Question

If the ionization constant of ${H_2}C{O_3}$ as an acid in aqueous solution at room temperature is $X$. If the first and second ionisation constants of${H_2}C{O_3}$ are ${X_1}$ and ${X_2}$ respectively, then what will be the equation relating $X$ to ${X_1}$ and ${X_2}$ ?
(A) $X = \dfrac{{{X_1}}}{{{X_2}}}$
(B) $X = \dfrac{{{X_2}}}{{{X_1}}}$
(C) $X = {X_1}{X_2}$
(D) $X = \dfrac{{{X_1}{X_2}}}{2}$

Answer 1

Hint:As we know that the ionisation constant of an acid is the ratio of the concentration of the products and the concentrations of the reactants. It is represented as ${K_a}$ and informs about the strength of an acid. A strong acid is that acid which will completely ionise in the given aqueous solution.

Formula used: \[A + B \to C + D\]
Ionization constant $({K_a})$$ = \dfrac{{[C][D]}}{{[A][B]}}$$\dfrac{{{X_1}{X_2}}}{2}$

Complete step-by-step solution:An acid dissociation constant, ${K_a}$, is a measure of quantitative strength of an acid in solution. When the system achieves equilibrium it acts as the equilibrium constant for chemical reaction known as dissociation in the context of acid-base reaction.
The ionization of ${H_2}C{O_3}$ takes place in the following steps given below:
${H_2}C{O_3}$ + ${H_2}O$ $ \to $ $HC{O_3}^ - $+ ${H_3}{O^ + }$,
We can assume the ionisation constant for this reaction as: ${K_{a1}} = {X_1}$
$HC{O_3}^ - $ + ${H_2}O$ $ \to $ $C{O_3}^{2 - }$ + ${H_3}{O^ + }$
Let us assume the ionisation constant for this reaction as: ${K_{a2}} = {X_2}$
For step of ${H_2}C{O_3}$ ionisation, the ionisation constant is given as $X = \dfrac{{[C{O_3}^{2 - }][{H_3}{O^ + }]}}{{[{H_2}C{O_3}][{H_2}O]}}$
For${X_1}$, the ionisation constant would be: ${X_1} = \dfrac{{[HC{O_3}^ - ][{H_3}{O^ + }]}}{{[{H_2}C{O_3}][{H_2}O]}}$ -(1)
For${X_2}$, the ionisation constant would be: ${X_2} = \dfrac{{[C{O_3}^{2 - }][{H_3}{O^ + }]}}{{[HC{O_3}^ - ][{H_2}O]}}$ -(2)
After computing equations 1 and 2, we will get:
$X = \dfrac{{[HCO_3^ - ][{H_3}{O^ + }]}}{{[{H_2}C{O_3}][{H_2}O]}} \times \dfrac{{[CO_3^{2 - }][{H_3}{O^ + }]}}{{[HCO_3^ - ][{H_2}O]}}$
$X = \dfrac{{[CO_3^{2 - }][{H_3}{O^ + }]}}{{[{H_2}C{O_3}][{H_2}O]}}$
$\dfrac{{[CO_3^{2 - }][{H^ + }]}}{{[{H_2}C{O_3}]}} = \dfrac{{[CO_3^{2 - }][{H^ + }]}}{{[{H_2}C{O_3}]}}$
Both sides are equal so the correct option is $X = {X_1}{X_2}$.

Additional information:Like we can calculate the ionisation constant of the acid, similarly we can identify the ionisation constant of a base which is again the ratio of product to reactant concentrations and is represented by the symbol ${K_b}$ which informs us about the strength of a base in aqueous solution.

Therefore, the correct answer is (C).

Note:Always remember that the strong acids or strong bases are the one which completely ionises in an aqueous solution and a weak acid and a weak base are the one which will partially ionise in the aqueous solution and therefore the ionisation constant will be given as the ratio of product concentrations and reactant concentrations.