Ionic Strength and Activity Coefficients: The Debye–Hückel Limiting Law
In basic chemistry we write equilibrium constants using concentrations. This works well in very dilute solutions. In real solutions, ions attract and shield each other, so a dissolved ion behaves as if its concentration were lower than the amount actually present. The correct quantity is the activity. The Debye–Hückel limiting law gives a way to estimate how large the correction is. This is useful for electrochemistry, solubility and pH problems in JAM, GATE, CSIR-NET and CUET-PG, and for anyone working with natural waters or biological solutions.
Ionic strength
Ionic strength measures the total electrical "crowding" of ions in a solution. It counts charge squared, so highly charged ions matter much more.
The sum is over every ion in the solution. ci is its molar concentration (mol/L; strictly molality in the full theory) and zi is its charge number, which is positive or negative but is squared.
(a) 0.010 M NaCl gives Na+ 0.010 M and Cl− 0.010 M. I = ½(0.010 × 1² + 0.010 × 1²) = 0.010 M.
(b) 0.010 M CaCl2 gives Ca2+ 0.010 M and Cl− 0.020 M. I = ½(0.010 × 4 + 0.020 × 1) = ½(0.040 + 0.020) = 0.030 M.
(c) 0.0050 M Al2(SO4)3 gives Al3+ 0.010 M and SO42− 0.015 M. I = ½(0.010 × 9 + 0.015 × 4) = ½(0.090 + 0.060) = 0.075 M.
(d) 0.010 M Na2SO4 gives Na+ 0.020 M and SO42− 0.010 M. I = ½(0.020 × 1 + 0.010 × 4) = ½(0.060) = 0.030 M.
Answer: 0.010, 0.030, 0.075 and 0.030 M. Notice that the same 0.010 M of salt gives I = 0.010 for NaCl but 0.030 for CaCl2 and Na2SO4.
Activity and the activity coefficient
γ → 1 as the solution becomes infinitely dilute
Here a is the activity (dimensionless), γ is the activity coefficient and c° is the standard concentration of 1 mol/L. In a real solution γ is usually less than 1 at low to moderate ionic strength, so the activity is smaller than the concentration.
The Debye–Hückel limiting law
log10 γ± = −A |z+ z−| √I (mean ionic activity coefficient)
For water at 25 °C, A ≈ 0.509 (mol/kg)−1/2; many books round it to 0.51. A depends on the solvent and the temperature, so use the value given in the question. The law is a limiting law: it is reliable only in very dilute solutions, roughly I below about 0.01 M. Beyond that, extended forms such as the Debye–Hückel equation with a denominator (1 + B a √I), and more elaborate models, are used. We apply the limiting law to larger I below only to show the trend.
Single-ion activity coefficients cannot be measured separately on their own, because positive and negative ions always occur together. What can be measured is the mean ionic activity coefficient γ±, the geometric mean of the cation and anion values.
Step 1: I = 0.010 M, so √I = 0.100. |z+ z−| = 1.
Step 2: log γ± = −0.509 × 1 × 0.100 = −0.0509.
Step 3: γ± = 10−0.0509 = 0.889.
Answer: γ± ≈ 0.89. The effective concentration is about 11 % lower than the stoichiometric one.
Step 1: I = 0.030 M (from Example 1), so √I = 0.1732. |z+ z−| = 2 × 1 = 2.
Step 2: log γ± = −0.509 × 2 × 0.1732 = −0.1763.
Step 3: γ± = 10−0.1763 = 0.666.
Answer: γ± ≈ 0.67. At the same salt concentration, the doubly charged ion gives a bigger correction than NaCl. Note that I = 0.030 M is above the range where the limiting law is accurate, so treat this as an estimate.
Step 1: The simple answer uses concentration: pH = −log(0.0010) = 3.000.
Step 2: I = 0.0010 M (HCl gives H+ and Cl− at 0.0010 M each), so √I = 0.03162.
Step 3: For H+, log γ = −0.509 × 1 × 0.03162 = −0.01610, so γ = 0.9636.
Step 4: a(H+) = 0.9636 × 0.0010 = 9.636 × 10−4.
Step 5: pH = −log(9.636 × 10−4) = 3.016.
Answer: pH ≈ 3.016 with the activity correction, against 3.000 using concentration. The pH scale is defined by activity, not concentration, but the difference is small in dilute solutions.
The salt effect on solubility
When an inert salt such as KNO3 is added to a saturated solution of a sparingly soluble salt, the ionic strength rises, the activity coefficients of the ions fall, and more of the solid must dissolve to keep the activity-based product equal to Ksp. This is the "diverse-ion" or "salt" effect, and it makes a sparingly soluble salt slightly more soluble. It is the opposite direction to the common-ion effect, which lowers solubility.
Where this idea appears in practice
Any real measurement of pH in seawater, groundwater, blood plasma or an industrial brine involves ionic strength. Electrochemical cell potentials, conductivity and equilibrium constants all need activities for accurate results, which is why standard methods often fix the ionic strength of a solution with a background electrolyte. This article does not claim specific numbers for any particular natural water; those must be measured.
Common mistakes
- Forgetting to square the charge. It is z², so a 2+ ion contributes four times as much as a 1+ ion of equal concentration.
- Forgetting the factor ½ in I.
- Using the concentration of the salt rather than of each ion. CaCl2 gives twice as much Cl− as Ca2+.
- Using log instead of ln, or the reverse. The law is written with log10, and γ = 10(log γ).
- Using the limiting law at high concentrations. It is a limiting law for very dilute solutions only.
- Using single-ion values as though they were measured. Only γ± is directly measurable.
- Thinking γ is always below 1. At high ionic strength it can rise above 1. The limiting law does not capture this.
Exam relevance
| Question type | Key step |
|---|---|
| Ionic strength of a mixture | List every ion with its own concentration, then I = ½ Σ c z² |
| Activity coefficient from the limiting law | Compute log γ = −A z² √I, then γ = 10log γ |
| Effect on Ksp, pH or cell potential | Replace concentrations with activities |
| Salt effect vs common-ion effect | Inert salt raises solubility; common ion lowers it |
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