`=>` Consider a weak acid `color{red}(HX)` that is partially ionized in the aqueous solution. The equilibrium can be expressed by:

`color{red}(HX (aq) + H_2O (l) ⇌ H_3 O^+ (aq) + X^- (aq))`

`color{green}("Initial concentration (M)")`

`color{red}(c \ \ \ \ \ \ \ \ 0 \ \ \ \ \ \ \ \ \ \ \ 0)`

`color{green}("Let α be the extent of ionization
Change (M)")`

`color{red}(-c alpha \ \ \ \ \ \ \ \ \ \ +c alpha \ \ \ \ \ \ \ \ \ \ +c alpha)`

`color{green}("Equilibrium concentration (M)")`

`color{red}(c- calpha \ \ \ \ \ \ \ \ \ \ c alpha \ \ \ \ \ \ \ \ \ \ c alpha)`

Here, `color{red}(c)` = initial concentration of the undissociated acid, `color{red}(HX)` at time, `color{red}(t = 0. α =)` extent up to which `color{red}(HX)` is ionized into ions. Using these notations, we can derive the equilibrium constant for the above discussed acid dissociation equilibrium:

`color{red}(K_a = c^2 alpha^2 // c (1-alpha) = c alpha^2 // 1- alpha)`

`color{red}(K_a)` is called the dissociation or ionization constant of acid `color{red}(HX)`. It can be represented alternatively in terms of molar concentration as follows,

`color{red}(K_a = [H^+] [X^- ] // [HX])` ................(7.30)


`=>` At a given temperature `color{red}(T, K_a)` is a measure of the strength of the acid `color{red}(HX)` i.e., larger the value of `color{red}(K_a)`, the stronger is the acid. `color{red}(K_a)` is a dimensionless quantity with the understanding that the standard state concentration of all species is `1M`.


The values of the ionization constants of some selected weak acids are given in Table 7.6. The `color{red}(pH)` scale for the hydrogen ion concentration has been so useful that besides `color{red}(pK_w)`, it has been extended to other species and quantities. Thus, we have:

`color{red}(pK_a = -log(K_a))` .........(7.31)

Knowing the ionization constant, `color{red}(K_a)` of an acid and its initial concentration, `color{red}(c)`, it is possible to calculate the equilibrium concentration of all species and also the degree of ionization of the acid and the `color{red}(pH)` of the solution.

`=>` A general step-wise approach can be adopted to evaluate the `color{red}(pH)` of the weak electrolyte as follows:

Step 1. The species present before dissociation are identified as Brönsted-Lowry acids / bases.

Step 2. Balanced equations for all possible reactions i.e., with a species acting both as acid as well as base are written.

Step 3. The reaction with the higher `color{red}(K_a)` is identified as the primary reaction whilst the other is a subsidiary reaction.

Step 4. Enlist in a tabular form the following values for each of the species in the primary reaction

`color{red}(a)` Initial concentration, `color{red}(c)`.

`color{red}(b)` Change in concentration on proceeding to equilibrium in terms of `color{red}(α)`, degree of ionization.

`color{red}(c)` Equilibrium concentration.

Step 5. Substitute equilibrium concentrations into equilibrium constant equation for principal reaction and solve for `color{red}(α)`.

Step 6. Calculate the concentration of species in principal reaction.

Step 7. Calculate `color{red}(pH = – log[H_3O^+])`

`=>` The ionization of base `color{red}(MOH)` can be represented by equation:

`color{red}(MOH (aq) ⇌ M^(+) (aq) + OH^(-) (aq))`

In a weak base there is partial ionization of `color{red}(MOH)` into `color{red}(M^+)` and `color{red}(OH^–)`, the case is similar to that of acid-dissociation equilibrium. The equilibrium constant for base ionization is called base ionization constant and is represented by `color{red}(K_b)`. It can be expressed in terms of concentration in molarity of various species in equilibrium by the following equation:

`color{red}(K_b = [M^+] [OH] // [MOH])` ...........(7.33)

Alternatively, if `color{red}(c =)` initial concentration of base and `color{red}(α)` = degree of ionization of base i.e. the extent to which the base ionizes. When equilibrium is reached, the equilibrium constant can be written as :

`color{red}(K_c = ( c alpha)^2 //c (1-alpha) = c alpha^2//(1-alpha))`

`=>` The values of the ionization constants of some selected weak bases, `color{red}(K_b)` are given in Table 7.7.

`=>` Many organic compounds like amines are weak bases. Amines are derivatives of ammonia in which one or more hydrogen atoms are replaced by another group. For example, methylamine, codeine, quinine and nicotine all behave as very weak bases due to their very small `color{red}(K_b)`. Ammonia produces `color{red}(OH^–)` in aqueous solution :

`color{red}(NH_3(aq) + H_2O(l) ⇌ NH_4^+ (aq) + OH^– (aq))`

The `color{red}(pH)` scale for the hydrogen ion concentration has been extended to get:

`color{red}(pK_b = - log (K_b))` ...............(7.34)

`=>` As seen, `color{red}(K_a)` and `color{red}(K_b)` represent the strength of an acid and a base, respectively. In case of a conjugate acid-base pair, they are related in a simple manner so that if one is known, the other can be deduced. Considering the example of `color{red}(NH_4^+)` and `color{red}(NH_3)`

we see,

`color{red}(NH_4^+ (aq) +H_2O (l) ⇌ H_3O^+ (aq) + NH_3(aq))`

`color{red}(K_a = [H_3O^+][ NH_3] // [NH_4^+] = 5.6 × 10^(–10))`

`color{red}(NH_3(aq) + H_2O(l) ⇌ NH_4^+(aq) + OH^(–)(aq))`

`color{red}(K_b =[ NH_4+][ OH^–] // NH_3 = 1.8 × 10^(–5))`

Net: `color{red}(2 H_2O(l) ⇌ H_3O^+(aq) + OH^–(aq))`

`color{red}(K_w = [H_3O^+][ OH^– ] = 1.0 × 10^(–14) M)`

Where, `color{red}(K_a)` represents the strength of `color{red}(NH_4^+)` as an acid and `color{red}(K_b)` represents the strength of `color{red}(NH_3)` as a base.

It can be seen from the net reaction that the equilibrium constant is equal to the product of equilibrium constants `color{red}(K_a)` and `color{red}(K_b)` for the reactions added. Thus,

`color{red}(K_a × K_b = {[H_3O^+][ NH_3] // [NH_4^+ ]} × {[NH_4^+ ][ OH^–] // [NH_3]})`

`color{red}(= [ H_3 O^+ ] // [OH] = K_w)`

`color{red}(= (5.6x10^(–10)) × (1.8 × 10^(–5)) = 1.0 × 10^(–14) M)`

This can be extended to make a generalisation. The equilibrium constant for a net reaction obtained after adding two (or more) reactions equals the product of the equilibrium constants for individual reactions:

`color{red}(K_(NET) = K_1 xx K_2 xx ...............)` (3.35)

Similarly, in case of a conjugate acid-base pair

`color{red}(K_a xx K_b = K_w)` ..................(7.36)

Knowing one, the other can be obtained. It should be noted that a strong acid will have a weak conjugate base and vice-versa.

Alternatively, the above expression `color{red}(K_w = K_a × K_b)`, can also be obtained by considering the base-dissociation equilibrium
reaction:

`color{red}(B (aq) + H_2O (l) ⇌ BH^+(aq) + OH^–(aq))`

`color{red}(K_b = [BH^+] [OH^-] // [B])`

As the concentration of water remains constant it has been omitted from the denominator and incorporated within the
dissociation constant. Then multiplying and dividing the above expression by `color{red}([H^+])`, we get:

`color{red}(K_b = [ BH^+] [OH^-] [H^+] // [B] [H^+])`

`color{red}( = { [OH^-][H^+]}{[BH^+]//[B][H^+]})`

` color{red}(= K_w / K_a)`

or `color{red}(K_a xx K_b = K_w)`

It may be noted that if we take negative logarithm of both sides of the equation, then `color{red}(pK)` values of the conjugate acid and base are related to each other by the equation:

`color{red}(pK_a + pK_b = pK_w = 14)` (at 298K)

`=>` Some of the acids like oxalic acid, sulphuric acid and phosphoric acids have more than one ionizable proton per molecule of the acid. Such acids are known as polybasic or polyprotic acids.

The ionization reactions for example for a dibasic acid `color{red}(H_2X)` are represented by the equations :

`color{red}(H_2X (aq) ⇌ H^+(aq) + HX^–(aq))`

`color{red}(HX^–(aq) ⇌ H^+(aq) + X^(2–)(aq))`
And the corresponding equilibrium constants are given below:
`color{red}(K_(a_1) = {[H^+][HX^–]} // [H_2X])` and `color{red}(K_(a_2) = {[H^+][X^(2-)]} // [HX^-])`


Here, ` color{red}(K_(a_1))` and `color{red}(K_(a_2))` are called the first and second ionization constants respectively of the acid `color{red}(H_2X)`. Similarly, for tribasic acids like `color{red}(H_3PO_4)` we have three ionization constants.

`=>` The values of the ionization constants for some common polyprotic acids are given in Table 7.8.

`=>` It can be seen that higher order ionization constants `color{red}(( K_(a_2) ,K_(a_3))` are smaller than the lower order ionization constant `color{red}( K_(a_1) )` of a polyprotic acid. The reason for this is that it is more difficult to remove a positively charged proton from a negative ion due to electrostatic forces. This can be seen in the case of removing a proton from the uncharged `color{red}(H_2CO_3)` as compared from a negatively charged `color{red}(HCO_3^–)`. Similarly, it is more difficult to remove a proton from a doubly charged `color{red}(HPO_4^(2–))` anion as compared to `color{red}(H_2PO_4^–)`.

`=>` Polyprotic acid solutions contain a mixture of acids like `color{red}(H_2A, HA^–)` and `color{red}(A^(2–))` in case of a diprotic acid. `color{red}(H_2A)` being a strong acid, the primary reaction involves the dissociation of `color{red}(H_2 A)`, and `color{red}(H_3O^+)` in the solution comes mainly from the first dissociation step.

`=>` Having discussed quantitatively the strengths of acids and bases, we come to a stage where we can calculate the `color{red}(pH)` of a given acid solution. But, the curiosity rises about why should some acids be stronger than others? What factors are responsible for making them stronger? The answer lies in its being a complex phenomenon. But, broadly speaking we can say that the extent of dissociation of an acid depends on the strength and polarity of the `color{red}(H-A)` bond.

`=>` In general, when strength of `color{red}(H-A)` bond decreases, that is, the energy required to break the bond decreases, `color{red}(HA)` becomes a stronger acid. Also, when the `color{red}(H-A)` bond becomes more polar i.e., the electronegativity difference between the atoms `color{red}(H )`and `color{red}(A)` increases and there is marked charge separation, cleavage of the bond becomes easier thereby increasing the acidity.


`=>` But it should be noted that while comparing elements in the same group of the periodic table, `color{red}(H-A)` bond strength is a more important factor in determining acidity than its polar nature. As the size of A increases down the group, `color{red}(H-A)` bond strength decreases and so the acid strength increases. For example,

`color{green}("Size increases")`

`color{red}(underset(→) overset(→)( HF < < HCl < < HBr < < HI ))`

`color{green}("Acid strength increases")`
Similarly, `color{red}(H_2S)` is stronger acid than `color{red}(H_2O)`.

But, when we discuss elements in the same row of the periodic table, `color{red}(H-A)` bond polarity becomes the deciding factor for determining the acid strength. As the electronegativity of A increases, the strength of the acid also increases. For example,

`color{green}("Electronegativity of A increases ")`

`color{red}(underset(→) overset(→) (CH_4 < NH_3 < H_2O < HF))`

`color{green}("Acid strength increases")`