Problem 186 - A Complete Resource Book in Chemistry for JEE Main

Question

The limiting molar conductivities \(\Lambda^{\circ}\) for \(\mathrm{NaCl}, \mathrm{KBr}\) and \(\mathrm{KCl}\) are 126,152 and \(150 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) respectively. The \(\Lambda^{0}\) for \(\mathrm{NaBr}\) is \(\quad[2004]\) (a) \(278 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (b) \(178 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (c) \(128 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\) (d) \(306 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The limiting molar conductivity of NaBr is 128 S cm²/mol (option c).

Step 1: Understanding Limiting Molar Conductivity

Limiting molar conductivity (0) is the conductivity of an electrolyte at infinite dilution, where each ion moves independently. To find the value for a salt like NaBr, we can use the known 0 values for similar ions from other known salts.

Step 2: Use Kohlrausch's Law of Independent Migration

According to Kohlrausch's law, 0 of an electrolyte is equal to the sum of the limiting molar conductivities of its constituent ions. That means:\[4(NaBr) = 4(Na^+) + 4(Br^-)\]

Step 3: Identify Relevant Ion Conductivities

We know 0 values for NaCl, KBr, and KCl. Using these, we'll isolate conductivities for the specific ions: - For NaCl: 0(Na^+) + 0(Cl^-) = 126 - For KBr: 0(K^+) + 0(Br^-) = 152 - For KCl: 0(K^+) + 0(Cl^-) = 150.

Step 4: Calculate Ion Conductivities

From KCl and KBr equations, calculate separately 0(Br^-) and 0(Cl^-): 1. Subtract KCl from KBr: - 0(Br^-) - 0(Cl^-) = 152 - 150 = 2. 2. With this, solve: - 0(Br^-) = 0(Cl^-) + 2.

Step 5: Solve for NaBr

With ion values known or expressed, and knowing 0(Na^+), we find:\[4(NaBr) = 4(Na^+) + (4(Cl^-) + 2)\]Solve for actual numbers:- 4(Na^+) + 4(Cl^-) = 126 - Plug 4(Cl^-) = 4(Na^+) + 2 into the equation:\[4(Na^+) + 4(Br^-) = 126 + 2 = 128\]Thus, 4(NaBr) = 128 S cm²/mol.

Key Concepts

Kohlrausch's Law

Kohlrausch's Law plays a vital role in understanding the behavior of ionic solutions, especially concerning their conductive properties. It states that the limiting molar conductivity of an electrolyte at infinite dilution is equal to the sum of the limiting molar conductivities of its individual ions. This is expressed as \[ \Lambda^{\circ} = \lambda^{\circ}_{+} + \lambda^{\circ}_{-} \]where \( \lambda^{\circ}_{+} \) and \( \lambda^{\circ}_{-} \) are the limiting molar conductivities of the cation and anion, respectively.

This law is particularly useful because it allows scientists to predict the conductivity of a salt not directly measured by using known values from similar salts. By breaking down the conductivity into contributions from individual ions, one can calculate the expected behavior of newly formed compounds. This principle of additive contributions is what makes Kohlrausch's Law a cornerstone in the study of electrolytes and their conductive characteristics.

Ion Conductivity

Ion conductivity refers to the ability of ions to conduct electric current through a solution. Each ion contributes uniquely to the overall conductivity of the solution, depending on its mobility and charge. Highly mobile ions like protons or hydroxide ions typically result in higher conductivity values.

Understanding ion conductivity helps in determining the conduction mechanism within a solution. For example:

• Cations move towards the cathode, contributing positively to the current.
• Anions move towards the anode, also aiding in the flow of electricity.

Ion conductivity is also influenced by factors like temperature and the nature of the solvent, with higher temperatures often increasing conductive capacity as ion movement becomes more vigorous. Leveraging this knowledge, scientists can design solutions with desired electrolytic properties for various applications from batteries to bioengineering.

Electrolyte

The term 'electrolyte' refers to a substance that produces an electrically conducting solution when dissolved in a polar solvent, such as water. Electrolytes are essential in many processes in both nature and technology.

Electrolytes can be classified into two major types based on their dissociation:

• **Strong Electrolytes:** These completely dissociate into ions in solution. Examples include salts like NaCl and acids like HCl.
• **Weak Electrolytes:** These only partially dissociate, resulting in an equilibrium between ions and unionized molecules in solution. Examples are acetic acid and ammonia.

The presence of electrolytes in a solution is crucial for enabling electric flow, with the movement of ions permitting the transfer of charges. This property is harnessed in a wide array of applications, including the medical field for replenishing ions in the human body, and in energy storage systems like batteries to enable electron flow within circuits.