Calculate activity coefficients, ionic strength, and explore non-ideal behavior in electrolyte solutions
Use absolute value (e.g., 2 for Ca²⁺ or SO₄²⁻)
Activity vs Concentration: In non-ideal solutions, ions don't behave as if they're at their analytical concentration due to ionic interactions. Activity (a = γc) is the "effective concentration" that properly describes thermodynamic behavior.
Activity Coefficient (γ): Quantifies deviation from ideality. γ = 1 means ideal behavior, γ < 1 (common in ionic solutions) means attractive interactions reduce effective concentration, γ > 1 (rare, seen at high concentrations) means repulsive interactions dominate.
Ionic Strength: I = ½Σcizi² measures total ionic environment. Higher charge ions contribute more heavily (z² dependence). High I → stronger ion-ion interactions → larger deviation from ideality.
Debye-Hückel Theory: Models ionic atmosphere around each ion. Limiting law (log γ = -Az²√I) works for dilute solutions (I < 0.01 M). Extended equation includes ion size parameter å for moderate concentrations (I up to ~0.1 M).
Activity coefficients quantify the deviation of real solutions from ideal behavior. In ideal solutions, particles don't interact with each other, and concentration alone determines thermodynamic properties. Real ionic solutions, however, exhibit significant ion-ion interactions (Coulombic attractions and repulsions) that cause the "effective concentration" — called activity — to differ from the analytical concentration.
Activity:
a = γ × c
Where a is activity (effective concentration), γ is the activity coefficient, and c is molar concentration
Ionic Strength:
I = ½ Σ cizi²
Sum over all ions i, where ci is concentration and zi is charge number
Debye-Hückel Limiting Law:
log γ± = -A|z+z-|√I
Valid for I < 0.01 M. A ≈ 0.509 (mol/kg)⁻¹/² at 25°C in water
Extended Debye-Hückel:
log γ = -Az²√I / (1 + Bå√I)
Includes ion size å (in Å). Valid for I up to ~0.1 M. B ≈ 0.328 (mol/kg)⁻¹/² Å⁻¹ at 25°C
The distinction between activity and concentration is crucial for accurate thermodynamic calculations:
Ionic strength (I) is a measure of the total concentration of ions in solution, weighted by their charges:
I = ½ Σ cizi²
Sum includes all ions in solution
Examples:
Why z² matters:
| Electrolyte | Dissociation | I / c | Example (c = 0.01 M) |
|---|---|---|---|
| NaCl (1:1) | Na⁺ + Cl⁻ | I = c | I = 0.01 M |
| CaCl₂ (2:1) | Ca²⁺ + 2Cl⁻ | I = 3c | I = 0.03 M |
| MgSO₄ (2:2) | Mg²⁺ + SO₄²⁻ | I = 4c | I = 0.04 M |
| AlCl₃ (3:1) | Al³⁺ + 3Cl⁻ | I = 6c | I = 0.06 M |
| K₃Fe(CN)₆ (3:3) | 3K⁺ + Fe(CN)₆³⁻ | I = 6c | I = 0.06 M |
Problem:
Calculate the mean ionic activity coefficient (γ±) for a 0.01 M NaCl solution at 25°C using the Debye-Hückel limiting law.
Solution:
Step 1: Calculate ionic strength
For NaCl → Na⁺ + Cl⁻:
I = ½[c(Na⁺) × 1² + c(Cl⁻) × 1²]
I = ½[0.01 × 1 + 0.01 × 1]
I = 0.01 M
Step 2: Apply Debye-Hückel limiting law
log γ± = -A|z+z-|√I
At 25°C, A = 0.509 (mol/kg)⁻¹/². For NaCl, z+ = +1, z- = -1
Step 3: Substitute values
log γ± = -0.509 × |1 × (-1)| × √0.01
log γ± = -0.509 × 1 × 0.1
log γ± = -0.0509
Step 4: Calculate γ±
γ± = 10⁻⁰·⁰⁵⁰⁹
γ± = 0.889
Step 5: Interpret the result
Answer: γ± = 0.889 for 0.01 M NaCl
Experimental value: γ± ≈ 0.902 (Debye-Hückel slightly underestimates at this concentration)
Debye-Hückel theory explains activity coefficients by modeling the ionic atmosphere — the time-averaged distribution of ions surrounding each central ion:
Around each positive ion, negative ions are statistically more likely to be nearby (and vice versa). This creates an "atmosphere" of oppositely charged ions that:
| Feature | Limiting Law | Extended Equation |
|---|---|---|
| Equation | log γ = -Az²√I | log γ = -Az²√I / (1 + Bå√I) |
| Valid range | I < 0.01 M | I < 0.1 M (sometimes up to 0.2 M) |
| Parameters | A only (temperature dependent) | A, B (temp dependent), å (ion size) |
| Accuracy | Good for very dilute solutions | Better at moderate concentrations |
| Typical å values | N/A | 3-9 Å (e.g., H⁺: 9, K⁺: 3, SO₄²⁻: 4) |
Electrode potentials depend on activities, not concentrations: E = E° - (RT/nF)ln(a). Neglecting γ can lead to errors of 10-30 mV in dilute solutions, more in concentrated ones. Critical for accurate pH measurement, ion-selective electrodes, and battery voltage calculations.
Thermodynamic equilibrium constants (K°) use activities: K° = Π ai. Converting to concentrations requires activity coefficients. Essential for accurate solubility products, complex formation constants, and speciation calculations in seawater, biological fluids, and industrial processes.
Natural waters (seawater I ≈ 0.7 M, groundwater I ≈ 0.001-0.1 M) require activity corrections for mineral solubility, metal speciation, and carbonate equilibria. PHREEQC and other speciation models use extended Debye-Hückel or Pitzer equations for high ionic strength.
Biological fluids (blood I ≈ 0.16 M, intracellular I ≈ 0.15 M) have significant ionic strength. Enzyme kinetics (Michaelis-Menten), protein-ligand binding, and drug solubility all depend on activities. Neglecting γ can lead to 10-20% errors in Kd or IC₅₀ values.
Crystallization, precipitation, and ion exchange depend on activities. Calculating supersaturation ratios, predicting scale formation in boilers/pipelines, and optimizing separation processes require accurate activity coefficients. Salt effects in organic reactions also involve ionic strength.
pH buffers, complexometric titrations (EDTA), and solubility-based separations all require activity corrections. The Davies equation (extension of Debye-Hückel) is commonly used for I up to 0.5 M. Ionic strength adjusters (ISA) maintain constant I to simplify calibrations.
Wrong: "For 0.1 M NaCl: I = 0.1 × 1² + 0.1 × 1² = 0.2 M"
Correct: "I = ½[0.1 × 1² + 0.1 × 1²] = ½ × 0.2 = 0.1 M"
Wrong: "For SO₄²⁻: z = -2e = -3.2 × 10⁻¹⁹ C"
Correct: "z is the charge number, a dimensionless integer. For SO₄²⁻: z = -2 (or use |z| = 2 in formulas)"
Wrong: "Using log γ = -Az²√I for 0.5 M NaCl"
Correct: "Limiting law fails above I ≈ 0.01 M. For 0.5 M, use extended equation or empirical data (experimental γ± ≈ 0.66 for NaCl at this concentration)."
Wrong: "γ(Na⁺) and γ(Cl⁻) are independent and can be measured separately."
Correct: "Single-ion activity coefficients cannot be measured individually. We measure γ± = [γ(+)^ν₊ γ(-)^ν₋]^(1/(ν₊+ν₋)). For symmetric electrolytes (z₊ = |z₋|), Debye-Hückel gives identical γ values for both ions."
For ionic strengths above 0.1 M or mixed electrolytes, more sophisticated models are needed: