JEE Main & Advanced Chemistry Equilibrium / साम्यावस्था 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 \[pH=\log \frac{1}{[{{H}^{+}}]}\]

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

\[pOH=-\log [O{{H}^{-}}]\]

Considering the relationship, \[[{{H}^{+}}][O{{H}^{-}}]={{K}_{w}}=1\times {{10}^{-14}}\]

Taking log on both sides, we have

\[\log [{{H}^{+}}]+\log [O{{H}^{-}}]=\log {{K}_{w}}=\log (1\times {{10}^{-14}})\]or

\[-\log [{{H}^{+}}]-\log [O{{H}^{-}}]=-\log {{K}_{w}}=-\log (1\times {{10}^{-14}})\]

or \[pH+pOH=p{{K}_{w}}=14\]

pH of some materials

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). A \[4\times {{10}^{-5}}N\ HCl\] is twice concentrated of a \[2\times {{10}^{-5}}N\ HCl\] solution, but the \[pH\] 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 \[{{10}^{-8}}N\], can not have pH 8, 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 \[O{{H}^{-}}\] 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.

\[p{{K}_{a}}=-\log {{K}_{a}}\] and \[p{{K}_{b}}=-\log {{K}_{b}}\]

Weak acids have higher \[p{{K}_{a}}\] values. Similarly weak bases have higher \[p{{K}_{b}}\] values

     For any conjugate acid-base pair in aqueous solution, \[{{K}_{a}}\times {{K}_{b}}={{K}_{w}}\]

\[p{{K}_{a}}+p{{K}_{b}}=p{{K}_{w}}=14\] (at 298^o K)

Calculation of the pH of \[{{10}^{-8}}M\ HCl\] & \[{{10}^{-8}}M\,NaOH\]

If we use the relation \[pH=-\log [{{H}_{3}}{{O}^{+}}]\] we get \[pH\] equal to 8, but this is not correct because an acidic solution connot have \[pH\] greater than 7. In this condition \[{{H}^{+}}\]concentration of water cannot be neglected.

Therefore, \[{{[{{H}^{+}}]}_{total}}=H_{Acid}^{+}+H_{water}^{+}\]

Since \[HCl\] is strong acid and completely ionised,

\[{{[{{H}^{+}}]}_{HCl}}=1\times {{10}^{-8}}\], \[{{[{{H}^{+}}]}_{{{H}_{2}}O}}={{10}^{-7}}\]

\[{{[{{H}^{+}}]}_{total}}={{[{{H}^{+}}]}_{HCl}}+{{[{{H}^{+}}]}_{{{H}_{2}}O}}\]\[={{10}^{-8}}+{{10}^{-7}}\]\[={{10}^{-8}}\ [1+10]\]\[={{10}^{-8}}\times 11\]

\[pH=-\log {{10}^{-8}}+\log 11=6.958\]

Similarly if \[NaOH\] concentration is \[{{10}^{-8}}M\]

Then,  \[{{[O{{H}^{-}}]}_{total}}={{[{{10}^{-8}}]}_{NaOH}}+{{[{{10}^{-7}}]}_{{{H}_{2}}O}}\]

\[[O{{H}^{-}}]={{10}^{-8}}\times 11\] ; \[pOH=6.96\] \[pH=7.04\]