Debye Huckel Equation

The Debye Huckel equation is a mathematical expression developed to explain certain properties of electrolyte solutions, or substances found in solutions in the form of charged particles (ions). The Debye Huckel equation accounts for the interactions between the different ions, which are the primary cause of differences between the properties of dilute electrolyte solutions and those of so-called ideal solutions.

Introduction

The entropy, rather than the enthalpy, drives the mixing of the solutions. By definition, while an ideal gas does not have the interactions between particles, an ideal solution assumes that there are interactions. Without interactions, the solution would not be in the liquid phase. Rather, the ideal solutions can be defined as having an enthalpy of either mixing or enthalpy of the solution, which is equal to zero (ΔHmixing or ΔHsolution = 0).

Since the interactions between the two liquids, A-B, are calculated as the sum of the A-A and B-B interactions, this is the case. The average A-A and B-B interactions are similar in the ideal solution, so there is no distinction between the average A-B and the A-A/B-B interactions.

Since in chemistry and biology, the average interactions between A and B are not always equivalent to the interactions of A or B alone, the enthalpy of mixing is not zero. As a result, a new definition for the concentration of molecules in solution has emerged. The effective concentration, a1, is calculated by taking into account the deviation from ideal action, with the operation of the ideal solution equal to one.

To convert from the mole fraction of the solute, x1 (as the unit of concentration, mole fraction can be determined from other concentration units like molality, molarity, or percent by weight) to activity, a1, an activity coefficient, 1, is used.

a\[_{1}\] = γ\[_{1}\]x\[_{1}\]

Debye Huckel Theory

This theory is related to electrochemistry. This theory was proposed by Peter debye and Eric huckel to explain the departure of electrolytic solution from ideality, because the electrolytic solution shows the deviation from the ideal behavior.

Colligative properties of ideal solutions are directly proportional to the concentration of solute. But this does not happen in the case of real solutions. In Case of real solutions we cannot take the concentration term as a whole. In case of real solutions the concentration term is replaced by another term which is known as activity. Activity is represented by a.

So we can say that this activity is directly proportional to the concentration c. Or we can say a is equal to gamma (Greek letter) multiplied by c, where Gamma is the activity coefficient. 

For an ideal solution the value of Gamma is 1. If we take a dilute solution, they are near to ideality but concentrated solution shows departure from ideality that shows non ideality.

When we take an electrolytic solution, we know that in case of strong electrolyte they have the tendency to dissociate into cations and anions. You also know that the opposite charges attract each other while the same charges repel each other. So the cations will have the tendency to go near anions and the anions will have the tendency to go near cations. So we cannot say that the ions are randomly distributed throughout the solution, in case of real solutions.

We cannot measure the activity coefficient of a single ion. So we will measure the mean activity coefficient. There are formulas to calculate mean activity coefficient.

By Debye Huckel theory we can calculate the activity coefficient of a single ion. And we know that the departure from the ideal is due to the electrostatic interactions which are between the ions.

With the help of coulomb's law we can calculate the forces of interaction between two electrical charges which are separated by a distance r. So in real solutions, anions are more likely to be found near cations and vice versa.

Debye Huckel Limiting Law

See Debye–Hückel theory for the concepts that were used to derive this equation.

To measure the activity a\[_{C}\] of an ion C in a solution, the concentration and activity coefficient must be known:

a\[_{C}\] = γ \[\frac{[C]}{[C^{\Theta}]}\],

This is also called the debye huckel limiting law equation.

Where, 

γ is given as the activity coefficient of C,

\[\left [C \right ]\] is the measure of concentration of C,

[[C^{\Theta}]] is given as the concentration of chosen standard state, for example, 1 mol/kg if molality is used.

Dividing \[\left [C \right]\] with [[C^{\Theta}]] gives the dimensionless quantity.

The Debye–Hückel limiting law can be used to calculate an ion's activity coefficient in a dilute solution with a known ionic power. The following is the equation:

ln(γ₁) = \[\frac{Z_{i}^{2}q^{2}k}{8 \prod \epsilon_{r} \epsilon_{0} k_{b} T}\] = - \[\frac{Z_{i}^{2}q^{3}N_{A}^{1/2}}{4\prod (\epsilon_{r} \epsilon_{0} k_{B}T)^{3/2}}\] \[\sqrt{10^{3} \frac{1}{2}}\] = - AZ\[_{i}^{2}\]\[\sqrt{I}\]

where 

Z\[_{i}\] is given as the charge number of ion species i,

q is given as the elementary charge,

k is given as the  inverse of the Debye screening length,

\[\epsilon_{0}\] is the permittivity of free space,

\[\epsilon_{r}\] is given as the relative permittivity of the solvent,

k\[_{B}\] is given as the Boltzmann constant,

T is given as the temperature of the solution,

I is the ionic strength of the solution,

N\[_{A}\] is given as the Avogadro constant,

And A is the constant that is completely based on the temperature.

The scientists have determined that it is not that simple because the things such as the sizes of the ions including the amount of charge that they contain may affect properties like conductivity. This equation is their attempt to account for certain variables when assessing an ionic behaviour of the compound.

Experimental Verification of the Theory

Several experimental methods for calculating activity coefficients have been tried to check the validity of Debye–Hückel theory: the main problem is that we need extremely high dilutions. Measurements of the freezing point, vapour pressure, osmotic pressure (which is an indirect method), and electric potential in cells are just a few examples (which is a direct method). 

Using liquid membrane cells, it was possible to examine the aqueous media 104 M at high dilutions and obtain good results and it also has been found that for the ratio 1:1 electrolytes (as either KCl or NaCl) the Debye–Hückel equation is completely correct, but for the ratio 2:2 or 3:2 electrolytes it can be possible to find the negative deviation from Debye–Hückel limit law: this strange behavior may be noticed only in the very dilute area, and in many concentrate regions the deviation becomes positive.

It is also possible that Debye–Hückel equation is unable to foresee this particular behavior because of the linearization of Poisson–Boltzmann equation, or may be not: about this, studies have been started only during the last years of the 20th-century because prior to it, it was not possible to investigate the region of 10−4 M, so it can be possible that during the next years new theories will be come up.

Main Assumptions of the Debye Huckel Theory

• The solute or electrolyte is completely dissociated which means that the electrolyte is strong.

• Ions are spherical and the solvation of ions is ignored in this theory.

• According to this theory solvent plays no role. It only provides medium for the interactions.

• Individual ions surrounding a central ion can be represented by an average cloud of continuous charge density.

So there is a spherical haze around the Central ion having net charge equal in amount but opposite in sign and is called its ionic atmosphere. The energy and chemical potential of any given Central iron is lowered due to electrostatic interaction with its ionic atmosphere.

According to this model, at very low concentration, the activity coefficient is calculated by the debye huckel limiting law.