Debye–Hückel equation

In order to calculate the activity ${\displaystyle a_{C}}$  of an ion C in a solution, one must know the concentration and the activity coefficient: $${\displaystyle a_{C}=\gamma {\frac {\mathrm {[$$C]} }{\mathrm {[C^{\ominus }]} }},}   where

• ${\displaystyle \gamma }$  is the activity coefficient of C,
• ${\displaystyle \mathrm {[C^{\ominus }]} }$  is the concentration of the chosen standard state, e.g. 1 mol/kg if molality is used,
• ${\displaystyle \mathrm {[$C]} }   is a measure of the concentration of C.

Dividing ${\displaystyle \mathrm {[$C]} }   with ${\displaystyle \mathrm {[C^{\ominus }]} }$  gives a dimensionless quantity.

The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. The equation is^{[1]: section 2.5.2} $${\displaystyle \ln(\gamma _{i})=-{\frac {z_{i}^{2}q^{2}\kappa }{8\pi \varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=-{\frac {z_{i}^{2}q^{3}N_{\text{A}}^{1/2}}{4\pi (\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T)^{3/2}}}{\sqrt {10^{3}{\frac {I}{2}}}}=-Az_{i}^{2}{\sqrt {I}},}$$  where

• ${\displaystyle z_{i}}$  is the charge number of ion species i,
• ${\displaystyle q}$  is the elementary charge,
• ${\displaystyle \kappa }$  is the inverse of the Debye screening length ${\displaystyle \lambda _{\rm {D}}}$  (defined below),
• ${\displaystyle \varepsilon _{r}}$  is the relative permittivity of the solvent,
• ${\displaystyle \varepsilon _{0}}$  is the permittivity of free space,
• ${\displaystyle k_{\text{B}}}$  is the Boltzmann constant,
• ${\displaystyle T}$  is the temperature of the solution,
• ${\displaystyle N_{\mathrm {A} }}$  is the Avogadro constant,
• ${\displaystyle I}$  is the ionic strength of the solution (defined below),
• ${\displaystyle A}$  is a constant that depends on temperature. If ${\displaystyle I}$  is expressed in terms of molality, instead of molarity (as in the equation above and in the rest of this article), then an experimental value for ${\displaystyle A}$  of wateris${\displaystyle 1.172{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}}$  at 25 °C. It is common to use a base-10 logarithm, in which case we factor ${\displaystyle \ln 10}$ , so Ais${\displaystyle 0.509{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}}$ . The multiplier ${\displaystyle 10^{3}}$  before ${\displaystyle I/2}$  in the equation is for the case when the dimensions of ${\displaystyle I}$  are ${\displaystyle {\text{mol}}/{\text{dm}}^{3}}$ . When the dimensions of ${\displaystyle I}$  are ${\displaystyle {\text{mole}}/{\text{m}}^{3}}$ , the multiplier ${\displaystyle 10^{3}}$  must be dropped from the equation.

It is important to note that because the ions in the solution act together, the activity coefficient obtained from this equation is actually a mean activity coefficient.

The excess osmotic pressure obtained from Debye–Hückel theory is in cgs units:^[1] $${\displaystyle P^{\text{ex}}=-{\frac {k_{\text{B}}T\kappa _{\text{cgs}}^{3}}{24\pi }}=-{\frac {k_{\text{B}}T\left({\frac {4\pi \sum _{j}c_{j}q_{j}}{\varepsilon _{0}\varepsilon _{r}k_{\text{B}}T}}\right)^{3/2}}{24\pi }}.}$$  Therefore, the total pressure is the sum of the excess osmotic pressure and the ideal pressure ${\textstyle P^{\text{id}}=k_{\text{B}}T\sum _{i}c_{i}}$ . The osmotic coefficient is then given by $${\displaystyle \phi ={\frac {P^{\text{id}}+P^{\text{ex}}}{P^{\text{id}}}}=1+{\frac {P^{\text{ex}}}{P^{\text{id}}}}.}$$ 

The differential equation is ready for solution (as stated above, the equation only holds for low concentrations): $${\displaystyle {\frac {\partial ^{2}\varphi ($$r)}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial \varphi (r)}{\partial r}}={\frac {Iq\varphi (r)}{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=\kappa ^{2}\varphi (r).}  

Using the Buckingham π theorem on this problem results in the following dimensionless groups: $${\displaystyle {\begin{aligned}\pi _{1}&={\frac {q\varphi ($$r)}{k_{\text{B}}T}}=\Phi (R(r))\\\pi _{2}&=\varepsilon _{r}\\\pi _{3}&={\frac {ak_{\text{B}}T\varepsilon _{0}}{q^{2}}}\\\pi _{4}&=a^{3}I\\\pi _{5}&=z_{0}\\\pi _{6}&={\frac {r}{a}}=R(r).\end{aligned}}}   ${\displaystyle \Phi }$  is called the reduced scalar electric potential field. ${\displaystyle R}$  is called the reduced radius. The existing groups may be recombined to form two other dimensionless groups for substitution into the differential equation. The first is what could be called the square of the reduced inverse screening length, ${\displaystyle (\kappa a)^{2}}$ . The second could be called the reduced central ion charge, ${\displaystyle Z_{0}}$  (with a capital Z). Note that, though ${\displaystyle z_{0}}$  is already dimensionless, without the substitution given below, the differential equation would still be dimensional.

$${\displaystyle {\frac {\pi _{4}}{\pi _{2}\pi _{3}}}={\frac {a^{2}q^{2}I}{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=(\kappa a)^{2}}$$  $${\displaystyle {\frac {\pi _{5}}{\pi _{2}\pi _{3}}}={\frac {z_{0}q^{2}}{4\pi a\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=Z_{0}}$$ 

To obtain the nondimensionalized differential equation and initial conditions, use the ${\displaystyle \pi }$  groups to eliminate ${\displaystyle \varphi ($r)}   in favor of ${\displaystyle \Phi (R($r))}  , then eliminate ${\displaystyle R($r)}   in favor of ${\displaystyle r}$  while carrying out the chain rule and substituting ${\displaystyle {R^{\prime }}($r)=a}  , then eliminate ${\displaystyle r}$  in favor of ${\displaystyle R}$  (no chain rule needed), then eliminate ${\displaystyle I}$  in favor of ${\displaystyle (\kappa a)^{2}}$ , then eliminate ${\displaystyle z_{0}}$  in favor of ${\displaystyle Z_{0}}$ . The resulting equations are as follows: $${\displaystyle {\frac {\partial \Phi ($$R)}{\partial R}}{\bigg |}_{R=1}=-Z_{0}}   $${\displaystyle \Phi (\infty )=0}$$  $${\displaystyle {\frac {\partial ^{2}\Phi ($$R)}{\partial R^{2}}}+{\frac {2}{R}}{\frac {\partial \Phi (R)}{\partial R}}=(\kappa a)^{2}\Phi (R).}  

For table salt in 0.01 M solution at 25 °C, a typical value of ${\displaystyle (\kappa a)^{2}}$  is 0.0005636, while a typical value of ${\displaystyle Z_{0}}$  is 7.017, highlighting the fact that, in low concentrations, ${\displaystyle (\kappa a)^{2}}$  is a target for a zero order of magnitude approximation such as perturbation analysis. Unfortunately, because of the boundary condition at infinity, regular perturbation does not work. The same boundary condition prevents us from finding the exact solution to the equations. Singular perturbation may work, however.

• Strong electrolyte
• Weak electrolyte
• Ionic atmosphere
• Debye–Hückel theory
• Poisson–Boltzmann equation

• ^ Debye P.; Hückel E. (1923). "Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen" [The theory of electrolytes. I. Lowering of freezing point and related phenomena]. Physikalische Zeitschrift. 24: 185–206. Archived from the original (PDF) on 2019-12-20. Alt URL
• ^ Hamann, Hamnett, and Vielstich (1998). Electrochemistry. Weinheim: Wiley-VCH Verlag GmbH. ISBN 3-527-29096-6.{{cite book}}: CS1 maint: multiple names: authors list (link)
• ^ Harris, Daniel C. (2003). Quantitative Chemical Analysis (6th ed.). W. H. Freeman & Company. ISBN 0-7167-4464-3.
• ^ Skoog, Douglas A. (July 2003). Fundamentals of Analytical Chemistry. ISBN 0-534-41796-5.
• ^ Malatesta, F., and Zamboni, R. (1997). Activity and osmotic coefficients from the EMF of liquid membrane cells, VI – ZnSO₄, MgSO₄, CaSO₄ and SrSO₄ in water at 25 °C. Journal of Solution Chemistry, 26, 791–815.

• For easy calculation of activity coefficients in (non-micellar) solutions, check out the IUPAC open project Aq-solutions^{[permanent dead link]} (freeware).
• Gold Book definition