Question

A monoprotic acid in $1.00$M solution is $0.001$ % ionised. The dissociation constant of acid is:
A. $1 \times \,{10^{ - 3}}$
B. $1 \times \,{10^{ - 6}}$
C. $1 \times \,{10^{ - 8}}$
D. $1 \times \,{10^{ - 10}}$

Answer 1

The dissociation constant is determined as the product of the square of the degree of dissociation and concentration. We will use the dissociation constant formula to determine the dissociation constant. It is given that the acid is monoprotic so the van’t Hoff factor will remain one. We can use the simple dissociation constant formula.

Formula used: ${{{K}}_{{a}}}{{ = }}\,\dfrac{{{{C}}{{{\alpha }}^{{2}}}}}{{{{1}} - {{\alpha }}}}$

Complete step-by-step answer:
The formula which relates the degree of dissociation with dissociation constant is as follows:
${{{K}}_{{a}}}{{ = }}\,\dfrac{{{{C}}{{{\alpha }}^{{2}}}}}{{{{1}} - {{\alpha }}}}$
Where,
${{{K}}_{{a}}}$is the acid dissociation constant.
C is the concentration.
${{\alpha }}$is the degree of dissociation
$1.00$M solution is $0.001$ % ionised

On substituting $1.00$M for the concentration and $0.001$ for degree of dissociation.
$\Rightarrow {{{K}}_{{a}}}{{ = }}\,\dfrac{{{{1}}{{.00}}\, \times {{{{(0}}{{.001)}}}^{{2}}}}}{{{{1}} - 0.001}}$
$\Rightarrow {{{K}}_{{a}}}{{ = }}\,\dfrac{{{{1}} \times {{1}}{{{0}}^{ - 6}}}}{{0.99}}$
$\Rightarrow {{{K}}_{{a}}}{{ = }}\,{{1}} \times {{1}}{{{0}}^{ - 6}}$

Mostly weak electrolyte dissociates very less so, for weak electrolyte the degree of dissociation is very-very less than one,$\alpha < < 1$ so, the value of $1 - \alpha$ can be taken to equal to$1$so, the formula of the dissociation constant can be reduced as, ${K_a} = \,C{\alpha ^2}$.
$\Rightarrow {{{K}}_{{a}}}{{ = }}\,{{1}}{{.00}}\, \times {{{(0}}{{.001)}}^{{2}}}$
$\Rightarrow {{{K}}_{{a}}}{{ = }}\,{{1}} \times {{1}}{{{0}}^{ - 6}}$
So, the dissociation constant of acid is ${{1}} \times {{1}}{{{0}}^{ - 6}}$.

Therefore, option (B) $1 \times \,{10^{ - 6}}$, is correct.

Note: The degree of dissociation tells the dissociated amount of the weak electrolyte. The strong electrolyte dissociates completely, so it is not calculated for the strong electrolyte. A similar formula is used for the determination of the degree of dissociation for a weak base only the${{{K}}_{{a}}}$ is replaced with ${{{K}}_{{b}}}$where, ${{{K}}_{{b}}}$ is the base dissociation constant. The degree of dissociation can also be calculated by using equivalent conductance at a concentration and equivalent conductance at infinite dilution. The formula to calculate the degree of dissociation is $\alpha = \,\dfrac{{{\lambda _m}}}{{\lambda _m^ \circ }}$. Where, ${\lambda _m}$ is the equivalent conductance at a concentration and $\lambda _m^ \circ$is the equivalent conductance at infinite dilution.