Problem 195 - A Complete Resource Book in Chemistry for JEE Main

Question

The equivalent conductances of two strong electrolytes at infinite dilution in \(\mathrm{H}_{2} \mathrm{O}\) (where ions move freely through a solution) at \(25^{\circ} \mathrm{C}\) are given below: [2007] \(\Lambda^{\circ}\left(\mathrm{CH}_{3} \mathrm{COONa}\right)=91.0 \mathrm{~S} \mathrm{~cm}^{2} /\) equiv. \(\Lambda^{\circ}(\mathrm{HCl})=426.2 \mathrm{~S} \mathrm{~cm}^{2} / \mathrm{equiv}\) What additional information/quantity one needs to calculate \(\Lambda^{\circ}\) of an aqueous solution of acetic acid? (a) \(\Lambda^{\circ}\) of \(\mathrm{CH}_{3} \mathrm{COOK}\) (b) The limiting equivalent conductance of \(\mathrm{H}^{+}\left(\lambda^{\circ}\right)\) (c) \(\Lambda^{\circ}\) of chloroacetic acid \(\left(\mathrm{ClCH}_{2} \mathrm{COOH}\right)\) (d) \(\Lambda^{\circ}\) of \(\mathrm{NaCl}\)

Step-by-Step Solution

Answer

The required additional information is \( \Lambda^{\circ}(\text{NaCl}) \).

Step 1: Understanding the Problem

We need to find what additional information is required to calculate the limiting equivalent conductance (\( \Lambda^{\circ} \)) of acetic acid (\( \mathrm{CH}_3\mathrm{COOH} \)) in water at 25°C, given the \( \Lambda^{\circ} \) values of \( \mathrm{CH}_3\mathrm{COONa} \) and HCl.

Step 2: Using Kohlrausch's Law

Kohlrausch's Law of Independent Migration of Ions states that the limiting molar conductivity of an electrolyte can be expressed as the sum of contributions from its individual ions. Mathematically, \( \Lambda^{\circ}_{ ext{CH}_3 ext{COOH}} = \lambda^{\circ}( ext{CH}_3 ext{COO}^-) + \lambda^{\circ}( ext{H}^+) \).

Step 3: Relating Conductance Values

Given: \( \Lambda^{\circ}( ext{CH}_3 ext{COONa}) = \lambda^{\circ}( ext{CH}_3 ext{COO}^-) + \lambda^{\circ}( ext{Na}^+) \) and \( \Lambda^{\circ}( ext{HCl}) = \lambda^{\circ}( ext{H}^+) + \lambda^{\circ}( ext{Cl}^-) \). To find \( \Lambda^{\circ}( ext{CH}_3 ext{COOH}) \), we need \( \lambda^{\circ}( ext{Na}^+) + \lambda^{\circ}( ext{Cl}^-) \) because these will allow isolating \( \lambda^{\circ}( ext{H}^+) \) from \( \Lambda^{\circ}( ext{HCl}) \).

Step 4: Identifying the Missing Component

To isolate \( \lambda^{\circ}( ext{H}^+) \), we can use \( \Lambda^{\circ}( ext{NaCl}) = \lambda^{\circ}( ext{Na}^+) + \lambda^{\circ}( ext{Cl}^-) \). This allows us to sum up and find \( \lambda^{\circ}( ext{H}^+) \), which is then used in the equation for \( \Lambda^{\circ}( ext{CH}_3 ext{COOH}) \).

Step 5: Conclusion

The missing component that is necessary to complete the calculation of \( \Lambda^{\circ}(\text{CH}_3\text{COOH}) \) is \( \Lambda^{\circ}(\text{NaCl}) \) because it provides the values for \( \lambda^{\circ}(\text{Na}^+) \) and \( \lambda^{\circ}(\text{Cl}^-) \). Therefore, the correct answer is option (d).

Key Concepts

Equivalent Conductance

Equivalent conductance is an important concept in electrochemistry that helps us understand the ability of ions to conduct electricity in a solution. The idea revolves around how effectively an equivalent amount of electrolyte can conduct electrical charge. It can be defined as the conductance of an electrolyte solution containing one gram equivalent of the electrolyte dissolved in a given volume. Mathematically, it is often represented as:\[\Lambda = \frac{k \times 1000}{C}\]where \(\Lambda\) is the equivalent conductance, \(k\) is the specific conductance, and \(C\) is the concentration of the solution.The conductance of a solution increases with dilution because the ions are more freely dispersed to carry charge. That's why at infinite dilution, the equivalent conductance reaches its maximum value, known as the limiting molar conductivity, which allows ions to move independently.

Strong Electrolytes

Strong electrolytes are substances that completely dissociate into ions in a solution. This property enables them to conduct electricity very efficiently. Common examples include salts such as sodium chloride (NaCl) and acids like hydrochloric acid (HCl). When dissolved, the ions are free to move and thus can carry an electric current. Strong electrolytes are characterized by high values of limiting molar conductivity since their ions are fully separated at infinite dilution. In calculations involving strong electrolytes, the contribution of each ion to the overall conductivity is crucial. For example, the conductivity of an acetic acid solution may require contributions from both acetate ions and hydrogen ions, with the help of other substances like sodium acetate and hydrochloric acid for identifying each ion's contribution.

Limiting Molar Conductivity

Limiting molar conductivity is an essential concept linked to the behavior of ions when a solution is infinitely diluted. It represents the maximum conductivity achievable when each ion acts independently, without interactions. Kohlrausch's Law of Independent Migration of Ions explains that the limiting molar conductivity of an electrolyte is the sum of the ionic conductivities of the cations and anions. For instance, if you need to find the limiting molar conductivity of acetic acid, understanding the independent movement of acetate and hydrogen ions is essential. By knowing the limiting molar conductivities of sodium acetate and HCl, and subtracting the contributions from sodium and chloride ions via NaCl, you can isolate and calculate the required values for acetic acid.