# Debye Huckel Onsager Equation

Debye Huckel Onsager equation is a mathematical equation that relates the activity coefficient of an electrolyte solution to its ionic strength. It was first formulated by Peter Debye and Erich Huckel in the early 1920s, and later expanded upon by Lars Onsager in the 1930s. This equation has been widely used in physical chemistry to describe the behavior of ionic solutions.

Debye Huckel Onsager equation is based on the concept of electrolyte solutions. In case of weak electrolytes the increase in equivalent conductance with dilution can be easily explained on the basis of Arrhenius theory, according to which the conductance increases due to increase in dissociation of weak electrolyte with dilution. But this explanation cannot be applied in case of strong electrolytes like NaCl as they are almost completely dissociated into constituent ions (Na^{+} and Cl^{−}) even at moderate concentration.

Peter Debye and Huckel in 1923 initially derived an equation to describe how the equivalent conductance of electrolytic solution changes on dilution. Later this equation was further improved by Onsager and now this equation is known as Debye Huckel Onsager equation.

Debye-Huckel-Onsager Theory is based on the following Assumptions-

1. Strong electrolytes are completely dissociate into their constituent ions in solution.

2. Due to Coulornbic forces between the charges of the ions, they do not behave like molecules in their transport and thermodynamic properties.

3. Since the solution as a whole is neutral, the total number of positive charges must be equal to the total number of negative charges.

4. It was assumed that the electric field causes the charge cloud to be distorted away from spherical symmetry.

5. A given ion will have more ions of the opposite sign (i.e. charge) close to it than ions of the same sign. This cluster of ions is called the ionic atmosphere. The ionic atmosphere is assumed to be spehrical symmetry.

6. The charge density of ionic atmosphere is the greatest in the immediate vicinity of its central ion and it gradually decreases with increasing distance.

7. The electrostatic interaction is assumed to be small in comparison with the energy of thermal movement.

8. The electrostatic potential in the solution can be described by an equation is known as the Poisson and Boltzmann equation.

When electric field is applied to a solution of strong electrolyte, ionic atmosphere produces two effects which prevent the ions to move with the expecteed speed leading to decreased values of conductance

1. Asymmetry Effect or Relaxation Effect

2. Electrophoretic Effect

**1. Asymmetry Effect or Relaxation Effect**

According to Debye and Huckel theory of strong electrolytes each ion is surrounded by atmosphere of oppositely charged ions as shown in Figure.

In the absence of applied electric field the ionic atmosphere around the central ion is spherically symmetrical. At this point the electrostatic force of attraction on central ion from all the directions is same.

When an electric field is applied the central ion tries to move in a direction opposite to that of ionic atmosphere. That is, if central ions move in one direction, the ionic atmosphere moves opposite to it. During the movement of central ion, the ionic atmosphere drags it and the ionic atmosphere will be destroyed. In this process, the destruction of old and formation of new ionic atmosphere doesn't occur simultaneously. There is a time lag between destruction and formation of ionic atmosphere, and due to which mobility of central ion decreases and the effect is known as **relaxation effect**. During the movement of central ions, the ions that were present in front are left behind so that the symmetry of ionic atmosphere is broken and the effect is known as **asymmetric effect**. This phenomenon of retardation of mobility of ions is considered to be asymmetric effect or relaxation effect.

**2. Electrophoretic Effect**

The ionic atmosphere of a central ion always remains associated with solvent molecules. The ionic atmosphere is oppositely charged with respect to the central ion. When electric field is applied, a cation migrates towards cathode through medium, while the negatively charged ionic atmosphere along with solvent molecules moves towards anode. These counter movements causes a retarding influence on the movement of the central ion. This effect is known as **Electrophoretic effect**.

When dilution is increased, the electrophoretic effect, the relaxation effect becomes weaker so that ionic mobility increases ultimately increasing the equivalence conductance. Just opposite situation appears when concentration is increased.

**3. Viscous Effect or Frictional Force**

Not only relaxation effect and electrophoretic effect operates to decrease ionic mobility but also resistance offered by solvent plays an important role.

Viscous effect is simply the frictional resistance created by the viscosity of the solvent used. Viscous effect is determined by Stokes' law.
Any moving ions experience resistive force by medium. If central ion is moving with uniform velocity, it experiences frictional resistance or force by the solvent of the solution (medium) in turn its velocity in the solution is decreased and the conductivity of the solution too. In other words, an ion with its ionic atmosphere travels in the solution, then the medium of the solution offers the frictional resistance. This force depends upon viscosity of the medium and its dielectric constant. Ionic mobility decreases with the increase in the viscosity of the solvent used. Thus, the decrease in ionic mobility results in decrease in the conductance. So, we can say that less viscous solvents give higher conductance and vice-versa.

Fractional force = u_{i} K_{i}
u_{i} is Steady velocity (or mobility) of ith ion

K_{I} is Coefficient of frictional resistant of solvent opposing the motion of ith kind ion.

All the above three factors decreases the velocity of the ion and leads to decrease the equivalent conductivity. At infinite dilution the relaxation and electrophoretic effects are virtually zero and hence the velocity of the ions, and thereby the equivalent conductivity are determined only by frictional force of medium.

Thus, we conclude that the difference between the conductance at infinite dilution and at certain appreciable concentration is a direct consequence of the two electrical forces. In other words, all deviations from ideal behaviour are ascribed to electrical interactions between the ions.

Debye-Huckel with the help of Onsager gave a mathematical relation between equivalent conductance and concentration for strong electrolyte.

Where Λ_{c} = Equivalent conductance at concentration c

Λ_{o} = Equivalent conductance at infinite dilution

ε = Diectric constant of the medium

η = Coefficient of viscosity of the medium

T = Temperature of the solution in degree absolute

c = Concentration of the solution in moles/litre

As ε and η are constant for a particular solvent. Therefore, at constant temperature, the above equation can be written as-

Λ_{c} = Λ_{o} - (A + B Λ_{o}) √c

where A and B are constants for a particular solvent at particular temperature.

At a temperature of 25°C, if water is solvent then the value of A is taken as 60.20, and the value of B is taken as 0.229.

## Debye Huckel Onsager Equation Graph

When Λ_{c} is plotted against √C, a straight line having a negative slope (A + B Λ_{O}) and intercept Λ_{O} is obtained. Therefore, from the slope, we can determine the values of equivalent conductance of strong electrolytes at infinite dilution.

## Limitations of Debye Huckel Onsager Equation

The graph of Λ_{c} versus √C shows that the equivalent conductance of weak electrolyte decreases exponentially with concentration. So, the equivalent conductance at infinite dilution (Λ_{c}) for weak electrolyte can not be determined from the above curve on extrapolation. Therefore, the Onsager equation is limited only for strong electrolytes and invalid for weak electrolytes.

### Debye-Huckel Onsager equation for strong electrolytes

Λ = Λ_{o} − A √C

Which of the following equality holds?

A. Λ = Λ_{o} as C ⟶ √A

B. Λ = Λ_{o} as C ⟶ ∞ ✔

C. Λ = Λ_{o} as C ⟶ 0

D. Λ = Λ_{o} as C ⟶ 1

### The unit of A in Debye-Huckel-Onsager equation for strong electrolytes is

A. Ω^{-1}cm^{2} M^{-1/ 2}

B. Ω^{-1}cm^{2} M^{-1}

C. Ω^{-1}cm^{-2} M^{1/2}

D. M^{-1/2} ✔