Problem 25 - Chemistry in 30 days

Question

At infinite dilution, the aqueous solution of \(\mathrm{BaCl}_{2}\), molar conductivities of \(\mathrm{Ba}^{2+}\) and \(\mathrm{Cl}^{-}\) ions are \(127.32 \mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol}\) and \(76.34 \mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol}\) respectively. What is \(A_{\mathrm{m}}^{-}\)for \(\mathrm{BaCl}_{2}\) at same dilution ? (a) \(280 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (b) \(330.98 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (c) \(90.98 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (d) \(203.6 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

(a) 280 \(\mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\).

Step 1: Understand the Problem

We are given the molar conductivities of ions in an aqueous solution of \(\mathrm{BaCl}_{2}\) at infinite dilution. We need to find the molar conductivity of the salt \(A_{\mathrm{m}}^{-}\) under the same conditions. Molar conductivity at infinite dilution is the sum of the molar conductivities of the ions making up the salt.

Step 2: Identify Known Values

From the problem, we know:- Molar conductivity of \(\mathrm{Ba}^{2+}\) = 127.32 \(\mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol}\)- Molar conductivity of \(\mathrm{Cl}^{-}\) = 76.34 \(\mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol}\)

Step 3: Apply the Formula

Use the formula for molar conductivity at infinite dilution:\[ \Lambda_{\infty} = \lambda_{\infty}(\mathrm{Ba}^{2+}) + 2 \cdot \lambda_{\infty}(\mathrm{Cl}^{-}) \]

Step 4: Calculate Contribution from Ions

Substitute the given values into the formula:- Contribution from \(\mathrm{Ba}^{2+}\): 127.32 \(\mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol}\)- Contribution from \(\mathrm{Cl}^{-}\): 2 \times 76.34 = 152.68 \(\mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol}\)

Step 5: Compute Total Molar Conductivity

Add the contributions of each ion:\[ \Lambda_{\infty}(\mathrm{BaCl}_{2}) = 127.32 + 152.68 = 280 \mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol} \]

Step 6: Select the Correct Answer

The calculated molar conductivity of \(\mathrm{BaCl}_{2}\) at infinite dilution is 280 \(\mathrm{~S} \mathrm{~cm}^{2} / \mathrm{mol}\). Therefore, the correct answer is (a) 280 \(\mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\).

Key Concepts

Ion Conductivity

Ion conductivity is a measure of how well ions move in a solution. It is vital in understanding how electricity flows through ionic solutions, like salt dissolved in water. When we talk about ion conductivity, we are focusing on the movement of charged particles in a liquid, which is crucial for transferring charge in chemical reactions. In an aqueous solution, ions are surrounded by water molecules which allows them to act as carriers of electric current.
The conductivity of an ion is influenced by several factors:

• The charge and size of the ion – smaller, highly charged ions generally move faster.
• The viscosity of the solvent – a less viscous solvent means ions can move more freely.
• The concentration of ions – higher concentrations can lead to more interactions and potentially lower conductivity.

Ion conductivity is important in applications such as batteries, electroplating, and biochemical sensors, where the movement of ions is essential for efficacy.

Infinite Dilution

Infinite dilution is a scenario where a solute in a solution is so diluted that its concentration becomes negligible, effectively allowing ions to move without interfering with each other. At this point, every ion behaves independently as there is ample solvent to separate the ions from their partners.
The concept is theoretical and helps scientists understand how ions behave without interionic interactions impacting their movement. Molar conductivity at infinite dilution is a key concept, as it indicates the maximum conductive potential of a solution.
At infinite dilution, the molar conductivity is the sum of individual ion conductivities, expressed mathematically as:\[\Lambda_{\infty} = \sum \lambda_{ions}\]This ideal condition aids in simplifying calculations and provides a baseline to compare real-world conductivity measurements, where interactions are inevitable.

Aqueous Solution

An aqueous solution is any solution where water acts as the solvent. It is a ubiquitous concept in chemistry, given water's role as a universal solvent. The properties of aqueous solutions are crucial in numerous chemical reactions. Water molecules can break apart ionic compounds, facilitating the movement of ions which is central to the conductivity of the solution.
In these solutions:

• The solute, often an ionic compound like salts, dissociates into cations and anions.
• Water's polar nature helps stabilize these ions, allowing them to remain separate and mobile.
• This mobility is what makes the solution conductive—important in both natural processes and industrial applications.

Understanding the behavior of aqueous solutions is crucial for fields such as environmental chemistry, medicine, and food science.

Molecular Formulas

Molecular formulas express the actual number and types of atoms present in a molecule. It provides a basic but essential insight into the composition of a chemical substance. For example, in \( \text{BaCl}_2 \), the formula indicates the molecule consists of one barium (Ba) and two chlorine (Cl) atoms.
Understanding molecular formulas helps in determining properties such as molar mass, chemical reactivity, and stoichiometry in reactions. This is particularly relevant when computing properties like molar conductivity at infinite dilution, as each ion's contribution is considered: \[ \text{Contribution from } \text{Ba}^{2+}: \lambda_{\text{Ba}^{2+}}\] \[\text{Contribution from } \text{Cl}^{-}: 2 \times \lambda_{\text{Cl}^{-}}\] Thus, the molecular formula helps establish the proportions of ions, key in calculating properties related to ionic solutions.