Question

What is the hydronium ion concentration of a 0.02 M solution of ${\text{C}}{{\text{u}}^{2 + }}$ solution of copper(II) perchlorate? The acidity constant of the following reaction is$5 \times {10^{ - 9}}$.
${\text{C}}{{\text{u}}^{2 + }}(aq.) + 2{{\text{H}}_2}{\text{O}}({\text{I}}) \rightleftharpoons {\text{Cu}}{({\text{HO}})^ + }({\text{aq}}.) + {{\text{H}}_3}{{\text{O}}^ + }(aq.)$
A.$1 \times {10^{ - 5}}$
B.$7 \times {10^{ - 4}}$
C.$5 \times {10^{ - 4}}$
D.$1 \times {10^{ - 4}}$

Answer 1

Hint: Ostwald's dilution law is a mathematical expression of the law of mass behaviour that expresses the relationship between the equilibrium constant/dissociation constant, the degree of dissociation, and concentration at constant temperature.
Formula used:
$\alpha = \sqrt {\dfrac{K}{C}} $
C = Concentration
K =dissociation constant of a weak acid
\[\alpha \]= degree of dissociation.

Complete answer:
Ostwald's dilution law relates the weak electrolyte's dissociation constant to the degree of dissociation ($\alpha $) and the concentration of the weak electrolyte.
The electrolyte AB is a binary electrolyte that dissociates between ${A^ + }$ and ${B^ - }$ ions.
${\text{AB}} \rightleftharpoons {{\text{A}}^ + } + {{\text{B}}^ - }$
Here ${\mathbf{C}}{{\mathbf{u}}^{2 + }} + {\mathbf{2}}{{\mathbf{H}}_2}{\mathbf{O}} \rightleftharpoons {\mathbf{Cu}}{({\mathbf{OH}})^ + } + {{\mathbf{H}}_3}{{\mathbf{O}}^ + }$
From dilution law we deduce that,
$(K = C{\alpha ^2}){\text{ }}$
$\alpha = \sqrt {\dfrac{K}{C}} $
Now from the given question we find the values to be
C = 0.02
K = $5 \times {10^{ - 9}}$
Now upon substituting the values we get
$\alpha = \sqrt {\dfrac{{\text{K}}}{{\text{C}}}} $
Hence
$\alpha = \sqrt {\dfrac{{5 \times {{10}^{ - 9}}}}{{0.02}}} $
Solving the denominator we get
$\alpha = \sqrt {\dfrac{{5 \times {{10}^{ - 9}} \times {{10}^2}}}{2}} $
$\alpha = \sqrt {25 \times {{10}^{ - 8}}} $
$ \Rightarrow \alpha = {\mathbf{5}} \times {\mathbf{1}}{{\mathbf{0}}^{ - 4}}$
So, Now
${\text{ }}{\mathbf{C\alpha }} = \left[ {{{\mathbf{H}}_3}{{\text{O}}^ + }} \right] = {\mathbf{0}}.{\mathbf{02}} \times {\mathbf{5}} \times {\mathbf{1}}{{\mathbf{0}}^{ - 4}}$
$ \Rightarrow C\alpha = {10^{ - 5}}$
Hence option A is correct.
The concentration dependency of the conductivity of weak electrolytes including $C{H_3}COOH$ and \[N{H_4}OH\] is well described by the Ostwald law of dilution. The incomplete dissociation of weak electrolytes into ions is the primary cause of molar conductivity variation. However, Lewis and Randall realised that the rule fails miserably for solid electrolytes, since the assumed equilibrium constant is far from constant. Since the dissociation of strong electrolytes into ions is almost complete below a concentration threshold value, this is the case.

Note:
The equation isn't exact except for weak electrolytes. The true equilibrium constant, according to chemical thermodynamics, is a ratio of thermodynamic operations, and each concentration must be compounded by an operation coefficient. Because of the heavy forces between ionic charges, this correction is critical for ionic solutions. At low concentrations, the Debye–Hückel principle provides an approximation of their values.