Finding the molar conductivity of a weak electrolyte involves two key aspects: measuring its conductivity at a specific concentration and, crucially, utilizing Kohlrausch's Law to determine its limiting molar conductivity at infinite dilution.
Molar conductivity ($\Lambda_m$) indicates how efficiently an electrolyte conducts electricity per mole of substance. For weak electrolytes, which only partially dissociate in solution, this value depends significantly on concentration.
Understanding Molar Conductivity
Molar conductivity ($\Lambda_m$) is defined as the conductivity of a solution divided by its molar concentration. It quantifies the conducting power of all the ions produced by one mole of an electrolyte.
The formula for molar conductivity is:
$\Lambda_m = \frac{\kappa}{C}$
Where:
- $\Lambda_m$ is the molar conductivity (S cm$^2$ mol$^{-1}$ or S m$^2$ mol$^{-1}$)
- $\kappa$ (kappa) is the conductivity or specific conductance of the solution (S cm$^{-1}$ or S m$^{-1}$)
- $C$ is the molar concentration of the electrolyte (mol cm$^{-3}$ or mol m$^{-3}$)
Key Differences Between Weak and Strong Electrolytes
| Feature | Strong Electrolyte | Weak Electrolyte |
|---|---|---|
| Dissociation | Complete dissociation into ions in solution | Partial dissociation into ions in solution |
| Conductivity | High conductivity | Low conductivity (at comparable concentrations) |
| Concentration Effect | Molar conductivity decreases slightly with dilution | Molar conductivity increases significantly with dilution |
| Limiting Molar Conductivity | Determined by extrapolation to zero concentration | Cannot be determined by direct extrapolation |
Determining Molar Conductivity at a Given Concentration
To find the molar conductivity ($\Lambda_m$) of a weak electrolyte at a specific concentration, you generally follow an experimental approach:
- Prepare the Solution: Accurately prepare a solution of the weak electrolyte at a known molar concentration ($C$).
- Measure Resistance: Use a conductivity cell and a Wheatstone bridge (or a modern conductivity meter) to measure the resistance ($R$) of the solution.
- Calculate Cell Constant: First, determine the cell constant ($G^$) of the conductivity cell. This is typically done by measuring the resistance of a standard solution with a known conductivity (e.g., KCl solution).
$G^ = \kappa{\text{standard}} \times R{\text{standard}}$ - Calculate Specific Conductance ($\kappa$): Using the measured resistance of your weak electrolyte solution and the cell constant, calculate the specific conductance.
$\kappa = \frac{G^*}{R}$ - Calculate Molar Conductivity ($\Lambda_m$): Finally, use the formula to find the molar conductivity.
$\Lambda_m = \frac{\kappa \times 1000}{C}$ (if $\kappa$ is in S cm$^{-1}$ and $C$ in mol L$^{-1}$ to get $\Lambda_m$ in S cm$^2$ mol$^{-1}$)
Example: If a 0.01 M solution of acetic acid (a weak electrolyte) has a specific conductance ($\kappa$) of 1.65 x 10$^{-4}$ S cm$^{-1}$, its molar conductivity would be:
$\Lambda_m = \frac{(1.65 \times 10^{-4} \text{ S cm}^{-1}) \times 1000}{0.01 \text{ mol L}^{-1}} = 16.5 \text{ S cm}^2 \text{ mol}^{-1}$
Calculating Limiting Molar Conductivity ($\Lambda_m^0$) for Weak Electrolytes
The most challenging aspect for weak electrolytes is determining their limiting molar conductivity ($\Lambda_m^0$ or $\Lambda_m^\infty$), which is the molar conductivity at infinite dilution (when the concentration approaches zero). Unlike strong electrolytes, whose $\Lambda_m$ vs. $\sqrt{C}$ graphs can be linearly extrapolated to find $\Lambda_m^0$, weak electrolytes show a sharp, non-linear increase in $\Lambda_m$ at low concentrations, making direct extrapolation unreliable.
For weak electrolytes, the limiting molar conductivity must be determined indirectly using Kohlrausch's Law of Independent Migration of Ions. This law states that the limiting molar conductivity of an electrolyte can be represented as the sum of the individual contributions of the anion and cation of the electrolyte.
Mathematically, for any electrolyte, XY:
$\Lambda_m^0 = \lambda_X^0 + \lambda_Y^0$
Where $\lambda_X^0$ and $\lambda_Y^0$ are the limiting ionic conductivities of the cation and anion, respectively.
Applying Kohlrausch's Law to Weak Electrolytes
Since we cannot directly measure $\Lambda_m^0$ for a weak electrolyte, we use the $\Lambda_m^0$ values of strong electrolytes that contain the relevant ions. The principle is to combine the limiting molar conductivities of strong electrolytes in a way that effectively isolates the contributions of the ions forming the weak electrolyte.
Steps to find $\Lambda_m^0$ for a weak electrolyte (e.g., acetic acid, CH₃COOH):
-
Identify Strong Electrolytes: Choose strong electrolytes that, when combined, yield the ions of the weak electrolyte.
- A strong salt containing the anion of the weak electrolyte (e.g., Sodium Acetate, CH₃COONa, for CH₃COOH).
- A strong acid that shares the cation of the weak electrolyte (e.g., Hydrochloric Acid, HCl, for CH₃COOH).
- A strong salt that cancels out the "extra" ions introduced by the first two strong electrolytes (e.g., Sodium Chloride, NaCl, for CH₃COOH).
-
Formulate the Equation: Combine their limiting molar conductivities using Kohlrausch's Law.
For acetic acid (CH₃COOH):
$\Lambda_m^0 (\text{CH}_3\text{COOH}) = \Lambda_m^0 (\text{CH}_3\text{COONa}) + \Lambda_m^0 (\text{HCl}) - \Lambda_m^0 (\text{NaCl})$Let's break this down by ionic contributions:
- $\Lambda_m^0 (\text{CH}3\text{COONa}) = \lambda{\text{CH}3\text{COO}^-}^0 + \lambda{\text{Na}^+}^0$
- $\Lambdam^0 (\text{HCl}) = \lambda{\text{H}^+}^0 + \lambda_{\text{Cl}^-}^0$
- $\Lambdam^0 (\text{NaCl}) = \lambda{\text{Na}^+}^0 + \lambda_{\text{Cl}^-}^0$
Substituting these into the equation:
$\Lambda_m^0 (\text{CH}3\text{COOH}) = (\lambda{\text{CH}3\text{COO}^-}^0 + \lambda{\text{Na}^+}^0) + (\lambda{\text{H}^+}^0 + \lambda{\text{Cl}^-}^0) - (\lambda{\text{Na}^+}^0 + \lambda{\text{Cl}^-}^0)$
$\Lambda_m^0 (\text{CH}3\text{COOH}) = \lambda{\text{CH}3\text{COO}^-}^0 + \lambda{\text{H}^+}^0$This effectively sums the limiting ionic conductivities of the acetate ion and the hydrogen ion, which are the ions of acetic acid.
Practical Example
Let's calculate the limiting molar conductivity of acetic acid ($\Lambda_m^0 (\text{CH}_3\text{COOH})$) using the following known $\Lambda_m^0$ values at 298 K:
- $\Lambda_m^0 (\text{CH}_3\text{COONa}) = 91.0 \text{ S cm}^2 \text{ mol}^{-1}$
- $\Lambda_m^0 (\text{HCl}) = 426.2 \text{ S cm}^2 \text{ mol}^{-1}$
- $\Lambda_m^0 (\text{NaCl}) = 126.5 \text{ S cm}^2 \text{ mol}^{-1}$
Using the formula:
$\Lambda_m^0 (\text{CH}_3\text{COOH}) = \Lambda_m^0 (\text{CH}_3\text{COONa}) + \Lambda_m^0 (\text{HCl}) - \Lambda_m^0 (\text{NaCl})$
$\Lambda_m^0 (\text{CH}_3\text{COOH}) = 91.0 + 426.2 - 126.5$
$\Lambda_m^0 (\text{CH}_3\text{COOH}) = 390.7 \text{ S cm}^2 \text{ mol}^{-1}$
Significance of Limiting Molar Conductivity for Weak Electrolytes
The calculated $\Lambda_m^0$ for a weak electrolyte is crucial for:
- Determining the Degree of Dissociation ($\alpha$):
$\alpha = \frac{\Lambda_m}{\Lambda_m^0}$
Where $\Lambda_m$ is the molar conductivity at a given concentration $C$. This allows us to quantify how much of the weak electrolyte dissociates at a particular concentration. - Calculating the Dissociation Constant ($K_a$ or $K_b$): Once $\alpha$ is known, the dissociation constant for a weak acid or base can be calculated:
$K_a = \frac{C \alpha^2}{1-\alpha}$ (for a weak acid)
By combining experimental measurements at specific concentrations with the theoretical framework of Kohlrausch's Law, we can fully characterize the conductive behavior of weak electrolytes.
