Understanding the Calculation of the Activity Coefficient in Thermodynamics

The activity coefficient quantifies deviations from ideal solution behavior in mixtures. It is essential for accurate thermodynamic modeling and process design.

This article explores detailed methods to calculate activity coefficients, including formulas, tables, and real-world applications. Expect comprehensive technical insights and practical examples.

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Calculate the activity coefficient of ethanol in water at 25°C using the Wilson model.
Determine the activity coefficient for a binary mixture of benzene and toluene at 50°C.
Compute the activity coefficient of NaCl in aqueous solution at varying ionic strengths.
Evaluate the activity coefficient of CO2 in a gas mixture using the fugacity coefficient approach.

Comprehensive Tables of Common Activity Coefficient Values

Activity coefficients vary widely depending on the system, temperature, pressure, and composition. Below are extensive tables presenting typical values for common binary mixtures and electrolytes under standard conditions.

Mixture	Temperature (°C)	Composition (Mole Fraction)	Activity Coefficient (γ₁)	Activity Coefficient (γ₂)	Model Used
Ethanol – Water	25	x_ethanol = 0.1	1.25	1.05	Wilson
Ethanol – Water	25	x_ethanol = 0.5	1.10	1.10	Wilson
Benzene – Toluene	50	x_benzene = 0.3	1.02	1.03	Margules
Benzene – Toluene	50	x_benzene = 0.7	1.04	1.01	Margules
NaCl in Water (Electrolyte)	25	Ionic Strength = 0.1 M	γ_NaCl = 0.85		Debye-Hückel
NaCl in Water (Electrolyte)	25	Ionic Strength = 1.0 M	γ_NaCl = 0.60		Debye-Hückel Extended
CO2 in Gas Mixture	40	Partial Pressure = 1 atm	γ_CO2 = 0.95		Fugacity Coefficient
CO2 in Gas Mixture	100	Partial Pressure = 5 atm	γ_CO2 = 0.80		Fugacity Coefficient

These values serve as benchmarks for engineers and scientists when modeling phase equilibria and reaction kinetics in chemical processes.

Fundamental Formulas for Calculating Activity Coefficients

Activity coefficients (γ) quantify non-ideality in mixtures and are calculated using various thermodynamic models. Below are the primary formulas and detailed explanations of each variable involved.

1. Definition of Activity Coefficient

The activity coefficient γ_i of component i is defined as:

γi = ai / xi

a_i: Activity of component i (dimensionless)
x_i: Mole fraction of component i in the mixture

Activity represents the “effective concentration” accounting for interactions, while mole fraction is the actual concentration.

2. Margules Equation (Two-Parameter Model)

Used for binary mixtures to estimate activity coefficients:

ln γ1 = A x22

ln γ2 = A x12

A: Margules parameter (energy interaction parameter, J/mol)
x₁, x₂: Mole fractions of components 1 and 2

Typical values of A range from 0 to 5,000 J/mol depending on system polarity and molecular size differences.

3. Wilson Model

Wilson’s equation accounts for molecular size and energy differences:

ln γ1 = -ln(x1 + Λ12 x2) + x2 [Λ12 / (x1 + Λ12 x2) – Λ21 / (x2 + Λ21 x1)]

ln γ2 = -ln(x2 + Λ21 x1) – x1 [Λ12 / (x1 + Λ12 x2) – Λ21 / (x2 + Λ21 x1)]

Λ₁₂ = (V₂/V₁) exp(-Δλ₁₂/RT)
Λ₂₁ = (V₁/V₂) exp(-Δλ₂₁/RT)
V_i: Molar volume of component i (m³/mol)
Δλ_ij: Energy parameter between components i and j (J/mol)
R: Universal gas constant (8.314 J/mol·K)
T: Temperature (K)

Wilson parameters are typically obtained from regression of experimental data.

4. Non-Random Two-Liquid (NRTL) Model

The NRTL model incorporates non-randomness in molecular interactions:

ln γi = Σj τji Gji xj / Σk Gki xk + Σj xj Gij / Σk Gkj xk [τij – Σm xm τmj Gmj / Σk Gkj xk]

τ_ij = Δg_ij / RT, interaction energy parameter
G_ij = exp(-α_ij τ_ij), non-randomness factor
α_ij: Non-randomness parameter (typically 0.2–0.47)
x_i: Mole fraction of component i

NRTL is widely used for highly non-ideal mixtures, including associating and polar compounds.

5. Debye-Hückel Equation for Electrolytes

For dilute electrolyte solutions, the Debye-Hückel limiting law estimates activity coefficients:

log γ± = -A z2 √I / (1 + B a √I)

γ_±: Mean ionic activity coefficient
A: Debye-Hückel constant (depends on temperature and solvent)
B: Constant related to ion size and solvent dielectric constant
z: Ionic charge
I: Ionic strength (mol/L), defined as 0.5 Σ c_i z_i²
a: Effective ion size parameter (Å)

Extended versions include specific ion interactions for higher ionic strengths.

6. Fugacity Coefficient and Gas Phase Activity Coefficient

In gas mixtures, the activity coefficient relates to the fugacity coefficient (φ):

γi = φi P / Psati

φ_i: Fugacity coefficient of component i
P: Total pressure
P^sat_i: Saturation vapor pressure of pure component i

Fugacity coefficients are calculated from equations of state such as Peng-Robinson or Soave-Redlich-Kwong.

Detailed Real-World Examples of Activity Coefficient Calculation

Example 1: Activity Coefficient of Ethanol in Water at 25°C Using Wilson Model

Consider a binary mixture of ethanol (component 1) and water (component 2) at 25°C with mole fraction x_ethanol = 0.2. Given molar volumes and energy parameters:

V_ethanol = 58.4 cm³/mol
V_water = 18.0 cm³/mol
Δλ₁₂ = 1200 J/mol
Δλ₂₁ = 800 J/mol
R = 8.314 J/mol·K
T = 298 K

Calculate Λ₁₂ and Λ₂₁:

Λ12 = (Vwater / Vethanol) × exp(-Δλ12 / RT)

= (18.0 / 58.4) × exp(-1200 / (8.314 × 298))

= 0.308 × exp(-0.484) = 0.308 × 0.616 = 0.190

Λ21 = (Vethanol / Vwater) × exp(-Δλ21 / RT)

= (58.4 / 18.0) × exp(-800 / (8.314 × 298))

= 3.244 × exp(-0.323) = 3.244 × 0.724 = 2.35

Calculate ln γ_ethanol:

ln γ1 = -ln(x1 + Λ12 x2) + x2 [Λ12 / (x1 + Λ12 x2) – Λ21 / (x2 + Λ21 x1)]

= -ln(0.2 + 0.190 × 0.8) + 0.8 [0.190 / (0.2 + 0.190 × 0.8) – 2.35 / (0.8 + 2.35 × 0.2)]

Calculate denominators:

0.2 + 0.190 × 0.8 = 0.2 + 0.152 = 0.352
0.8 + 2.35 × 0.2 = 0.8 + 0.47 = 1.27

Calculate terms:

-ln(0.352) = 1.044
0.190 / 0.352 = 0.54
2.35 / 1.27 = 1.85
Difference = 0.54 – 1.85 = -1.31
Multiply by 0.8 = -1.05

Sum terms:

ln γ1 = 1.044 – 1.05 = -0.006

Therefore, γ_ethanol = exp(-0.006) ≈ 0.994, indicating near ideal behavior at this composition.

Example 2: Activity Coefficient of NaCl in Aqueous Solution at Ionic Strength 0.1 M Using Debye-Hückel

Calculate the mean ionic activity coefficient γ_± for NaCl at 25°C with ionic strength I = 0.1 M. Constants for water at 25°C:

A = 0.509 mol^-1/2·L^1/2
B = 0.328 Å^-1·mol^-1/2·L^1/2
z = 1 (Na⁺ and Cl^– both monovalent)
a = 4.0 Å (effective ion size)

Apply Debye-Hückel limiting law:

log γ± = -A z2 √I / (1 + B a √I)

= -0.509 × 12 × √0.1 / (1 + 0.328 × 4.0 × √0.1)

= -0.509 × 0.316 / (1 + 0.328 × 4.0 × 0.316)

= -0.161 / (1 + 0.414) = -0.161 / 1.414 = -0.114

Convert to γ_±:

γ± = 10log γ± = 10-0.114 = 0.77

This indicates significant deviation from ideality due to ionic interactions at this concentration.

Additional Considerations and Advanced Topics

Accurate calculation of activity coefficients is critical in chemical engineering, environmental science, and materials research. Several factors influence these calculations:

Temperature and Pressure Dependence: Most models require temperature-dependent parameters; pressure effects are significant in gas mixtures.
Association and Hydrogen Bonding: Systems with strong molecular interactions (e.g., alcohols, acids) require models like NRTL or UNIQUAC.
Electrolyte Solutions: Extended Debye-Hückel or Pitzer models are necessary for concentrated solutions.
Data Regression: Experimental vapor-liquid equilibrium (VLE) or liquid-liquid equilibrium (LLE) data are used to regress model parameters.
Software Tools: Commercial simulators (Aspen Plus, HYSYS) implement these models with extensive databases.

For further reading and authoritative resources, consult:

Understanding and applying these principles ensures precise thermodynamic predictions, optimizing industrial processes and research outcomes.