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Chemistry

PAPER No. 6: Physical Chemistry -II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)

MODULE No. 26: Debye – Hückel Theory - I

Subject Chemistry

Paper No and Title 6, Physical Chemistry -II (Statistical

Thermodynamics, Chemical Dynamics,

Electrochemistry and Macromolecules)

Module No and

Title

The Debye Hückel Theory- I

Module Tag CHE_P6_M26

Chemistry



TABLE OF CONTENTS

1. Learning Outcomes

2. Introduction

3. The Debye- Hückel theory of mean ionic coefficients

3.1 Introduction

3.2 Assumptions

4. Mathematical Treatment of Debye- Hückel theory - Introduction

5. Summary

Chemistry



1. Learning Outcomes

After studying this module, you shall be able to :

Learn the Debye Hückel Theory , its postulates and its mathematical treatment.

2. Introduction

In a non-electrolytic solution, the solute particles are uncharged, therefore long range

columbic interactions are not observed in these type of solutions. The short-range

interactions, arising from dipole-dipole or dispersion forces become significant only

when the concentration of the solute is high. Thus, in a dilute non electrolytic solution

there are no interactions between solute particles. On the other hand electrolytic solution

contains strong ion-ion interactions and ion-solvent interactions. Such solutions where

ion-ion interactions cannot be ignored are termed as real / non-ideal solutions.

Non – ideality in these solutions arises because, oppositely charged species attract each

other due to electrostatic forces while ions of same charge repel each other.

We can appropriately say that there are no interactions between solute particles in dilute

electrolytic solutions Long-range columbic interactions between ions is ignored and only

ion-solvent interactions are considered.

Since in non-ideal solutions, ion-ion interactions are present, it is not just the analytical

concentration of the species that can be used but its effective concentration is used. We

define this effective concentration as activity ( ) of the species.

Mathematically, it is given by:

a = γ m (1)

Where:

a = activity

γ= activity coefficient (concentration dependent)

m = molality (mol/kg)

Chemistry



Individual ionic-activity coefficients cannot be determined experimentally. This is

because in an electrolytic solution, positive and negative ions cannot be treated

separately, as the cations are always accompanied by anions and vice versa. Therefore,

the individual contributions of the ionic species to the concentration cannot be

determined. Thus, activity coefficients of individual ions, are inaccessible to

experimental measurement. We can only measure the activity coefficient of the net

electrolyte. Therefore we define mean ionic terms for the ions in solution of strong

electrolytes.

Mathematically,

Mean Ionic Activity is given as:

(2)

where:

m = molality

∴ Debye-Hückel suggested a theory which relates the theoretical individual activity

coefficients ( ) to the experimentally accessible mean ionic-activity coefficient ( ).

With the help of this theory, the distribution of charges around ions in solution was

clearly understood.

Chemistry



3. The Debye – Hückel Theory

The Debye Hückel theory for determination of mean activity coefficients ( ) was

proposed by Peter Debye and Erich Hückel in 1923. This theory is applicable to strong

electrolytes (salts). Strong electrolytes are neutral and dissocial completely. It is a

theoretical explanation for deviation from ideality in solutions of strong electrolytes. This

theory is an analytical approach to the factors influencing activity coefficients of ionic

solutes.

Debye and Hückel developed this theory with which individual activity coefficients can

be calculated, and using this mean activity coefficients is calculated which could be

tested against experimental data.

A detailed description of Debye-Hückel theory will include discussion of the

assumptions, is mathematical development, limitations and applications.

Postulates of Debye-Hückel Theory.

The basis of the Debye-Hückel Theory is laid on strong electrolytes. The electrolytic

solution consists of solvated ions and water molecules. The basic representation of an

idealized electrolytic solution is shown in fig.1. The ions are represented as spheres with

unit electrical charge. The solvent (pale blue) is shown as a uniform medium. Each ion is

surrounded more closely by ions of opposite charge than by ions of same charge. The

principal assumption is that departure from ideality is due to electrostatic interactions

between ions (which varies as 1/r2 )

Chemistry



Fig. 1: Ideal representation of a 1:1 electrolyte

The postulates of the Debye-Hückel theory are as follows:

1. The ion were treated as point charges

2. The ion-ion interactions were assumed to be long range coulombic forces (which

varies as 1/r2) while short range non-couloumbic forces (dispersion forces) play a

negligible role.

3. The role of solvent is to provide a medium which is a continuum of dielectric

constant (ɛ) which is necessary for the operation of interionic forces (In the

present case the water molecules are looked upon as a continuous dielectric

medium)

4. The concept of charge density was introduced which measured charge density (ρ)

as a function of distance (r).

5. All the ions in the electrolytic solution are free to contribute to charge density.

Bulk charge density was assumed to zero, which means that the solution is

overall electrically neutral.

The basic percept of this theory is that because of attraction between positive and

negative ions, there are, on an average in the vicinity of a particular ion more ions of

opposite sign and this will reduce the ‘effective concentration of ions’. Consequently if

the solution is diluted the separation of ions involves doing an additional work to

overcome these interionic interactions and this represents deviation from an ideal

Chemistry



solution. Ultimately, Debye- Hückel theory generates a relationship between mean

activity (γ± ) coefficient and strength (I).

4. Mathematical Treatment

The first step in the Debye-Hückel approach is the arbitrary selection of reference or

central ion from the assembly of ions in solution. The solution contains ions surrounded

by water molecules. Only the reference ion carries a discrete charge i.e. zieo. In this case,

water is acting as the solvent, therefore, water molecules provide a continuum dielectric

medium. The remaining ions of the solution are spread out into a continuous spatial

distribution of charge (fig. 2)

Fig 2: A schematic comparison of (a) the assembly of ions and solvent molecules in a real electrolytic

solution (b) the Debye Hückel picture in which reference ion is surrounded by net charge density (ρ)

The total charge into the atmosphere and on the reference ion is exactly equal but has

opposite sign.

Chemistry



Now, we will discuss ion-ion interactions in detail in order to evolve a quantitative

measure of these interactions.

Consider an initial state, where ion-ion interactions are absent. This state is referred as the

discharged state (switched off). We also have a final state, where these ion-ion

interactions come into play. This state is known as charged state (switched on) (fig 3)

Fig 3: The free energy change of ion- ion interactions in going from discharged state to charge

state

Then the free-energy charge in going from the initial state of the final state is considered

as the free energy charge , for ion-ion interactions.

The initial state is an ideal case, but it is a hypothetical situation, as we cannot take ions

in vacuum, and moreover when these ions enter the solvent, there will be ion-solvent

interactions. Thus, initial state is a imaginary state of non-interacting ions which implies

that it is an assembly of discharged ions; in this state the reference ion is in discharged

state i.e., its charge is zero and thus ion-ion interactions are zero with respect to the

reference ion.

On the other hand, the charged state is the real/actual case, where the reference ion

carries a charge equal to zieo and ion-ion interactions are present. (fig. 4)

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Fig 4: The free energy change of ion – ion interactions is the electrostatic work of taking an

imaginary assembly of discharged ions and charging them up to obtain a solution of charged ions

Thus, the process of going from an initial state of non interacting ions to a final state of

ion-ion interactions is equivalent to taking an assembly of discharged ions and charging

them up and this electrostatic charging work is taken equal to the free energy ∆Gi-i of ion-

ion interactions.

Our main aim is to isolate the contribution to the free energy of ion-ion interactions

arising from one ionic species ‘i’ only. By definition, this partial free – energy change is

the chemical potential change ∆μί-ί arising from the interactions of one ionic species with

the ionic assembly.

During the entire charging process both the positively charged and negatively charged

ionic species are charged. Therefore we obtain a free energy change which involves

the contribution from all the ionic spices which constitute the electrolyte.

Chemistry



Fig 5: The chemical potential ∆μί-ί arising from the inter ionic interactions of an ionic species i with

the electrolytic solution.

If we want to calculate the work of charging up a mole of the reference ion from a state

of zero change to its final charge , then the charging work is W times the Avagadro

Number (NA) and this is equal to the partial molar free energy of ion-ion interactions i.e.,

the chemical potential of ion-ion interactions. Mathematically,

...(3)

The chemical potential ( ) of a non-electrolyte solution is represented using the

classical thermodynamic formula:

...(4)

where:

is the concentration of the solute in mole fraction.

= chemical potential in the standard state

For dilute solutions i.e when the concentration of solute is low, the short range

interactions become insignificant. It is referred as an ideal case.

Previously ionic solutions were treated in the same way as non electrolytic solutions even

though ionic solutions contain charged species. But as, we know in ionic solutions, there

exists ion-ion and ion solvent interactions therefore we cannot use the same classical

thermodynamic formula as we have used for non-electrolytic solutions. Here the

concentration term is modified and we use activity for such type of solutions.

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Thus chemical potential is given as:

…(5)

Since:

For ideal case

Therefore for Ideal case, equation 5:

...(6)

On the other hand, when the concentration of solute is high the short range interactions

which arise from dipole-dipole or dispersion forces become significant. This is referred

as the real case and chemical potential can be represented as:

...(7)

The change in chemical potential from real to ideal case is given by:

…(8)

Using equation (6) and (7) in equation (8):

...(9)

where:

change in chemical potential or Gibb’s flow energy per mole or the work done in

charging the reference ion from 0 to .

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Rearranging equation (9), we get

...(10)

Using equation (3) in equation (10) we get,

Thus, equation 10 fetches us with a relationship between the mean activity coefficient (γ)

and the work done in charging the reference ion from a discharged state to a charged state

(w).

Chemistry



5. Summary

In this module we have learnt:

Ionic solutes when present in high concentration show deviations from ideality.

Electrolytic solutions are non-ideal because of the powerful long range forces between

ions. Thus electrostatic forces are considered to be the predominant forces acting upon

ions in solutions

The Debye- Hückel approach has been proposed in order to calculate mean activity

coefficient of ions in solutions. Thus this theory analyses the factors that influence the

ionic activity of ions in solution

The Debye- Hückel theory assumes a number of conditions although many of these

assumptions are questionable

A hypothetical state in the electrolytic solution is considered where ion-ion interactions

do not operate and another state where the interactions come into play due to charging of

ions. Thus the free energy Gi-i of ion-ion interactions is the electrostatic work of

taking an imaginary assembly of discharged ions and charging them up to obtain a

solution of charged ions.

The chemical potential Δμi-i arises from interactions of an ionic specie with the solution

and this is equal to the Avogadro number (NA) times the electrostatic work of taking an

imaginary solution in which one reference ion is alone and discharged and charging this

reference ion.

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