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The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of the activity coefficients of the solution. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient {\displaystyle \gamma }\gamma . This factor takes into account the interaction energy of ions in solution.
The Equation 5.8.1 is known as the Debye-Hückel Limiting Law. The ionic strength is calculated by the following relation:
I=12∑imiz2i
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Explanation:
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 {[C]}{[C^{\ominus }]}},}
where
{\displaystyle \gamma } is the activity coefficient of C,{\displaystyle [C^{\ominus }]} is the concentration of the chosen standard state, e.g. 1 mol/kg if molality is used,{\displaystyle [C]} is a measure of the concentration of C.
Dividing {\displaystyle [C]} with {\displaystyle [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 (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 water is {\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 A is {\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{mole}}/{\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