Question 9.2: An assumption of the Debye–Hückel theory is that the concent...
An assumption of the Debye–Hückel theory is that the concentration distribution as a function of distance of the charged particles i around a chosen central ion is given as the first-power term of a power series of the function
c_{i}(r)=\bar{c}_{i} \exp \left[-\frac{e z_{i} \varphi(r)}{k T}\right]Here, \bar{c}_{i} is the average concentration of the charged particles i in the solution, e is the elementary charge, z_{i} is the charge number of the particles, φ(r) is the electric potential at a distance r from the central ion, and k is the Boltzmann constant. To get the radial charge density around the central ion, the above function is summed for all charged particles in the solution.
Based on these considerations, argue why the theory works better for a solution of pure KCl than for that of pure CaBr_{2}.
Let us write the charge density ρ(r) around the central ion as a power series expansion up to the second-order term in φ(r) summing for all the ionic species, multiplied by the respective ionic charge z_{i}F:
\rho(r)=\sum\limits_{i} \bar{c}_{i} z_{i} F-\sum\limits_{i} \frac{\bar{c}_{i} F^{2} z_{i}^{2}}{R T} \varphi(r)+\sum\limits_{i} \frac{\bar{c}_{i} F^{3} z_{i}^{3}}{R^{2} T^{2}} \varphi^{2}(r)The first term of the above expression is the charge density at the position of the central ion (at r = 0). This is always zero, regardless of the nature of the electrolyte – based on the electroneutrality principle as expressed in (9.48).
\sum\limits_{i=1}^{J} c_{i} z_{i}=0 ; \quad \sum\limits_{i=1}^{J} m_{i} z_{i}=0 . (9.48)
The second term is identical to the approximation used by Debye and Hückel. By inspecting the third term, we can state that for 1:1 electrolytes (and every higher term containing even powers of φ) it is also zero as odd powers of the charge number z_{i} sum to zero. This does not apply for electrolytes such as CaBr_{2}, where the absolute values of the charge numbers are not identical for the two ions. Thus, the truncation of the power series is a worse approximation than for 1:1 electrolytes.