# 10.5: Derivation of the Debye-Hückel Theory

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Debye and Hückel derived Eq. 10.4.1 using a combination of electrostatic theory, statistical mechanical theory, and thermodynamics. This section gives a brief outline of their derivation.

The derivation starts by focusing on an individual ion of species $$i$$ as it moves through the solution; call it the central ion. Around this central ion, the time-average spatial distribution of any ion species $$j$$ is not random, on account of the interaction of these ions of species $$j$$ with the central ion. (Species $$i$$ and $$j$$ may be the same or different.) The distribution, whatever it is, must be spherically symmetric about the central ion; that is, a function only of the distance $$r$$ from the center of the ion. The local concentration, $$c'_j$$, of the ions of species $$j$$ at a given value of $$r$$ depends on the ion charge $$z_j e$$ and the electric potential $$\phi$$ at that position. The time-average electric potential in turn depends on the distribution of all ions and is symmetric about the central ion, so expressions must be found for $$c'_j$$ and $$\phi$$ as functions of $$r$$ that are mutually consistent.

Debye and Hückel assumed that $$c'_j$$ is given by the Boltzmann distribution $$c'_j = c_j e^{-z_j e \phi/ kT} \tag{10.5.1}$$ where $$z_j e\phi$$ is the electrostatic energy of an ion of species $$j$$, and $$k$$ is the Boltzmann constant ($$k = R/N\subs{A}$$). As $$r$$ becomes large, $$\phi$$ approaches zero and $$c'_j$$ approaches the macroscopic concentration $$c_j$$. As $$T$$ increases, $$c'_j$$ at a fixed value of $$r$$ approaches $$c_j$$ because of the randomizing effect of thermal energy. Debye and Hückel expanded the exponential function in powers of $$1/T$$ and retained only the first two terms: $$c'_j \approx c_j(1 - z_j e\phi/kT)$$. The distribution of each ion species is assumed to follow this relation. The electric potential function consistent with this distribution and with the electroneutrality of the solution as a whole is $$\phi = (z_i e / 4\pi\epsilon\subs{r}\epsilon_0 r) e^{\kappa(a-r)} / (1 + \kappa a) \tag{10.5.2}$$ Here $$\kappa$$ is defined by $$\kappa^2 = 2N\subs{A}^2 e^2 I_c/\epsilon_r \epsilon_0 RT$$, where $$I_c$$ is the ionic strength on a concentration basis defined by $$I_c = (1/2)\sum_i c_i z_i^2$$.

The electric potential $$\phi$$ at a point is assumed to be a sum of two contributions: the electric potential the central ion would cause at infinite dilution, $$z_i e/4\pi \epsilon_r \epsilon_0 r$$, and the electric potential due to all other ions, $$\phi'$$. Thus, $$\phi'$$ is equal to $$\phi - z_i e/4\pi \epsilon_r \epsilon_0 r$$, or $$\phi' = (z_i e/4\pi\epsilon\subs{r}\epsilon_0 r) [e^{\kappa(a-r)}/(1+\kappa a)-1] \tag{10.5.3}$$ This expression for $$\phi'$$ is valid for distances from the center of the central ion down to $$a$$, the distance of closest approach of other ions. At smaller values of $$r$$, $$\phi'$$ is constant and equal to the value at $$r = a$$, which is $$\phi'(a) = -(z_i e/4\pi \epsilon_r \epsilon_0)\kappa/(1 + \kappa a)$$. The interaction energy between the central ion and the surrounding ions (the ion atmosphere) is the product of the central ion charge and $$\phi'(a)$$.

The last step of the derivation is the calculation of the work of a hypothetical reversible process in which the surrounding ions stay in their final distribution, and the charge of the central ion gradually increases from zero to its actual value $$z_i e$$. Let $$\alpha z_i e$$ be the charge at each stage of the process, where $$\alpha$$ is a fractional advancement that changes from $$0$$ to $$1$$. Then the work $$w'$$ due to the interaction of the central ion with its ion atmosphere is $$\phi'(a)$$ integrated over the charge: $$\begin{split} w' & = -\int_{\alpha=0}^{\alpha=1} [(\alpha z_i e / 4\pi\epsilon\subs{r}\epsilon_0)\kappa /(1+\kappa a)]\dif(\alpha z_i \epsilon) \cr & = -(z_i^2 e^2/8\pi\epsilon\subs{r}\epsilon_0) \kappa/(1+\kappa a) \end{split} \tag{10.5.4}$$ Since the infinitesimal Gibbs energy change in a reversible process is given by $$\dif G = -S\dif T + V\difp + \dw'$$ (Eq. 5.8.6), this reversible nonexpansion work at constant $$T$$ and $$p$$ is equal to the Gibbs energy change. The Gibbs energy change per amount of species $$i$$ is $$w'N\subs{A} = -(z_i^2 e^2 N\subs{A}/8\pi \epsilon\subs{r}\epsilon_0)\kappa/(1 + \kappa a)$$. This quantity is $$\Del G/n_i$$ for the process in which a solution of fixed composition changes from a hypothetical state lacking ion–ion interactions to the real state with ion–ion interactions present. $$\Del G/n_i$$ may be equated to the difference of the chemical potentials of $$i$$ in the final and initial states. If the chemical potential without ion–ion interactions is taken to be that for ideal-dilute behavior on a molality basis, $$\mu_i=\mu_{m,i}\rf + RT\ln(m_i/m\st)$$, then $$-(z_i^2 e^2 N\subs{A}/8\pi \epsilon\subs{r}\epsilon_0)\kappa/(1 + \kappa a)$$ is equal to $$\mu_i - [\mu_{m,i}\rf + RT\ln(m_i/m\st)] = RT\ln\g_{m,i}$$. In a dilute solution, $$c_i$$ can with little error be set equal to $$\rho\A^* m_i$$, and $$I_c$$ to $$\rho\A^*I_m$$. Equation 10.4.1 follows.

This page titled 10.5: Derivation of the Debye-Hückel Theory is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Howard DeVoe via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.