Question

Saturated solution $Co_2[Fe(CN)_6]$ is taken in an electrolytic cell of cell constant 0.3 $cm^{-1}$. If conductivity of saturated solution of $Co_2[Fe(CN)_6]$ & water used to make solution is $2.2 \times 10^{-6} S cm^{-1}$ and $4 \times 10^{-7} S cm^{-1}$ respectively, & molar conductance of $Co^{+2}$ & $[Fe(CN)_6]^{-4}$ ions are 80 $S cm^2 mol^{-1}$ & 440 $S cm^2 mol^{-1}$ respectively then

• A. Solubility of $Co_2[Fe(CN)_6]$ is $3 \times 10^{-6}$ $mole/dm^3$
• B. Ksp of $Co_2[Fe(CN)_6]$ is $1.08 \times 10^{-16}$ $mole^3/dm^9$
• C. Ionic mobility of $Co^{+2}$ of this solution increases on increasing potential difference across the terminal of the cell.
• D. On increasing temperature conductivity of this solution increases.

Answer 1

Answer: (A), (B), (D)

Solubility of $Co_2[Fe(CN)_6]$ is $3 \times 10^{-6}$ $mole/dm^3$, Ksp of $Co_2[Fe(CN)_6]$ is $1.08 \times 10^{-16}$ $mole^3/dm^9$, On increasing temperature conductivity of this solution increases.

Answer 2

The conductivity of the saturated solution of $Co_2[Fe(CN)_6]$ is $\kappa_{sol} = 2.2 \times 10^{-6} \, S \, cm^{-1}$.

The conductivity of water is $\kappa_{water} = 4 \times 10^{-7} \, S \, cm^{-1}$.

The conductivity due to the dissolved salt $Co_2[Fe(CN)_6]$ is $\kappa_{salt} = \kappa_{sol} - \kappa_{water} = 2.2 \times 10^{-6} - 0.4 \times 10^{-6} = 1.8 \times 10^{-6} \, S \, cm^{-1}$.

The molar conductance of $Co^{+2}$ is $\lambda_{m}(Co^{+2}) = 80 \, S \, cm^2 \, mol^{-1}$.

The molar conductance of $[Fe(CN)_6]^{-4}$ is $\lambda_{m}([Fe(CN)_6]^{-4}) = 440 \, S \, cm^2 \, mol^{-1}$.

The dissociation of $Co_2[Fe(CN)_6]$ is $Co_2[Fe(CN)_6](s) \rightleftharpoons 2Co^{+2}(aq) + [Fe(CN)_6]^{-4}(aq)$.

The molar conductivity of $Co_2[Fe(CN)_6]$ at infinite dilution (which is assumed for a saturated solution of a sparingly soluble salt) is given by Kohlrausch's Law: $\Lambda_m(Co_2[Fe(CN)_6]) = 2 \times \lambda_m(Co^{+2}) + 1 \times \lambda_m([Fe(CN)_6]^{-4})$ $\Lambda_m = 2 \times 80 \, S \, cm^2 \, mol^{-1} + 440 \, S \, cm^2 \, mol^{-1} = 160 + 440 = 600 \, S \, cm^2 \, mol^{-1}$.

Let $S$ be the solubility of $Co_2[Fe(CN)_6]$ in $mol/dm^3$ (or $mol/L$). The concentration of the salt in the saturated solution is $S$.

The relationship between conductivity, molar conductivity, and concentration is $\kappa = \Lambda_m \times C$.

We need to be consistent with units. If $\kappa$ is in $S \, cm^{-1}$ and $\Lambda_m$ is in $S \, cm^2 \, mol^{-1}$, the concentration $C$ must be in $mol/cm^3$.

$1 \, dm^3 = 1000 \, cm^3$, so $S \, mol/dm^3 = S/1000 \, mol/cm^3$.

$\kappa_{salt} = \Lambda_m \times \frac{S}{1000}$ $1.8 \times 10^{-6} \, S \, cm^{-1} = 600 \, S \, cm^2 \, mol^{-1} \times \frac{S}{1000} \, mol/cm^3$ $1.8 \times 10^{-6} = 0.6 \times S$ $S = \frac{1.8 \times 10^{-6}}{0.6} = 3 \times 10^{-6} \, mol/dm^3$.

Statement 1: Solubility of $Co_2[Fe(CN)_6]$ is $3 \times 10^{-6}$ $mole/dm^3$. This is true based on our calculation.

Statement 2: Ksp of $Co_2[Fe(CN)_6]$ is $1.08 \times 10^{-16}$ $mole^3/dm^9$. For the equilibrium $Co_2[Fe(CN)_6](s) \rightleftharpoons 2Co^{+2}(aq) + [Fe(CN)_6]^{-4}(aq)$, if the solubility is $S$, the concentrations are $[Co^{+2}] = 2S$ and $[Fe(CN)_6]^{-4}] = S$. $Ksp = [Co^{+2}]^2 [[Fe(CN)_6]^{-4}] = (2S)^2 (S) = 4S^3$. Using $S = 3 \times 10^{-6} \, mol/dm^3$: $Ksp = 4 \times (3 \times 10^{-6})^3 = 4 \times (27 \times 10^{-18}) = 108 \times 10^{-18} = 1.08 \times 10^{-16} \, (mol/dm^3)^3$. Statement 2 is true.

Statement 3: Ionic mobility of $Co^{+2}$ of this solution increases on increasing potential difference across the terminal of the cell. Ionic mobility is defined as the drift velocity of an ion per unit electric field strength. It is a property of the ion, solvent, temperature, and pressure. Under normal conditions, ionic mobility is independent of the applied electric field or potential difference. Increasing the potential difference increases the electric field, which increases the drift velocity, but the ratio (mobility) remains constant. Statement 3 is false.

Statement 4: On increasing temperature conductivity of this solution increases. The conductivity of electrolytic solutions generally increases with increasing temperature. This is because the viscosity of the solvent decreases, and the kinetic energy and mobility of ions increase. For sparingly soluble salts, solubility also typically increases with temperature, leading to a higher concentration of ions. All these factors contribute to increased conductivity. Statement 4 is true.