Problem 76 - A Complete Resource Book in Chemistry for JEE Main

Question

The equivalent conductance of \(\mathrm{Ba}^{2+}\) and \(\mathrm{Cl}^{-}\)are respectively 127 and \(76 \mathrm{ohm}^{-1} \mathrm{~cm}^{2}\) equiv \(^{-1}\) at infinite dilution. The equivalent conductance (in ohm \(^{-1} \mathrm{~cm}^{2}\) equiv \(^{-1}\) ) of \(\mathrm{BaCl}_{2}\) at infinite dilution will be (a) \(139.5\) (b) 203 (c) 279 (d) \(101.5\)

Step-by-Step Solution

Answer

The equivalent conductance of \(\text{BaCl}_2\) at infinite dilution is 279 ohm\(^{-1} \text{ cm}^{2}\) equiv\(^{-1}\).

Step 1: Understanding Equivalent Conductance

Equivalent conductance at infinite dilution of an electrolyte is the sum of the equivalent conductances of the cation and anion. It is measured in units of ohm\(^{-1} \text{ cm}^{2} \text{ equiv}^{-1}\). For a compound like \(\text{BaCl}_2\), which dissociates into \(\text{Ba}^{2+}\) and 2 \(\text{Cl}^{-}\), we need to account for both ions.

Step 2: Identify Individual Ion Contributions

We are given the equivalent conductance of \(\text{Ba}^{2+}\) as 127 \(\text{ohm}^{-1} \text{ cm}^{2} \text{ equiv}^{-1}\), and \(\text{Cl}^{-}\) as 76 \(\text{ohm}^{-1} \text{ cm}^{2} \text{ equiv}^{-1}\). Each molecule of \(\text{BaCl}_2\) provides 1 \(\text{Ba}^{2+}\) and 2 \(\text{Cl}^{-}\) ions.

Step 3: Calculate Total Conductance of \(\text{BaCl}_2\)

The equivalent conductance of \(\text{BaCl}_2\) can be calculated by adding the equivalent conductance of 1 \(\text{Ba}^{2+}\) ion and 2 \(\text{Cl}^{-}\) ions:\[\lambda_{\text{BaCl}_2} = \lambda_{\text{Ba}^{2+}} + 2\lambda_{\text{Cl}^{-}}\]Substitute the given values:\[\lambda_{\text{BaCl}_2} = 127 + 2 \times 76 = 127 + 152\]

Step 4: Final Calculation

Continuing from the previous step:\[\lambda_{\text{BaCl}_2} = 127 + 152 = 279\]Therefore, the equivalent conductance of \(\text{BaCl}_2\) at infinite dilution is 279 \(\text{ohm}^{-1} \text{ cm}^{2} \text{ equiv}^{-1}\).

Key Concepts

Infinite Dilution

Infinite dilution is a term used in chemistry to describe the state where an electrolyte is dissolved in a solvent to such an extent that it is completely dissociated into its ions. This state is crucial for measuring true properties of electrolytes like equivalent conductance. At infinite dilution, interactions between ions of the electrolyte are negligible due to the vast amount of solvent.
Understanding infinite dilution helps chemists determine the true conductance of an electrolyte, as it is unaffected by the intermolecular forces that might exist at higher concentrations. These interactions can skew conductance measurements by facilitating or impeding ion movement. Thus, physicists and chemists rely on the concept of infinite dilution to extract accurate intrinsic properties of ions within solutions. These measurements provide vital parameters in electrolyte chemistry and help in calculating factors such as equivalent conductance.

Electrolyte Conductivity

Electrolyte conductivity refers to the ability of ions in a solution to conduct electricity. This property is essential in understanding how solutions work, especially ionic solutions used in various technological applications. It is typically measured by assessing how well a substance allows the flow of electric current.
Conductivity is influenced by several factors:

• Ion concentration: More ions mean higher conductivity.
• Types of ions: Different ions carry different charges, affecting conductivity levels.
• Temperature: Higher temperatures typically increase conductivity.

When electrolytes are dissolved in water, they break into cations and anions, which carry electric charge through the solution. By measuring the electrolyte conductivity at infinite dilution, scientists can evaluate the maximum capacity of ions to conduct electricity without ionic interactions interfering. This is extremely beneficial in industries like battery production and water purification.

Ionic Contributions

In a compound, the overall conductivity is determined by the individual contributions of each ion present. For example, in the compound \(\mathrm{BaCl}_2\), dissociation results in one \(\mathrm{Ba}^{2+}\) ion and two \(\mathrm{Cl}^-\) ions. Each ion type has its own equivalent conductance, a unique measurement highlighting its efficiency in conducting electricity within the solution.
To find the total equivalent conductance of a compound at infinite dilution, as demonstrated in the solution, the equivalent conductance of all individual ions must be summed up according to their stoichiometric proportions in the compound. Therefore:

• Calculate the contribution of each ion.
• Consider stoichiometry: multiply each ion’s equivalent conductance by the number of times it appears in the formula.

For \(\mathrm{BaCl}_2\), the conductance formula becomes \(\lambda_{\text{BaCl}_2} = \lambda_{\mathrm{Ba}^{2+}} + 2\lambda_{\mathrm{Cl}^-}\). This calculation reflects the combined effect of all ions in the solution, providing a complete picture of the electrolyte's conductive properties at infinite dilution.