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The Debye–Hückel limiting law estimates an ion activity coefficient from charge and ionic strength in a very dilute solution.
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Formula
I in mol/kg · A in (kg/mol)1ᐟ2 · z and γ dimensionless; in water at 25 °C A≈0.509 (kg/mol)1ᐟ2
| Symbol | Meaning | Unit |
|---|---|---|
| Activity coefficient of ion i | dimensionless | |
| Temperature- and solvent-dependent Debye–Hückel constant | (kg/mol)1ᐟ2 | |
| Charge number of ion i | dimensionless | |
| Molal ionic strength of the solution | mol/kg |
\log_{10}\gamma_i=-A z_i^2\sqrt{I}This limiting law applies to very dilute aqueous electrolytes at a stated temperature and molality convention. Single-ion activity coefficients depend on convention and are not independently measurable. At moderate ionic strength, use an extended Debye–Hückel or Davies model rather than extrapolating the square-root law. Ionic strength weights each ionic species by the square of its charge; the activity correction therefore responds more strongly to multivalent ions than to monovalent ions.
Limiting law for very dilute aqueous electrolytes at a stated temperature and molality convention. Single-ion activity coefficients depend on convention; use an extended model at moderate ionic strength.
Electrostatic interactions shift an ion’s effective concentration; in the dilute limit the logarithmic correction is proportional to z2√I.
Ionic strength from activity coefficient
Use only within the validity range and with molal ionic strength.
For water at 25 °C: I=0.010 mol/kg, z=−1, A=0.509. Find γ.
log10γ=−0.509·12·√0.010=−0.0509; γ≈0.889.
Use I in mol/L with a constant for molality, or use charge linearly.
Match the unit convention and use z2; each ion contributes through the square of its charge.
Formula cluster
Ionic strength is the input; Debye–Hückel maps it to γ in the dilute limit.
For a monovalent ion in water at 25 °C with I=0.010 mol/kg, log10γ=−0.509√0.010=−0.0509, so γ=10^(−0.0509)≈0.889. Check: for z=2 the logarithm has four times the magnitude and γ≈10^(−0.2036)≈0.626. This is a limiting-law estimate for low ionic strength.
Interpret activity corrections to equilibria in very dilute electrolyte solutions.
Here is how to work through a typical Debye–Hückel Limiting Law (log₁₀ γᵢ = −A zᵢ²√I) task step by step:
Task
For water at 25 °C: I=0.010 mol/kg, z=−1, A=0.509. Find γ.
Solution path
log10γ=−0.509·12·√0.010=−0.0509; γ≈0.889.
It estimates an ion activity coefficient γi from charge and ionic strength in a very dilute aqueous solution. For a monovalent ion at 25 °C and I=0.010 mol/kg, using A=0.509 gives log10γ=−0.0509 and γ≈0.889. The effective activity is therefore below concentration. This is a limiting approximation, not a general equation for concentrated salt solutions. Single-ion values also depend on convention; mean activity coefficients are more directly accessible experimentally. Always state the temperature and reference standard explicitly. For z=0, no electrostatic correction would occur.
In Debye–Hückel theory, electrostatic interaction scales with the square of ionic charge. A divalent ion therefore receives a logarithmic correction four times as large as a monovalent ion at the same ionic strength. For I=0.010 mol/kg and A=0.509, z=2 gives log10γ≈−0.204 and γ≈0.626. This does not mean the ion is four times as concentrated. It is an activity correction within the model. Counterions and every other ionic species also affect I, even when their coefficients do not appear directly in γi.
It is intended for very dilute electrolyte solutions, where long-range electrostatic effects dominate and specific interactions are small. As ionic strength rises, the simple expression fits data less well; finite ion size and short-range interactions matter. Extended Debye–Hückel or Davies models can cover a wider range, but they are not universal either. Check solvent, temperature, concentration convention and the source’s stated validity range. Accurate work at higher salt concentrations requires experimentally supported activity models. State the concentration convention and validity range before reporting a correction.
Activity augments a reference quantity such as concentration with coefficient γ. On a molality convention, write approximately ai=γi(mi/m°), where m° is the standard state; the activity is then dimensionless. In the ideal dilute limit γ approaches 1, so activity and normalized concentration coincide. Debye–Hückel estimates the deviation for very dilute solutions. Do not confuse γ with a fraction or concentration, and use activities consistently in equilibrium expressions. Use the same standard state when inserting activities into an equilibrium expression. Activities remain dimensionless.
No. The limiting law does not calculate K; it approximates γi. You can use it to estimate activities from concentrations and insert them into a thermodynamic reaction quotient. This requires every relevant ion, charge and the molal ionic strength to be included correctly. Equilibrium also requires a balanced reaction and specified temperature. At concentrated conditions, a small error in γ can shift the result substantially; state the activity model and its limits explicitly. Higher ionic strength requires measurements or an extended activity model.
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Which formula describes Debye–Hückel Limiting Law?
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How do you rearrange Debye–Hückel Limiting Law for Ionic strength from activity coefficient?
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Which common mistake happens with Debye–Hückel Limiting Law?
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