviors requires accurate activity models, which particula provides through the Binary Activity Thermodynamics (BAT) framework.

Ideal Activity (Raoult's Law)

In an ideal solution, the activity of each component equals its mole fraction. This relationship is known as Raoult's Law:

\[a_i = x_i\]

where:

• \(a_i\) is the thermodynamic activity of component \(i\)
• \(x_i\) is the mole fraction of component \(i\)

Raoult's Law assumes that all molecular interactions (solute-solute, solvent-solvent, and solute-solvent) are equivalent. This holds true only for:

• Dilute solutions
• Mixtures of very similar molecules (e.g., isotopes)
• Ideal gases

For organic-water mixtures in aerosols, Raoult's Law is rarely valid because organic molecules and water have very different properties.

Activity Coefficients

Real mixtures deviate from ideal behavior, and we quantify this deviation using the activity coefficient \(\gamma_i\):

\[a_i = \gamma_i \cdot x_i\]

The activity coefficient captures all non-ideal effects:

• \(\gamma_i = 1\): Ideal behavior (Raoult's Law)
• \(\gamma_i > 1\): Positive deviation (repulsive interactions, higher volatility)
• \(\gamma_i < 1\): Negative deviation (attractive interactions, lower volatility)

For organic-water mixtures, activity coefficients can vary significantly with composition and temperature, making accurate models essential for aerosol thermodynamics.

Binary Activity Thermodynamics (BAT) Model

The Binary Activity Thermodynamics (BAT) model provides a computationally efficient approach to estimate activity coefficients for organic-water mixtures. Developed by Gorkowski et al. (2019), the BAT model uses fits derived from the comprehensive AIOMFAC thermodynamic model.

Key Features

The BAT model:

1. Reduces complexity: Instead of computing detailed molecular interactions, BAT uses empirical fits based on the oxygen-to-carbon (O:C) ratio
2. Covers realistic ranges: Valid for O:C ratios from 0 to 2, covering most atmospheric organics
3. Captures phase separation: Predicts when mixtures separate into distinct liquid phases

Model Inputs

The BAT model requires:

Parameter	Description	Typical Range
molar_mass_ratio	Ratio of water to organic molecular weight	0.05 - 0.5
organic_mole_fraction	Mole fraction of organic in mixture	0 - 1
oxygen2carbon	Oxygen-to-carbon ratio of organic	0 - 2
density	Mixture density	1000 - 2000 kg/m^3

Mathematical Formulation

The BAT model computes activity coefficients from the excess Gibbs free energy of mixing. The activity coefficients are derived as:

\[\ln \gamma_w = g^E - x_{org} \cdot \frac{\partial g^E}{\partial x_{org}}\]

\[\ln \gamma_{org} = g^E + (1 - x_{org}) \cdot \frac{\partial g^E}{\partial x_{org}}\]

where:

• \(g^E\) is the dimensionless excess Gibbs energy of mixing
• \(x_{org}\) is the organic mole fraction
• \(\gamma_w\) and \(\gamma_{org}\) are the activity coefficients for water and organic

The activities are then:

\[a_w = \gamma_w \cdot (1 - x_{org})\]

\[a_{org} = \gamma_{org} \cdot x_{org}\]

Fit Coefficients

The BAT model uses three sets of fit coefficients depending on the O:C ratio:

O:C Range	Fit Set	Description
Low (< 0.3)	G19_FIT_LOW	Hydrophobic organics
Mid (0.3 - 0.6)	G19_FIT_MID	Moderately oxygenated
High (> 0.6)	G19_FIT_HIGH	Highly oxygenated organics

These coefficients are interpolated smoothly to provide continuous predictions across the full O:C range.

Activity Calculation Flow

The following diagram shows how particula calculates activities using the BAT model:

flowchart TD
    A[Input: Mass Fractions, O:C, Density] --> B[Convert to Mole Fractions]
    B --> C{Activity Model}
    C -->|Ideal| D["a = x (Raoult's Law)"]
    C -->|BAT Model| E[Compute Gibbs Mixing Weights]
    E --> F[Calculate gamma from BAT Coefficients]
    F --> G["a = gamma × x"]
    D --> H[Activity Values]
    G --> H
    H --> I[Use in Partitioning/Equilibria]

Gibbs Free Energy and Thermodynamics

The activity coefficient is fundamentally connected to the Gibbs free energy of mixing (\(\Delta G_{mix}\)), which describes the thermodynamic driving force for mixing:

\[\Delta G_{mix} = RT \sum_i x_i \ln(a_i)\]

For an ideal mixture:

\[\Delta G_{mix}^{ideal} = RT \sum_i x_i \ln(x_i)\]

The excess Gibbs energy captures non-ideal contributions:

\[G^E = \Delta G_{mix} - \Delta G_{mix}^{ideal} = RT \sum_i x_i \ln(\gamma_i)\]

The BAT model parameterizes \(G^E\) using empirical fits, allowing efficient computation of activity coefficients without solving complex molecular interaction equations.

Functional Group Corrections

Some organic compounds with specific functional groups (alcohols, carboxylic acids, ethers) behave differently than predicted by O:C ratio alone. The BAT model can optionally apply OH-equivalent corrections to account for these effects:

from particula.activity import bat_activity_coefficients

# With functional group correction
a_w, a_org, m_w, m_org, g_w, g_org = bat_activity_coefficients(
    molar_mass_ratio=0.09,
    organic_mole_fraction=0.3,
    oxygen2carbon=0.4,
    density=1400.0,
    functional_group="alcohol",  # Optional correction
)

This converts the functional group information to an OH-equivalent form before computing activities.

Implementation in Particula

The BAT activity calculations are implemented in the particula.activity module:

Function	Purpose
bat_activity_coefficients()	Main entry point for BAT calculations
gibbs_mix_weight()	Computes Gibbs mixing weights
coefficients_c()	Applies BAT fit coefficients

Example Usage

from particula.activity import bat_activity_coefficients

# Calculate activities for a water-organic mixture
activity_water, activity_organic, mass_water, mass_organic, gamma_water, gamma_organic = (
    bat_activity_coefficients(
        molar_mass_ratio=0.09,  # M_water / M_organic
        organic_mole_fraction=0.3,
        oxygen2carbon=0.4,
        density=1400.0,
    )
)

print(f"Water activity: {activity_water:.3f}")
print(f"Organic activity: {activity_organic:.3f}")
print(f"Water activity coefficient: {gamma_water:.3f}")
print(f"Organic activity coefficient: {gamma_organic:.3f}")

References

Gorkowski, K., Preston, T. C., & Zuend, A. (2019). Relative-humidity-dependent organic aerosol thermodynamics via an efficient reduced-complexity model. Atmospheric Chemistry and Physics, 19(19), 13383-13407. https://doi.org/10.5194/acp-19-13383-2019

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Next: Equilibria Theory - Learn how activities are used in gas-particle partitioning calculations.