Ionic Equilibrium : Ostwald dilution law, Ka & Kb

Electrical conductors

Substances, which allow electric current to pass through them, are known as conductors or electrical conductors. Conductors can be divided into two types,

(1)     Metallic or Electronic conductors : Those conductors which conduct electricity without undergoing any chemical change are known as metallic or electronic conductors. As the name indicates, the conduction in this case is due to flow of electrons. Metals (e.g., copper, iron, aluminium, silver etc.), non-metals like graphite and certain minerals are common examples.

(2)     Electrolytic conductors or Electrolytes : Those conductors which undergo decomposition (a chemical change) when an electric current is passed through them are known as electrolytic conductors. In this case the flow of electricity is due to the movement of ions. Solutions of acids, bases and salts in water, fused salt etc. are common examples. Electrolytes are further divided into two types on the basis of their strengths,

(i)      Strong electrolytes : Those substances which are almost completely ionize into ions in their aqueous solution are called strong electrolytes. Degree of ionization for this type of electrolyte is one i.e., α ≈ 1.

For example : HCl, H2SO4, NaCl, HNO3, KOH, NaOH, HNOAgNO3, AgNO3 etc. means all strong acids, bases and all types of salts.

(ii)     Weak electrolytes : Those substances which are ionize to a small extent in their aqueous solution are known as weak electrolytes. Degree of ionization for this types of electrolytes is α <<<1.

For example : H2O. CH3COOH, H2O, CH3COOH, NH4OH, HCN, HCOOH Liq. SO2 etc. Means all weak acids and bases.

Arrhenius theory of electrolytic dissociation

In order to explain the behaviour of electrolytes (substances decomposable by electricity) in solution, Arrhenius postulated the existence of electrically charged fragments (ions) in the form of theory commonly known as Arrhenius theory of dissociation or ionization.

(1)     Postulates of Arrhenius theory

(i)      In aqueous solution, the molecules of an electrolyte undergo spontaneous dissociation to form positive and negative ions.

                             NaOH → Na+ + OH ; KCl → K+ + Cl

(ii)     The degree of ionization or dissociation (the fraction of the total amount which dissociates) increases with dilution until at infinite dilution, it approaches unity. Mathematically,

          Degree of ionization (α) \[=\,\,\frac{\text{Number}\,\,\text{of}\,\,\text{dissociated}\,\,\text{molecules}}{\text{Total}\,\,\text{number}\,\,\text{of}\,\,\text{molecules}\,\,\text{of}\,\,\text{electrolyte}\,\,\text{before}\,\,\text{dissociation}}

(iii)    At moderate concentrations, there exists an equilibrium between the ions and undissociated molecules, such as,

         NaOH ⇌ Na+ + OH ; KCl  ⇌ K+ + Cl ; Al2(SO4)3 ⇌ 2Al+3 + 3SO4–2

          This equilibrium state is called ionic equilibrium.

(iv)    Each ion produced, due to electrolytic dissociation, produces the same effect on osmotic pressure and other colligative properties as an undissociated molecule. In other words, each ion behaves osmotically as a molecule.

 

(2)     Factors affecting degree of ionisation

(i)      Nature of electrolyte : As mentioned earlier, solutions of different electrolytes (strong and weak) of same concentration at constant temperature ionize to a different extent. Thus at normal dilution, value of  is nearly 1 for strong electrolytes, while it is very less than 1 for weak electrolytes.

(ii)     Nature of solvent : Dielectric constant of a solvent is a measure of its tendency to weaken the forces of attraction between oppositely charged ions of the electrolyte. Higher the dielectric constant of a solvent more is its ionising power. Water is the most powerful ionising solvent as its dielectric constant is highest.

(iii)    Concentration of the solution

          \[\text{Degree}\,\,\text{of}\,\,\text{ionisation}\,\,\propto \,\,\frac{\text{1}}{\text{Concentration}\,\,\text{of}\,\,\text{solution}}  \[\propto \frac{\text{1}}{\text{Amount of solute in given volume or wt}\text{. of solution}}

          ∞ Dilution of solution  Amount of solvent

(iv)    Temperature : Degree of ionisation of an electrolyte in solution increases with rise in temperature.

(v)     Presence of common ion : The degree of ionisation of an electrolyte decreases in the presence of a strong electrolyte having a common ion. For example, ionisation of CH3COOH is supressed in presence of HCl due to common H+ ions.

 

Ostwald’s dilution law

When acetic acid (a weak electrolyte) is dissolved in water, it dissociates partially into H+ or H3O+ and CH3COO ions and the following equilibrium is obtained, CH3COOH + H2O ⇌ CH3COO + H3O+

Applying law of chemical equilibrium, \[K=\frac{[C{{H}_{3}}CO{{O}^{-}}]\times [{{H}_{3}}{{O}^{+}}]}{[C{{H}_{3}}COOH]\times [{{H}_{2}}O]} .

In dilute solution, [H2O] is constant. The product of K and constant [H2O] is denoted as Ka, the ionization constant or dissociations constant of the acid is,

\[{{K}_{a}}=\frac{[C{{H}_{3}}CO{{O}^{-}}]\times [{{H}_{3}}{{O}^{+}}]}{[C{{H}_{3}}COOH]}                        …..(i)

If ‘C’ represents the initial concentration of the acid in moles L–1 and α, the degree of dissociation, then equilibrium concentration of the ions (CH3COO and H3O+ is equal to Cα and that of the undissociated acetic acid = C(1–α) i.e., we have

                   CH3COOH + H2O ⇌ CH3COO + H3O+

Initial conc.                  C                0                 0

Conc. at eqb.             C(1–α)           Cα            Cα

Substituting the values of the equilibrium concentrations in equation (i), we get

\[{{K}_{a}}=\frac{C\alpha .C\alpha }{C(1-\alpha )}=\frac{{{C}^{2}}{{\alpha }^{2}}}{C(1-\alpha )}=\frac{C{{\alpha }^{2}}}{1-\alpha }                 …..(ii)

In case of weak electrolytes, the value of  is very small and can be neglected in comparison to 1 i.e., 1 – α = 1. Hence, we get

                             Ka = Cα2 or $\alpha =\sqrt{\frac{{{K}_{a}}}{C}                             …..(iii)

The degree of dissociation, α can therefore be calcualted at a given concentration, C if Ka is known. Further, if V is the volume of the solution in litres containing 1 mole of the electrolyte, C = 1/V. Hence we have

                            \[\alpha =\sqrt{{{K}_{a}}V}                                                    …..(iv)

Similarly, for a weak base like NH4OH, we have

                            \[\alpha =\sqrt{{{K}_{b}}/C}=\sqrt{{{K}_{b}}V}                                    …..(v)

The above equations lead to the following result

“For a weak electrolyte, the degree of ionisation is inversely proportional to the square root of molar concentration or directly proportional to the square root of volume containing one mole of the solute.”

This is called Ostwald’s dilution law.

 

(1)     Experimental verification of Ostwald’s dilution law : For verifying the Ostwald’s dilution law, the degree of dissociation (α) of a binary electrolyte at different dilutions is determined and the values are placed in equation (ii) and thus values of K at different dilutions are calculated. If these values come out to be nearly constant, the law is correct.

The degree of dissociation, α, is determined by conductance measurements since the degree of ionisation α, of a weak electrolyte at a particular dilution is related as below,

\[\alpha =\frac{{{\lambda }_{v}}}{{{\lambda }_{\infty }}}                            

Where λV is the equivalent conductance at V- dilution. It can be calculated by conductance measurements and λ can be caculated by Kohlrausch’s law as,

λv = λc + λa 

Where, λ is the equivalent conductance at infinite dilution.

          λ is equivalent conductance of cation at infinite dilution.

          λ is  equivalent conductance of anion at infinite dilution.

 

(2)     Limitations of Ostwald’s dilution law : It was observed that,

(i)      The value of K comes out to be constant only for weak electrolytes and that too when the concentration of the solution is not too high, i.e., for dilute solutions only.

(ii)     The value of K does not comes out to be constant for strong electrolytes like HCl, KOH, KCl, etc.

Thus, it does not hold good for strong electrolytes.

(3)     Applications of Ostwald’s dilution law

(i)      It is useful in the calculation of the dissociation constants (K) of the weak acids and weak bases, by determining the degree of dissociation (α) from conductance measurements (λv) at any concentration (C).

(ii)     Knowing the value of K which is constant for a particular weak acid or weak base at a particular temperature, the degree of dissociation (α) of the weak acid or weak base can be calculated at any concentration (C).

 

Dissociation constants of acids and bases

(1)     Dissociation constant for weak acid : Consider an acid HA which when dissolved in water ionizes as,

HA ⇌ H+ + A

Applying the law of mass action, to this equilibrium,

\[{{K}_{a}}=\frac{[{{H}^{+}}][{{A}^{-}}]}{[HA]}

Where, Ka is the dissociation constant of the acid, HA. It has constant value at definite temperature and does not change with the change of concentration.

Dissociation Constant for polybasic acid : Polybasic acids ionise stepwise as, for example, orthophosphoric acid ionises in three steps and each step has its own ionisation constant.

                             H3PO4 ⇌H+ + H2PO4                    (I step)

                             H­2PO4 ⇌ H+ + HPO4–2                  (II step)

                             HPO4–2 ⇌ H+ + PO4–3                     (III step)

Let K1, K2 and K3 be the ionization constants of first, second and third steps respectively. Thus,

 \[{{K}_{2}}=\frac{[{{H}^{+}}][HPO_{4}^{-2}]}{[{{H}_{2}}PO_{4}^{-}]}  and  \[{{K}_{3}}=\frac{[{{H}^{+}}][PO_{4}^{-3}]}{[HPO_{4}^{-2}]}

In general, K1 is always much higher than K2 and K2 is higher than K3, i.e., K1 > K2 > K3

The decrease in the dissociation constants of successive steps of a polybasic acid is due to the fact that in the first step positively charged proton comes from a neutral molecule while in subsequent steps the proton comes from negatively charged molecules. The presence of negative charges makes process difficult for the loss of proton.

The overall dissociation constant (K) is given by the following relation,

                                      K = K1 × K2 × K3        

Ionisation constants of some acids at 25oC in water

Acid K1 K2 K3
HCN 1.8 × 10–10
CH3COOH 1.75 × 10–5
C6H5COOH 6.31 × 10–5
HCOOH 1.77 × 10–4
HNO2 4.6 × 10–4
H2S 6.3 × 10–8 1.3 × 10–12
H2CO3 4.5 × 10–7 5.62 × 10–11
H2C2O4 5.0 × 10–2 5.2 × 10–5
H2SO3 1.7 × 10–2 6.24 × 10–8
H3BO3 5.8 × 10–10 2.0 × 10–12
H3PO4 7.52 × 10–3 6.23 × 10–8 4.8 × 10–13
H3AsO4 5.0 × 10–5 8.3 × 10–8 6.0 × 10–11

 

(2)     Dissociation constant for weak base : The equilibrium of NH4OH (a weak base) can be represented as,      

NH4OH ⇌ NH4+ + OH

Applying law of mass action,

\[{{K}_{b}}=\frac{[NH_{4}^{+}][O{{H}^{-}}]}{[N{{H}_{4}}OH]}

Kb is constant at a definite temperature and does not change with the change of concentration.

 

Ionisation constants of some bases at 25oC

Base Kb Base Kb
NH4OH 1.81 × 10–5 C6H5NH2 4.1 × 10–10
CH3NH2­ 5.0 × 10–4 AgOH 1.1 ×10–4
(CH3)2NH 5.12 × 10–4 C2H5NH2 5.6 × 10–4
(CH3)3N 5.21 × 10–4 NH2NH2 3 × 10–6
HNO2 4.6 × 10–4

 

Note :     On the basis of Ka and Kb values, strength of acids and bases can be known. The stronger an acid (base), the larger is its Ka(Kb) and smaller is its pKa (pKb) value.