The pH and conductance of a weak acid (HX) was found to be 5 and 4 × 10⁻⁵ S, respectively. The value of the limiting molar conductivity is _____ Sm² mol⁻¹.
Q. The pH and conductance of a weak acid (HX) was found to be 5 and 4 × 10⁻⁵ S, respectively. The conductance was measured under standard condition using a cell where the electrode plates having a surface area of 1 cm² were at a distance of 15 cm apart. The value of the limiting molar conductivity is _____ Sm² mol⁻¹. (nearest integer)

(Given : degree of dissociation of the weak acid (α) ≪ 1 )
Correct Answer: 6

Step-by-Step Explanation

Step 1: Find [H⁺] concentration from pH.
Given \( \text{pH} = 5 \)
Using the formula \( [\text{H}^+] = 10^{-\text{pH}} \):
\( [\text{H}^+] = 10^{-5} \text{ mol/L} \)
Step 2: Relate concentration (C) and degree of dissociation (\( \alpha \)).
For a weak acid HX: \( \text{HX} \rightleftharpoons \text{H}^+ + \text{X}^- \)
\( [\text{H}^+] = C\alpha \)
From Step 1, \( C\alpha = 10^{-5} \) --- (Equation 1)
Step 3: Calculate Specific Conductance (\( \kappa \)).
Given Conductance \( G = 4 \times 10^{-5} \text{ S} \)
Area \( A = 1 \text{ cm}^2 = 1 \times 10^{-4} \text{ m}^2 \)
Distance \( l = 15 \text{ cm} = 0.15 \text{ m} \)
Formula: \( \kappa = G \times \frac{l}{A} \)
\( \kappa = (4 \times 10^{-5} \text{ S}) \times \frac{0.15 \text{ m}}{1 \times 10^{-4} \text{ m}^2} \)
\( \kappa = 4 \times 10^{-5} \times 1500 = 0.06 \text{ S/m} \)
Step 4: Calculate Molar Conductivity (\( \Lambda_m \)).
\( \Lambda_m = \frac{\kappa}{1000 \times C} \) (where C is in mol/L)
\( \Lambda_m = \frac{0.06}{1000 \times C} \) --- (Equation 2)
Step 5: Use the relation for \( \alpha \) and solve for \( \Lambda_m^\circ \).
We know \( \alpha = \frac{\Lambda_m}{\Lambda_m^\circ} \)
Substituting \( \Lambda_m \) from Equation 2:
\( \alpha = \frac{0.06}{1000 \times C \times \Lambda_m^\circ} \)
Rearranging: \( C\alpha = \frac{0.06}{1000 \times \Lambda_m^\circ} \)
From Step 2, \( C\alpha = 10^{-5} \):
\( 10^{-5} = \frac{0.06}{1000 \times \Lambda_m^\circ} \)
\( \Lambda_m^\circ = \frac{0.06}{1000 \times 10^{-5}} = \frac{0.06}{10^{-2}} \)
\( \Lambda_m^\circ = 0.06 \times 100 = 6 \text{ Sm}^2\text{mol}^{-1} \)

Final Answer: 6

Related Theory

Electrochemistry deals with the relationship between electrical energy and chemical change. One of the most important concepts for JEE Main is the Conductance of Electrolytic Solutions. In this problem, we bridge the gap between Ionic Equilibrium (pH) and Electrochemistry (Conductivity).

1. Conductance (G) vs. Specific Conductance (\( \kappa \))
Conductance is the ease with which current flows through a conductor, defined as the reciprocal of resistance (\( G = 1/R \)). Its SI unit is Siemens (S). However, conductance depends on the dimensions of the cell. To standardize this, we use Specific Conductance (\( \kappa \), Kappa), which is the conductance of a solution kept between two electrodes of unit area separated by a unit distance.

2. The Cell Constant (\( G^* \))
The ratio \( l/A \) is known as the cell constant. In the SI system, \( l \) is in meters and \( A \) is in \( m^2 \), so the unit is \( m^{-1} \). In the CGS system, it is \( cm^{-1} \). Converting units correctly is the most common place where students lose marks in JEE.

3. Molar Conductivity (\( \Lambda_m \))
It is the conducting power of all the ions produced by dissolving one mole of an electrolyte in solution. For a solution with concentration \( C \) (mol/L) and specific conductance \( \kappa \) (S/m), the formula is: \[ \Lambda_m = \frac{\kappa}{1000 \times C} \] This factor of 1000 is used to convert liters (dm³) to cubic meters (m³).

4. Kohlrausch's Law and Weak Electrolytes
For weak electrolytes like acetic acid or the generic HX used in this problem, molar conductivity increases sharply with dilution. At infinite dilution (concentration approaching zero), molar conductivity reaches a maximum value called Limiting Molar Conductivity (\( \Lambda_m^\circ \)). According to Ostwald's Dilution Law, the degree of dissociation \( \alpha \) is: \[ \alpha = \frac{\Lambda_m}{\Lambda_m^\circ} \] This relationship is only valid for weak electrolytes where \( \alpha \) is significantly less than 1.

5. Integration with pH
In competitive exams, examiners rarely give concentration (C) directly. They use pH to hide the data. Since \( [\text{H}^+] = C\alpha \), and pH gives you \( [\text{H}^+] \), you can effectively find the product of concentration and degree of dissociation. This "hidden variable" approach is a hallmark of JEE Advanced and Main level questions.

6. Common Mistakes
- Unit Conversion: Mixing cm and m. Always convert everything to SI (\( m, m^2, m^3 \)) or CGS (\( cm, cm^2, cm^3 \)) before starting calculations.
- Formula for \( \Lambda_m \): Using \( 1000 \times \kappa / C \) when \( \kappa \) is in S/cm vs using \( \kappa / (1000 \times C) \) when \( \kappa \) is in S/m.
- Approximation: Ignoring the given condition \( \alpha \ll 1 \). If \( \alpha \) was large, the relationship \( [\text{H}^+] = C\alpha \) would still hold, but the equilibrium constant calculation would change.

Frequently Asked Questions (FAQs)

1. What is the unit of Specific Conductance?
In SI units, it is S/m (Siemens per meter). In CGS, it is S/cm.
2. Why does Molar Conductivity increase with dilution?
For weak electrolytes, dilution increases the degree of dissociation (\( \alpha \)), producing more ions. For strong electrolytes, it decreases inter-ionic attractions.
3. What is a Cell Constant?
It is the ratio of the distance between electrodes (\( l \)) to their surface area (\( A \)). It is unique to each conductivity cell.
4. How is pH related to [H⁺]?
\( \text{pH} = -\log_{10}[\text{H}^+] \). Therefore, \( [\text{H}^+] = 10^{-\text{pH}} \).
5. Can we use Kohlrausch's law for weak electrolytes?
Yes, Kohlrausch's law of independent migration of ions is used specifically to calculate \( \Lambda_m^\circ \) for weak electrolytes by summing the limiting conductivities of individual ions.

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Related Covered Topics

jee mains jee advanced electrochemistry ionic equilibrium molar conductivity kohlrausch law cell constant specific conductance ostwald dilution law physical chemistry numerical engineering entrance exam neet chemistry weak acid dissociation standard conditions si unit conversions

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