Ionic product of water

“The product of concentrations of H⁺ and OH^– ions in water at a particular temperature is known as ionic product of water.” It is designated as K_w.

                               H₂O ⇌ H⁺ ; ΔH = +57.3 kJM^–1

                             

The value of K_w increases with the increase of temperature, i.e., the concentration H⁺ and OH^– ions increases with increase in temperature.

                     Temperature (^oC)                  Value of K_w

                               0                                    0.11 × 10^–4

                               10                                  0.31 × 10^–14

                               25                                  1.00 × 10^–14

                               100                                7.50 × 10^–14

The value of K_w at 25^oC is 1 × 10^–14. Since pure water is neutral in nature, H⁺ ion concentration must be equal to OH^– ion concentration.

[H⁺] = [OH^–] = x or [H⁺][OH^–] = x² = 1 × 10^–14 or x = 1 × 10^–7 M or [H⁺] = [OH^–] = 1 × 10^–7 mole litre^–1

This  shows that at 25^oC, in 1 litre only 10^–7 mole of water is in ionic form out of a total of approximately 55.5 moles.

Thus when, [H⁺] = [OH^–] ; the solution is neutral

[H⁺] > [OH^–] ; the solution is acidic

[H⁺] < [OH^–] ; the solution is basic

 

Hydrogen ion concentration – pH scale

Sorensen, a Danish biochemist developed a scale to measure the acidity in terms of concentrations of H⁺ in a solution. As defined by him, “pH of a solution is the negative logarithm to the base 10 of the concentration of H⁺ ions which it contains.”

                     pH = –log[H⁺] or 

A pH scale ranges from 0 to 14

 

Just as pH indicates the hydrogen ion concentration, the pOH represents the hydroxyl ion concentration, i.e.,

                        pOH = –log[OH^–]

Considering the relationship, [H⁺][OH^–] = K_w = 1 × 10^–14

Taking log on both sides, we have

log[H⁺] + log[OH^–] = logK_w = log(1 × 10^–14) or

–log[H⁺] –log[OH^–] = –log K_w = –log(1 × 10^–14)

Or pH + pOH = pK_w = 14

i.e., sum of pH and pOH is equal to 14 in any aqueous solution at 25^oC. The above discussion can be summarised in the following manner,

 

	[H+]	[OH–]	pH	pOH
Acidic solution	> 10–7	< 10–7	< 7	> 7
Neutral solution	10–7	10–7	7	7
Basic solution	< 10–7	> 10–7	> 7	< 7

 

pH of some materials

Material	pH	Material	pH
Gastric juice	1.4	Rain water	6.5
Lemon juice	2.1	Pure water	7.0
Vinegar	2.9	Human saliva	7.0
Soft drinks	3.0	Blood plasma	7.4
Beer	4.5	Tears	7.4
Black coffee	5.0	Egg	7.8
Cow’s milk	6.5	Household ammonia	11.9

 

Limitations of pH scale

(i)      pH values of the solutions do not give us immediate idea of the relative strengths of the solutions. A solution of pH = 1 has a hydrogen ion concentration 100 times that of a solution pH = 3 (not three times). 4 × 10^–5N HCl is twice concentrated of a 2 × 10^–5N HCl solution, but the  values of these solutions are 4.40 and 4.70 (not double).

(ii)     pH value zero is obtained in 1N solution of strong acid. If the concentration is 2N, 3N, 10N, etc. the respective pH values will be negative.

(iii)    A solution of an acid having very low concentration, say 10^–8N, can not have pH8, as shown by pH formula but the actual pH value will be less than 7.

pK value : P stands for negative logarithm. Just as H⁺ and OH^– ion concentrations range over many negative powers of 10, it is convenient to express them as pH or pOH, the dissociation constant (K) values also range over many negative powers of 10 and it is convenient to write them as pK. Thus, pK is the negative logarithm of dissociation constant.

pK_a = –logK_a and pK_b = – logK­_b

Weak acids have higher pK_a values. Similarly weak bases have higher pK_b values

For any conjugate acid-base pair in aqueous solution, K_a × K_b = K_w

pK_a + pK_b = pK_w = 14 (at 298 K)

 

Calculation the pH of 10^–8 M HCl

If we use the relation pH = –log[H₃O⁺] we get pH equal to 8, but this is not correct because an acidic solution connot have  greater than 7. In this condition concentration of water cannot be neglected.

Therefore, [H⁺]_total [H⁺]_total = [H⁺]_total = H⁺_Acid + H⁺ _water

Since  is strong acid and completely ionised,

[H⁺]_HCl = 1 × 10^–8 [H⁺]_H2O = 10^–7

[H⁺]_total = [H⁺]_HCl + [H⁺]_H2O = 10^–8 + 10^–7 = 10^–8 [1+10] = 10^–8 × 11

 pH = –log10^–8 + log11 = 6.958

Similarly if NaOH concentration is 10^–8 M

Then, [OH^–]_total = [10^–8]_NaOH + [10^–7]_H2O

[OH^–] = 10^–8 × 11 ; pOH = 6.96    pH = 7.04

Note :

Solutions may have negative pH [C_H⁺ = 10M, pH = λ – 1] and even pH above 14[C_OH^– = 10M, pH = 15, but in such concentrated solutions the activity coefficients should be used. Hence the pH range at 298K is taken as 0 to 14 for most of the practical purposes:

Acidity of an aqueous solution α 1/pH

pH of pure water decreases with increase of temperature e.g., pH of boiling water is 6.56.

pH of the solution of a weak acid or a weak base depends upon its degree of ionization (a). The relationship may be derived as follows taking the example of weak acid HA.

HA ⇌ H⁺ + A^–

Initial conc.        C

Conc. at eqn.  C(1–α)  Cα     Cα

[H⁺] = Cα  pH = – log[H⁺] = –logCα

But K_a= Cα² or C = K_a/α²

pH = –log(K_a/α)

• If pK_a (or pK_b) value is negative, it indicates that the acid (or base) is completely ionised e., acids and bases are strong.