Problem 135 - A Complete Resource Book in Chemistry for JEE Main

Question

The mineral (A) is \(\left[\mathrm{CuCl}_{2} \cdot \mathrm{xCu}(\mathrm{OH})_{2}\right]\). A \(45.05 \mathrm{ml}\) solution of \(0.5089 \mathrm{M} \mathrm{HCl}\) was required to react completely with \(1.6320 \mathrm{~g}\) of the compound (A) whose molar mass is 427 . Hence, \(x\) is ___ .

Step-by-Step Solution

Answer

The value of x is approximately 0.083.

Step 1: Determine Moles of HCl Used

To find the moles of HCl used in the reaction, use the molarity equation \( M = \frac{n}{V} \), where \( M \) is the molarity, \( n \) is the moles, and \( V \) is the volume in liters. First, convert 45.05 ml to liters: \( \frac{45.05}{1000} = 0.04505 \text{ L} \). Then use the molarity, 0.5089 M, to find the moles of HCl: \[ n = M \times V = 0.5089 \, \text{M} \times 0.04505 \, \text{L} = 0.02291 \, \text{moles} \]

Step 2: Reaction Stoichiometry

In the reaction, each CuCl2 unit reacts with 2 moles of HCl. Thus, the moles of CuCl2 reacting with HCl can be determined by halving the moles of HCl:\[ n_{\text{CuCl}_2} = \frac{0.02291}{2} = 0.011455 \, \text{moles} \]

Step 3: Calculate the Mass of CuCl2

Knowing the number of moles of CuCl2, the mass of CuCl2 in the compound can be calculated using its molar mass. The molar mass of CuCl2 is about 134.45 g/mol:\[ \text{mass of CuCl}_2 = 0.011455 \, \text{moles} \times 134.45 \, \text{g/mol} = 1.539 \, \text{g} \]

Step 4: Find Mass of Cu(OH)2

Subtract the mass of CuCl2 from the total mass of the compound to find the mass of Cu(OH)2 in it:\[ \text{mass of Cu(OH)}_2 = 1.6320 \, \text{g} - 1.539 \, \text{g} = 0.093 \, \text{g} \]

Step 5: Calculate the Moles of Cu(OH)2

The molar mass of Cu(OH)2 is approximately 97.57 g/mol. Use this to find the moles of Cu(OH)2:\[ n_{\text{Cu(OH)}_2} = \frac{0.093}{97.57} = 0.000953 \, \text{moles} \]

Step 6: Calculate x

The moles of Cu(OH)2 should equal x times the moles of CuCl2, according to the formula of the compound. Therefore, you can find x by:\[ x = \frac{0.000953}{0.011455} \approx 0.0832 \]

Step 7: Verify Calculation

Multiplying the moles of CuCl2 (0.011455 moles) by x should equal the moles of Cu(OH)2 calculated. If our calculations are precise, this relation holds, confirming the calculated value of x.

Key Concepts

Molarity calculations

Molarity is an essential concept in chemistry, especially when dealing with solutions. It is defined as the number of moles of a solute per liter of solution. The formula used to calculate molarity is \( M = \frac{n}{V} \), where:

• \( M \) is the molarity
• \( n \) is the number of moles of solute
• \( V \) is the volume of the solution in liters

To solve problems involving molarity, it's critical to ensure that volume is always converted to liters. In the provided exercise, the volume of the HCl solution is initially given in milliliters and must be converted to liters to use in molarity calculations.
Given that \( 1 ext{ L} = 1000 ext{ ml} \), 45.05 ml is 0.04505 liters. This conversion sets the stage for determining the number of moles of HCl involved in the reaction. Ultimately, by multiplying the molarity by the converted volume, the moles of solute—or in this case, HCl—can be accurately determined, which is crucial for further stoichiometric calculations.

Mineral Chemistry

Mineral chemistry explores the composition and structure of minerals, which are naturally occurring substances with a defined chemical composition. In the context of this exercise, we're examining a mineral compound \( \left[\mathrm{CuCl}_{2} \cdot \mathrm{xCu} (\mathrm{OH})_{2}\right] \).
This compound's formula indicates it contains two distinct chemical components:

• Copper(II) chloride \( \text{CuCl}_2 \)
• Copper(II) hydroxide \( \text{Cu(OH)}_2 \)

Understanding mineral formulas is important as it allows us to decipher the relationship between these components. Here, "xCu(OH)_{2}" signifies multiple units of copper(II) hydroxide associated with copper(II) chloride.
A key part of mineral chemistry is utilizing these stoichiometric relationships to solve for unknowns—in this case, determining the value of "x," the stoichiometric coefficient, which tells us how many units of copper(II) hydroxide are present relative to one unit of copper(II) chloride.

Reaction Stoichiometry

Reaction stoichiometry involves calculating the amounts of reactants and products in chemical reactions. It utilizes the balanced chemical equation to provide ratios of how substances react.
In our mineral chemistry exercise, stoichiometry helps us understand how \( \text{HCl} \) reacts with each \( \text{CuCl}_2 \) unit. The stoichiometric relationship tells us that 2 moles of \( \text{HCl} \) are needed to fully react with 1 mole of \( \text{CuCl}_2 \).

• From this, we can determine how many moles of \( \text{CuCl}_2 \) are in the compound by halving the moles of \( \text{HCl} \) used.
• This stoichiometric ratio is crucial as it lays the foundation for subsequent calculations, such as determining the content of \( \text{Cu(OH)}_2 \).

By mastering reaction stoichiometry, one can predict product yields or find the limiting reactant in reactions, showcasing its vital role in chemical analysis.