Debye–Hückel equation

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 activities 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 $$\gamma$$. This factor takes into account the interaction energy of ions in solution.

Debye–Hückel limiting law
In order to calculate the activity $$a_C$$ of an ion C in a solution, one must know the concentration and the activity coefficient: $$a_C = \gamma \frac\mathrm{[C]}\mathrm{[C^\ominus]},$$ where Dividing $$\mathrm{[C]}$$ with $$\mathrm{[C^\ominus]}$$ gives a dimensionless quantity.
 * $$\gamma$$ is the activity coefficient of C,
 * $$\mathrm{[C^\ominus]}$$ is the concentration of the chosen standard state, e.g. 1 mol/kg if molality is used,
 * $$\mathrm{[C]}$$ is a measure of the concentration of C.

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 $$\ln(\gamma_i) = -\frac{z_i^2 q^2 \kappa}{8 \pi \varepsilon_r \varepsilon_0 k_\text{B} T} = -\frac{z_i^2 q^3 N^{1/2}_\text{A}}{4 \pi (\varepsilon_r \varepsilon_0 k_\text{B} T)^{3/2}} \sqrt{10^3\frac{I}{2}} = -A z_i^2 \sqrt{I},$$ where
 * $$z_i$$ is the charge number of ion species i,
 * $$q$$ is the elementary charge,
 * $$\kappa$$ is the inverse of the Debye screening length $$\lambda_{\rm D}$$ (defined below),
 * $$\varepsilon_r$$ is the relative permittivity of the solvent,
 * $$\varepsilon_0$$ is the permittivity of free space,
 * $$k_\text{B}$$ is the Boltzmann constant,
 * $$T$$ is the temperature of the solution,
 * $$N_\mathrm{A}$$ is the Avogadro constant,
 * $$I$$ is the ionic strength of the solution (defined below),
 * $$A$$ is a constant that depends on temperature. If $$I$$ 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 $$A$$ of water is $$1.172\text{ mol}^{-1/2}\text{kg}^{1/2}$$ at 25 °C. It is common to use a base-10 logarithm, in which case we factor $$\ln 10$$, so A is $$0.509\text{ mol}^{-1/2}\text{kg}^{1/2}$$. The multiplier $$10^3$$ before $$I/2$$ in the equation is for the case when the dimensions of $$I$$ are $$\text{mol}/\text{dm}^3$$. When the dimensions of $$I$$ are $$\text{mole}/\text{m}^3$$, the multiplier $$10^3$$ 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: $$P^\text{ex} = -\frac{k_\text{B} T \kappa_\text{cgs}^3}{24\pi} = -\frac{k_\text{B} T \left(\frac{4\pi \sum_j c_j q_j}{\varepsilon_0 \varepsilon_r k_\text{B} T }\right)^{3/2}}{24\pi}.$$ Therefore, the total pressure is the sum of the excess osmotic pressure and the ideal pressure $P^\text{id} = k_\text{B} T \sum_i c_i$. The osmotic coefficient is then given by $$\phi = \frac{P^\text{id} + P^\text{ex}}{P^\text{id}} = 1 + \frac{P^\text{ex}}{P^\text{id}}.$$

Nondimensionalization
The differential equation is ready for solution (as stated above, the equation only holds for low concentrations): $$\frac{\partial^2 \varphi(r) }{\partial r^2} + \frac{2}{r} \frac{\partial \varphi(r) }{\partial r} = \frac{I q \varphi(r)}{\varepsilon_r \varepsilon_0 k_\text{B} T} = \kappa^2 \varphi(r).$$

Using the Buckingham π theorem on this problem results in the following dimensionless groups: $$\begin{align} \pi_1 &= \frac{q \varphi(r)}{k_\text{B} T} = \Phi(R(r)) \\ \pi_2 &= \varepsilon_r \\ \pi_3 &= \frac{a k_\text{B} T \varepsilon_0}{q^2} \\ \pi_4 &= a^3 I \\ \pi_5 &= z_0 \\ \pi_6 &= \frac{r}{a} = R(r). \end{align}$$ $$\Phi$$ is called the reduced scalar electric potential field. $$R$$ 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, $$(\kappa a)^2$$. The second could be called the reduced central ion charge, $$Z_0$$ (with a capital Z). Note that, though $$z_0$$ is already dimensionless, without the substitution given below, the differential equation would still be dimensional.

$$\frac{\pi_4}{\pi_2 \pi_3} = \frac{a^2 q^2 I}{\varepsilon_r \varepsilon_0 k_\text{B} T} = (\kappa a)^2$$ $$\frac{\pi_5}{\pi_2 \pi_3} = \frac{z_0 q^2}{4 \pi a \varepsilon_r \varepsilon_0 k_\text{B} T} = Z_0$$

To obtain the nondimensionalized differential equation and initial conditions, use the $$\pi$$ groups to eliminate $$\varphi(r)$$ in favor of $$\Phi(R(r))$$, then eliminate $$R(r)$$ in favor of $$r$$ while carrying out the chain rule and substituting $${R^\prime}(r) = a$$, then eliminate $$r$$ in favor of $$R$$ (no chain rule needed), then eliminate $$I$$ in favor of $$(\kappa a)^2$$, then eliminate $$z_0$$ in favor of $$Z_0$$. The resulting equations are as follows: $$\frac{\partial \Phi(R) }{\partial R}\bigg|_{R=1} = - Z_0$$ $$\Phi(\infty) = 0$$ $$\frac{\partial^2 \Phi(R) }{\partial R^2} + \frac{2}{R} \frac{\partial \Phi(R) }{\partial R} = (\kappa a)^2 \Phi(R).$$

For table salt in 0.01 M solution at 25 °C, a typical value of $$(\kappa a)^2$$ is 0.0005636, while a typical value of $$Z_0$$ is 7.017, highlighting the fact that, in low concentrations, $$(\kappa a)^2$$ 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.