Kohlrausch's Law — Explained

Detailed Explanation

Electrolytic conductance is a fundamental property of solutions containing ions, enabling them to conduct electricity. The ability of an electrolyte solution to conduct electricity is quantified by its specific conductivity ($\kappa$) and molar conductivity ($\Lambda_m$).

Specific conductivity is the conductance of a unit volume (1 cm$^3$) of the solution, while molar conductivity is the conductivity of a solution containing one mole of electrolyte, placed between two electrodes 1 cm apart with a large enough area to contain the entire volume.

As concentration decreases, specific conductivity generally decreases because the number of ions per unit volume decreases. However, molar conductivity generally increases with dilution because the interionic attractions decrease, allowing ions to move more freely, and for weak electrolytes, the degree of dissociation increases.

\n\nFor strong electrolytes, which dissociate completely at all concentrations, the molar conductivity ($\Lambda_m$) increases with dilution, approaching a maximum value at infinite dilution, known as the limiting molar conductivity ($\Lambda_m^\circ$).

This variation is described by the Debye-Hückel-Onsager equation: $\Lambda_m = \Lambda_m^\circ - A\sqrt{C}$, where A is a constant that depends on the nature of the solvent and temperature, and C is the concentration.

This linear relationship allows us to determine $\Lambda_m^\circ$ for strong electrolytes by extrapolating the $\Lambda_m$ vs. $\sqrt{C}$ plot to zero concentration.\n\nHowever, for weak electrolytes, which only partially dissociate, the plot of $\Lambda_m$ vs.

$\sqrt{C}$ is a steep curve that does not extrapolate to a definite value at zero concentration. This is because, at very low concentrations, the degree of dissociation of a weak electrolyte increases significantly, leading to a sharp rise in the number of ions, making direct extrapolation unreliable.

This is where Kohlrausch's Law becomes indispensable.\n\nKohlrausch's Law of Independent Migration of Ions\nFriedrich Kohlrausch, through extensive experimental work, observed a remarkable regularity in the molar conductivities of various electrolytes at infinite dilution.

He proposed his law in 1876, stating: 'At infinite dilution, when dissociation is complete, each ion makes a definite contribution towards the molar conductivity of the electrolyte, irrespective of the nature of the other ion with which it is associated.

'\n\nThis law implies that at infinite dilution, the ions are so far apart that the attractive forces between them are virtually non-existent. Each ion moves independently under the influence of the electric field, and its contribution to the total conductivity is solely dependent on its own nature (charge, size, mobility) and not on its counter-ion.

Therefore, the limiting molar conductivity of an electrolyte is simply the sum of the limiting molar conductivities of its constituent ions, each multiplied by the number of times it appears in the electrolyte's formula unit.

\n\nMathematically, for an electrolyte $A_x B_y$ that dissociates into $x$ cations $A^{y+}$ and $y$ anions $B^{x-}$, its limiting molar conductivity is given by:\n

$$\Lambda_m^\circ = x\lambda_{A^{y+}}^\circ + y\lambda_{B^{x-}}^\circ$$

\nWhere:\n* $\Lambda_m^\circ$ is the limiting molar conductivity of the electrolyte.

\n* $x$ and $y$ are the stoichiometric coefficients of the cation and anion, respectively, in the electrolyte's formula.\n* $\lambda_{A^{y+}}^\circ$ is the limiting molar conductivity of the cation $A^{y+}$.

\n* $\lambda_{B^{x-}}^\circ$ is the limiting molar conductivity of the anion $B^{x-}$.\n\nApplications of Kohlrausch's Law\nKohlrausch's Law has several crucial applications, particularly in the study of weak electrolytes and sparingly soluble salts:\n\n1.

Calculation of Limiting Molar Conductivity of Weak Electrolytes: This is arguably the most significant application. Since $\Lambda_m^\circ$ for weak electrolytes cannot be determined by extrapolation, Kohlrausch's Law allows us to calculate it indirectly.

We can combine the limiting molar conductivities of strong electrolytes in such a way that the desired weak electrolyte's $\Lambda_m^\circ$ is obtained. For example, to find $\Lambda_m^\circ$ for acetic acid (CH$_3$COOH), a weak electrolyte, we can use the $\Lambda_m^\circ$ values of strong electrolytes like HCl, CH$_3$COONa, and NaCl:\n

$$\Lambda_m^\circ(\text{CH}_3\text{COOH}) = \Lambda_m^\circ(\text{CH}_3\text{COONa}) + \Lambda_m^\circ(\text{HCl}) - \Lambda_m^\circ(\text{NaCl})$$

\n This works because:\n $\Lambda_m^\circ(\text{CH}_3\text{COONa}) = \lambda_{\text{CH}_3\text{COO}^-}^\circ + \lambda_{\text{Na}^+}^\circ$\n $\Lambda_m^\circ(\text{HCl}) = \lambda_{\text{H}^+}^\circ + \lambda_{\text{Cl}^-}^\circ$\n $\Lambda_m^\circ(\text{NaCl}) = \lambda_{\text{Na}^+}^\circ + \lambda_{\text{Cl}^-}^\circ$\n So, $(\lambda_{\text{CH}_3\text{COO}^-}^\circ + \lambda_{\text{Na}^+}^\circ) + (\lambda_{\text{H}^+}^\circ + \lambda_{\text{Cl}^-}^\circ) - (\lambda_{\text{Na}^+}^\circ + \lambda_{\text{Cl}^-}^\circ) = \lambda_{\text{CH}_3\text{COO}^-}^\circ + \lambda_{\text{H}^+}^\circ = \Lambda_m^\circ(\text{CH}_3\text{COOH})$.

\n\n2. **Determination of Degree of Dissociation ($\alpha$) of Weak Electrolytes:** Once $\Lambda_m^\circ$ for a weak electrolyte is known (either directly for strong electrolytes or calculated using Kohlrausch's Law for weak ones), we can determine its degree of dissociation at any given concentration (C) using its molar conductivity at that concentration ($\Lambda_m$).

The degree of dissociation is given by:\n

$$\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$$

\n Where $\Lambda_m$ is the molar conductivity at concentration C, and $\Lambda_m^\circ$ is the limiting molar conductivity.

\n\n3. **Calculation of Dissociation Constant ($K_a$ or $K_b$) of Weak Electrolytes:** For a weak electrolyte, once $\alpha$ is known at a particular concentration C, its dissociation constant can be calculated using Ostwald's Dilution Law.

For a weak acid HA:\n

$$HA \rightleftharpoons H^+ + A^-$$

\n

$$K_a = \frac{[H^+][A^-]}{[HA]} = \frac{C\alpha \cdot C\alpha}{C(1-\alpha)} = \frac{C\alpha^2}{1-\alpha}$$

\n If $\alpha$ is very small (for very weak electrolytes), $1-\alpha \approx 1$, so $K_a \approx C\alpha^2$.

\n\n4. Determination of Solubility of Sparingly Soluble Salts: Sparingly soluble salts (like AgCl, BaSO$_4$) dissolve to a very small extent, forming saturated solutions that are effectively infinitely dilute.

In such solutions, the concentration of the dissolved salt is equal to its solubility (S). The molar conductivity of such a saturated solution can be measured, and since it's practically at infinite dilution, $\Lambda_m \approx \Lambda_m^\circ$.

The specific conductivity ($\kappa$) of the saturated solution can be related to its molar conductivity and solubility (concentration) by the formula:\n

$$\Lambda_m = \frac{\kappa \times 1000}{C}$$

\n Since $C = S$ (solubility) and $\Lambda_m \approx \Lambda_m^\circ$ for sparingly soluble salts:\n

$$\Lambda_m^\circ = \frac{\kappa \times 1000}{S}$$

\n Therefore, solubility $S = \frac{\kappa \times 1000}{\Lambda_m^\circ}$.

The $\Lambda_m^\circ$ for the sparingly soluble salt can be calculated using Kohlrausch's Law from the limiting molar conductivities of its constituent ions (e.g., $\Lambda_m^\circ(\text{AgCl}) = \lambda_{\text{Ag}^+}^\circ + \lambda_{\text{Cl}^-}^\circ$).

\n\nCommon Misconceptions and NEET-Specific Angle\n* Applicability: Kohlrausch's Law is strictly applicable at *infinite dilution*. Students often mistakenly try to apply it at finite concentrations, where interionic interactions are significant, and ions do not migrate independently.

\n* **Distinction between $\Lambda_m$ and $\Lambda_m^\circ$**: It's crucial to understand that $\Lambda_m$ is molar conductivity at a given concentration, while $\Lambda_m^\circ$ is the limiting molar conductivity (at infinite dilution).

Kohlrausch's law deals with $\Lambda_m^\circ$.\n* Stoichiometric Coefficients: Remember to multiply the limiting ionic conductivities by their respective stoichiometric coefficients ($x$ and $y$) as per the electrolyte's formula.

\n* Units: Pay close attention to units. Molar conductivity is typically in S cm$^2$ mol$^{-1}$, and specific conductivity in S cm$^{-1}$. Ensure consistency in calculations, especially when using the factor of 1000 (if volume is in cm$^3$ and concentration in mol L$^{-1}$).

\n* Numerical Problems: NEET frequently tests the application of Kohlrausch's Law in numerical problems involving:\n * Calculating $\Lambda_m^\circ$ for a weak electrolyte using values of strong electrolytes.

\n * Determining the degree of dissociation ($\alpha$) and dissociation constant ($K_a$) of weak electrolytes.\n * Calculating the solubility of sparingly soluble salts.\n Mastering these types of calculations is key for NEET.