Kohlrausch's Law — Revision Notes

⚡ 30-Second Revision

Kohlrausch's Law: — At infinite dilution, $\Lambda_m^\circ = x\lambda_+^\circ + y\lambda_-^\circ$
For weak electrolytes: — $\Lambda_m^\circ(\text{HA}) = \Lambda_m^\circ(\text{NaA}) + \Lambda_m^\circ(\text{HCl}) - \Lambda_m^\circ(\text{NaCl})$
Degree of Dissociation: — $\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$
Dissociation Constant: — $K_a = \frac{C\alpha^2}{1-\alpha}$ (for weak acid HA)
Solubility (S) of sparingly soluble salt: — $S = \frac{\kappa \times 1000}{\Lambda_m^\circ}$ (where $\kappa$ is specific conductivity in S cm$^{-1}$, S in mol L$^{-1}$)
Key condition: — Applicable only at infinite dilution (zero concentration).

2-Minute Revision

Kohlrausch's Law is crucial for understanding electrolyte behavior at infinite dilution. It states that at this extreme dilution, ions move independently, and the total limiting molar conductivity ($\Lambda_m^\circ$) of an electrolyte is simply the sum of the limiting ionic conductivities ($\lambda^\circ$) of its constituent ions, weighted by their stoichiometric coefficients.

This law is particularly vital for weak electrolytes, whose $\Lambda_m^\circ$ cannot be determined by direct extrapolation. Instead, we calculate it indirectly by combining the $\Lambda_m^\circ$ values of strong electrolytes.

For instance, $\Lambda_m^\circ(\text{CH}_3\text{COOH})$ can be found from $\Lambda_m^\circ(\text{CH}_3\text{COONa})$, $\Lambda_m^\circ(\text{HCl})$, and $\Lambda_m^\circ(\text{NaCl})$. Once $\Lambda_m^\circ$ is known, we can determine the degree of dissociation ($\alpha = \Lambda_m / \Lambda_m^\circ$) and the dissociation constant ($K_a = C\alpha^2 / (1-\alpha)$) for weak electrolytes.

Another important application is calculating the solubility of sparingly soluble salts, where the saturated solution is considered infinitely dilute, using the formula $S = (\kappa \times 1000) / \Lambda_m^\circ$.

Remember, the law's validity is strictly limited to infinite dilution.

5-Minute Revision

Kohlrausch's Law, or the Law of Independent Migration of Ions, is a cornerstone of electrochemistry, particularly relevant for NEET. It posits that at infinite dilution, where interionic attractions are negligible, each ion contributes a fixed, independent amount to the total molar conductivity of the electrolyte.

This means the limiting molar conductivity ($\Lambda_m^\circ$) of an electrolyte $A_x B_y$ is given by $\Lambda_m^\circ = x\lambda_+^\circ + y\lambda_-^\circ$, where $x$ and $y$ are stoichiometric coefficients and $\lambda^\circ$ are the limiting ionic conductivities.

\n\nWhy is it important? For strong electrolytes, $\Lambda_m^\circ$ can be found by extrapolating a $\Lambda_m$ vs. $\sqrt{C}$ plot. However, for weak electrolytes, this plot is non-linear and doesn't extrapolate, making direct determination impossible.

Kohlrausch's Law provides an indirect method. For example, to find $\Lambda_m^\circ$ for a weak acid like CH$_3$COOH, we can use strong electrolytes:\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\nKey Applications:\n1.

**Degree of Dissociation ($\alpha$):** For a weak electrolyte, $\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$. Here, $\Lambda_m$ is the molar conductivity at a given concentration, and $\Lambda_m^\circ$ is the limiting molar conductivity (often calculated using Kohlrausch's Law).

\n *Example:* If $\Lambda_m = 10$ S cm$^2$ mol$^{-1}$ and $\Lambda_m^\circ = 400$ S cm$^2$ mol$^{-1}$, then $\alpha = 10/400 = 0.025$.\n2. **Dissociation Constant ($K_a$ or $K_b$):** Once $\alpha$ is known, the dissociation constant can be found using Ostwald's Dilution Law: $K_a = \frac{C\alpha^2}{1-\alpha}$.

\n *Example:* For $C = 0.02$ M and $\alpha = 0.025$, $K_a = \frac{0.02 \times (0.025)^2}{1 - 0.025} \approx 1.28 \times 10^{-5}$.\n3. Solubility (S) of Sparingly Soluble Salts: Saturated solutions of these salts are effectively at infinite dilution.

We can measure their specific conductivity ($\kappa$) and calculate their $\Lambda_m^\circ$ from ionic conductivities. Then, $S = \frac{\kappa \times 1000}{\Lambda_m^\circ}$.\n *Example:* If $\kappa = 1.

826 \times 10^{-6}$S cm$^{-1}$and$\Lambda_m^\circ(\text{AgCl}) = 138.2$S cm$^2$mol$^{-1}$, then$S = \frac{1.826 \times 10^{-6} \times 1000}{138.2} \approx 1.321 \times 10^{-5}$mol L$^{-1}$.\n\nCrucial Point for NEET: Always remember that Kohlrausch's Law is strictly valid at infinite dilution.

Be careful with units and algebraic manipulations in numerical problems.

Prelims Revision Notes

Kohlrausch's Law (Law of Independent Migration of Ions) is a key concept for NEET. It states that at infinite dilution, the limiting molar conductivity ($\Lambda_m^\circ$) of an electrolyte is the sum of the limiting molar conductivities of its constituent ions, each multiplied by its stoichiometric coefficient.

For an electrolyte $A_x B_y$, the formula is $\Lambda_m^\circ = x\lambda_+^\circ + y\lambda_-^\circ$. This law is crucial because it allows for the determination of $\Lambda_m^\circ$ for weak electrolytes, which cannot be found by direct extrapolation from $\Lambda_m$ vs.

$\sqrt{C}$ plots. To calculate $\Lambda_m^\circ$ for a weak electrolyte (e.g., CH$_3$COOH), use an algebraic combination of strong electrolytes: $\Lambda_m^\circ(\text{CH}_3\text{COOH}) = \Lambda_m^\circ(\text{CH}_3\text{COONa}) + \Lambda_m^\circ(\text{HCl}) - \Lambda_m^\circ(\text{NaCl})$.

\n\nApplications to remember:\n1. **Degree of Dissociation ($\alpha$):** For weak electrolytes, $\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$. Remember $\Lambda_m$ is at a given concentration, and $\Lambda_m^\circ$ is at infinite dilution.

\n2. **Dissociation Constant ($K_a$ or $K_b$):** Use Ostwald's Dilution Law: $K_a = \frac{C\alpha^2}{1-\alpha}$. If $\alpha$ is very small, $1-\alpha \approx 1$, so $K_a \approx C\alpha^2$.\n3. Solubility (S) of Sparingly Soluble Salts: For a saturated solution of a sparingly soluble salt, its concentration is its solubility (S), and its molar conductivity is approximately $\Lambda_m^\circ$.

The relationship is $S = \frac{\kappa \times 1000}{\Lambda_m^\circ}$, where $\kappa$ is specific conductivity in S cm$^{-1}$ and S is in mol L$^{-1}$. Ensure consistent units, especially the factor of 1000.

\n\nKey points for NEET: The law is strictly for infinite dilution. Be careful with stoichiometric coefficients. Practice numerical problems extensively, especially those involving multiple steps or algebraic manipulation of $\Lambda_m^\circ$ values.

Vyyuha Quick Recall

Kohlrausch's LAW: Limiting Additive Weak-electrolytes. (At Limiting dilution, ionic contributions are Additive, helping Weak electrolytes.)