Question

The ${K_a}$ values of formic acid and acetic acid are respectively $1 \cdot 77 \times {10^{ - 4}}$ and $1 \cdot 75 \times {10^{ - 5}}$ The ratio of the acid strength of ${\text{0}} \cdot {\text{1 M}}$ acid is:
A) ${\text{10}}$
B) ${\text{3}} \cdot {\text{178}}$
C) ${\text{0}} \cdot {\text{3}}$
D) ${\text{0}} \cdot {\text{1}}$
E) ${\text{100}}$

Answer 1

Hint: The acid strength is the tendency of acid to dissociate into a proton and an anion. The values of dissociation constants of both acids are given which quantifies the value of acidic strength.

Complete step by step answer:
1) First of all let's see the dissociation of an acid where it forms an anion and proton. The general scheme of dissociation is,
$HA\overset {} \leftrightarrows {H^ + } + {A^ - }$
2) Now the dissociation constant value, ${K_a}$ can be described by the following formula,
${K_a} = \dfrac{{\left[ {{H^ + }} \right]\left[ {{A^ - }} \right]}}{{\left[ {HA} \right]}}$
3) Now we can take the values of ${C_a}$ as the concentrations of the ${H^ + }$ and ${A^ - }$, the $\alpha $ as the degree of dissociation. We get the equation as,
${K_a} = \dfrac{{{C_a} \times {C_a}}}{{C(1 - \alpha )}} = \dfrac{{{C_a}^2}}{{C(1 - \alpha )}}$
Where the value of $\alpha $ is very very small and therefore can be neglected. Hence, we get the relation between the dissociation constant and the concentration as follows,
$\alpha = \sqrt {\dfrac{{{K_a}}}{C}} $
The value of concentration, ${\text{C}}$, is constant, hence we can write the relation between the dissociation constant and acidic strength as below,
$\dfrac{{{\alpha _1}}}{{{\alpha _2}}} = \sqrt {\dfrac{{{K_{a1}}}}{{{K_{a2}}}}} $
4) Now let's put the values of the dissociation constants of both acids in the above equation,
$\dfrac{{{\alpha _1}}}{{{\alpha _2}}} = \sqrt {\dfrac{{1 \cdot 77 \times {{10}^{ - 4}}}}{{1 \cdot 75 \times {{10}^{ - 5}}}}} $
Which we can simplify the above equation by taking the power factor same as below,
$\dfrac{{{\alpha _1}}}{{{\alpha _2}}} = \sqrt {\dfrac{{1 \cdot 77 \times {{10}^{ - 4}}}}{{0 \cdot 175 \times {{10}^{ - 4}}}}} $
$\dfrac{{{\alpha _1}}}{{{\alpha _2}}} = \sqrt {10} $
$\dfrac{{{\alpha _1}}}{{{\alpha _2}}} = 3 \cdot 178$
5) Therefore, the ratio between the acid strength is ${\text{3}} \cdot {\text{178}}$ which shows option B as a correct choice.

Therefore, option B is the correct choice.


Note:
The acidic strength of an acid is directly proportional to the square root of the dissociation constant value of that acid. The more the value of dissociation constant means the acid readily forms ions in the solution which means the value of the acidic strength will also increase. The ratio value doesn’t have a unit.