Problem 32 - Chemistry in 30 days

Question

If equivalent conductances of \(\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) and \(\mathrm{CaCl}_{2}\) are \(x\) and \(y\) respectively, the values of their molar conductances are respectively (a) \(3 x\) and \(3 y\) (b) \(6 x\) and \(2 y\) (c) \(6 x\) and \(6 y\) (d) \(x\) and \(y\)

Step-by-Step Solution

Answer

The correct answer is (b) \( 6x \) and \( 2y \).

Step 1: Understand the Problem

The problem asks us to find the molar conductance given the equivalent conductance for two salts: \( \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \) and \( \mathrm{CaCl}_{2} \). We need to calculate molar conductance for multi-ion electrolytes.

Step 2: Recall the Relation between Molar and Equivalent Conductance

Molar conductance (\( \Lambda_m \)) is related to equivalent conductance (\( \lambda_{eq} \)) by the formula: \[ \Lambda_m = n \lambda_{eq} \] where \( n \) is the number of equivalents per mole for the electrolyte.

Step 3: Determine Equivalents for Sodium Aluminum Sulfate

For \( \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \), it dissociates as \[ \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \rightarrow 2\mathrm{Al}^{3+} + 3\mathrm{SO}_{4}^{2-} \]. Therefore, the number of equivalents per mole is \( 6 \), because it's based on the total charges (3 Al ions \( \cdot 3 \) charge each and 3 SO4 ions \cdot 2 charge each).

Step 4: Calculate Molar Conductance for Aluminum Sulfate

Since the number of equivalents per mole for \( \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \) is \( 6 \), the molar conductance \( \Lambda_m \) is \( 6x \) (where \( x \) is the equivalent conductance).

Step 5: Determine Equivalents for Calcium Chloride

For \( \mathrm{CaCl}_{2} \), it dissociates as \[ \mathrm{CaCl}_{2} \rightarrow \mathrm{Ca}^{2+} + 2\mathrm{Cl}^{-} \]. Therefore, the number of equivalents per mole is \( 2 \), based on the charge of calcium (2 Ca ions \cdot 1 charge each).

Step 6: Calculate Molar Conductance for Calcium Chloride

Since the number of equivalents per mole for \( \mathrm{CaCl}_{2} \) is \( 2 \), the molar conductance \( \Lambda_m \) is \( 2y \) (where \( y \) is the equivalent conductance).

Step 7: Decide the Correct Option

By calculations, we have the molar conductance for \( \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \) as \( 6x \) and for \( \mathrm{CaCl}_{2} \) as \( 2y \), thus the correct option is (b) \( 6x \) and \( 2y \).

Key Concepts

Equivalent Conductance

Equivalent conductance is a key concept in understanding the conductance of electrolytes, which are substances that can produce ions and conduct electricity when dissolved in a solution. It is represented by the symbol \( \lambda_{eq} \). Equivalent conductance is defined as the conductance of an electrolyte divided by the number of equivalents of the electrolyte in the solution.
For a given electrolyte, equivalent conductance provides insight into how well the ions created from dissociation contribute to the overall conductance. It's an important measure because it provides insight beyond just how many ions are present, focusing instead on the efficiency of ion transport.
The understanding of equivalent conductance helps in calculating the molar conductance of a substance, as it is directly linked through a multiplier related to the extent of electrolytic dissociation.

Electrolyte Dissociation

Electrolyte dissociation refers to the process by which an electrolyte splits into its constituent ions when dissolved in a solvent like water. This process is critical for the conductance of electricity in solutions, as only free-moving ions can carry charge.
When discussing dissociation in the context of equivalent and molar conductance, the focus is on the degree to which an electrolyte dissociates. In our exercise, we looked at the dissociation of \( \mathrm{Al}_2(\mathrm{SO}_4)_3 \) and \( \mathrm{CaCl}_2 \).
- For \( \mathrm{Al}_2(\mathrm{SO}_4)_3 \), the dissociation forms \( 2\mathrm{Al}^{3+} \) and \( 3\mathrm{SO}_4^{2-} \). The full dissociation of these compounds affects how many charge carriers are available.
- In \( \mathrm{CaCl}_2 \), it dissociates into \( \mathrm{Ca}^{2+} \) and two \( \mathrm{Cl}^- \) ions.
Understanding the complete dissociation helps in determining how equivalent conductance relates to molar conductance.

Chemical Equivalence

Chemical equivalence is an important principle in chemistry and is related to how substances interact and react in a chemically equivalent manner. It connects directly to equivalent conductance because it informs us about the number of reactive units in a compound.
In the given context:
- For \( \mathrm{Al}_2(\mathrm{SO}_4)_3 \), each molecule dissociates into six equivalents due to its ions: two \( \mathrm{Al}^{3+} \) ions and three \( \mathrm{SO}_4^{2-} \) ions, which collectively account for a charge total of 6 equivalents.
- \( \mathrm{CaCl}_2 \) has two equivalents, as the dissociation releases one \( \mathrm{Ca}^{2+} \) ion and two \( \mathrm{Cl}^- \) ions, collectively accounting for a charge total of 2 equivalents.
These equivalent counts are crucial in transforming equivalent conductance into molar conductance, helping us understand the effective concentration of charge carriers in the solution.

Al2(SO4)3 and CaCl2 Calculations

The calculations for \( \mathrm{Al}_2(\mathrm{SO}_4)_3 \) and \( \mathrm{CaCl}_2 \) start with understanding how each compound dissociates in solution and the number of equivalents they provide. This sets up how we calculate their respective molar conductances from known equivalent conductances \( x \) and \( y \).
For \( \mathrm{Al}_2(\mathrm{SO}_4)_3 \):
- Dissociation results in 6 equivalences as noted from 2\( \mathrm{Al}^{3+} \) and 3\( \mathrm{SO}_4^{2-} \). Therefore, the molar conductance is \( 6x \).
For \( \mathrm{CaCl}_2 \):
- Dissociation results in 2 equivalences from one \( \mathrm{Ca}^{2+} \) and two \( \mathrm{Cl}^- \) ions. Consequently, the molar conductance is \( 2y \).
These calculations illustrate the relationship between equivalent conductance to molar conductance, reflecting the standard approach in chemistry of relating individual ionic contributions to overall solution behavior.