Question

In an ideal electrolyte solution, thermodynamic properties of electrolyte solution such as equilibrium constants can be simply expressed as a function of concentrations of dissolved ions. In most cases, however, the behavior of electrolyte solutions is often far from ideal because of the ionic atmosphere, and the deviations can be corrected by introducing the concept of activity coefficient. Quatatively explain how the ionic atmosphere influences the interactions between electrolytes. Also describe and explain the relation between ionic strength of electrolyte of a solution and activity coefficients (10 pt) 1.

Answer 1

Ionic Atmosphere is a concept employed in Debye-Hückel theory which explains the conductivity behaviour of solutions. It can be generally defined as the area at which a charged entity is capable of attracting an entity of the opposite charge.

If an electrical potential is applied to an electrolytic solution, a positive ion will move towards the negative electrode and drag along an entourage of negative ions with it. The more concentrated the solution, the closer these negative ions are to the positive ion and thus the greater the resistance experienced by the positive ion. This influence on the speed of an ion is known as the "Asymmetry effect" because the ionic atmosphere moving around the ion is not symmetrical; the charge density behind is greater than in the front, slowing the motion of the ion. The time required to form a new ionic atmosphere on the right or time required for ionic atmosphere on the left to fade away is known as time of relaxation. The asymmetrization of ionic atmosphere does not occur in the case of Debye Falkenhagen effect due to high frequency dependence of conductivity.

ionic strength---I or µ ---a measure of the total ion concentration in solution but ions with more charge are counted more due to stronger electrostatic interactions with other ions! (I.e., can influence the increase “ionic atmosphere” greater than singly charged ions!)

$\mu =\frac{1}{2} \sum CiZi^{2}$

∑ where ci is conc. of ith species

and zi is the charge on ith species

Derivation

The deviation from ideality is taken to be a function of the potential energy resulting from the electrostatic interactions between ions and their surrounding clouds. To calculate this energy two steps are needed.

The first step is to specify the electrostatic potential for ion j by means of Poisson's equation

$\nabla ^{2}\psi _{j}(r)=-{\frac {1}{\epsilon _{0}\epsilon _{r}}}\rho _{j}(r)$

ψ(r) is the total potential at a distance, r, from the central ion and ρ(r) is the averaged charge density of the surrounding cloud at that distance. To apply this formula it is essential that the cloud has spherical symmetry, that is, the charge density is a function only of distance from the central ion as this allows the Poisson equation to be cast in terms of spherical coordinates with no angular dependence.

The second step is to calculate the charge density by means of a Boltzmann distribution.

zeur) ni ni exp

where k_B is Boltzmann constant and T is the temperature. This distribution also depends on the potential ψ(r) and this introduces a serious difficulty in terms of the superposition principle. Nevertheless, the two equations can be combined to produce the Poisson–Boltzmann equation.

KB

Solution of this equation is far from straightforward. Debye and Hückel expanded the exponential as a truncated Taylor series to first order. The zeroth order term vanishes because the solution is on average electrically neutral (so that ∑ n_i z_i = 0), which leaves us with only the first order term. The result has the form of the Helmholtz equation

${\displaystyle \nabla ^{2}\psi _{j}(r)=\kappa ^{2}\psi _{j}(r)\qquad {\text{with}}\qquad \kappa ^{2}={\frac {e^{2}}{\epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}\sum _{i}n_{i}z_{i}^{2}}$

{\displaystyle \nabla ^{2}\psi _{j}(r)=\kappa ^{2}\psi _{j}(r)\qquad {\text{with}}\qquad \kappa ^{2}={\frac {e^{2}}{\epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}\sum _{i}n_{i}z_{i}^{2}}${\displaystyle \nabla ^{2}\psi _{j}(r)=\kappa ^{2}\psi _{j}(r)\qquad {\text{with}}\qquad \kappa ^{2}={\frac {e^{2}}{\epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}\sum _{i}n_{i}z_{i}^{2}}$,

which has an analytical solution. This equation applies to electrolytes with equal numbers of ions of each charge. Nonsymmetrical electrolytes require another term with ψ². For symmetrical electrolytes, this reduces to the modified spherical Bessel equation

(02+, 2with solutions -A+A" 0 W1

The coefficients A' and A'' are fixed by the boundary conditions. As ${\displaystyle r\rightarrow \infty } , {\displaystyle \psi }$ $\psi$ must not diverge, so A''=0. At r=a₀, which is the distance of the closest approach of ions, the force exerted by the charge should be balanced by the force of other ions, imposing
TEEC
from which A' is found, yielding

zie ehdo bj(r)

The electrostatic potential energy, u_j$u_{j}$, of the ion at r=0$r=0$ is

zie 1

$u_{j}=z_{j}e{\Big (}\psi _{j}(a_{0})-{\frac {z_{j}e}{4\pi \varepsilon _{0}\varepsilon _{r}}}{\frac {1}{a_{0}}}{\Big )}=-{\frac {z_{j}^{2}e^{2}}{4\pi \varepsilon _{0}\varepsilon _{r}}}{\frac {\kappa }{1+\kappa a_{0}}}$

This is the potential energy of a single ion in a solution. The multiple-charge generalization from electrostatics gives an expression for the potential energy of the entire solution (see also: Debye–Hückel equation). The mean activity coefficient is given by the logarithm of this quantity as follows (see also: Extensions of the theory).

E -0.05 -0.1 0 0.2 0.4 lonic strength /M

Experimental $\log \gamma _{\pm }$ values for KBr at 25°C (points) and Debye–Hückel limiting law (coloured line)

$\log _{{10}}\gamma _{\pm }=-Az_{j}^{2}{\frac {{\sqrt I}}{1+Ba_{0}{\sqrt I}}} {\displaystyle A={\frac {e^{2}B}{2.303\times 8\pi \epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}} {\displaystyle B=\left({\frac {2e^{2}N}{\epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}\right)^{1/2}}$

where I is the ionic strength and a₀ is a parameter that represents the distance of closest approach of ions. For aqueous solutions at 25 °C A = 0.51 mol^−1/2dm^3/2 and B = 3.29 nm⁻¹mol^−1/2dm^3/2

The most significant aspect of this result is the prediction that the mean activity coefficient is a function of ionic strengthrather than the electrolyte concentration. For very low values of the ionic strength the value of the denominator in the expression above becomes nearly equal to one. In this situation the mean activity coefficient is proportional to the square root of the ionic strength. This is known as the Debye–Hückel limiting law.