Problem 61 - A Complete Resource Book in Chemistry for JEE Main

Question

The weight of \(1 \times 10^{22}\) molecules of \(\mathrm{CuSO}_{4} \cdot 2 \mathrm{H}_{2} \mathrm{O}\) is (a) \(42.42 \mathrm{~g}\) (b) \(41.42 \mathrm{~g}\) (c) \(44.44 \mathrm{~g}\) (d) \(48.94 \mathrm{~g}\)

Step-by-Step Solution

Answer

The options given are incorrect; the weight is approximately 29.41 g.

Step 1: Determine Molar Mass

Calculate the molar mass of CuSO₄ · 2H₂O by adding the atomic masses of each element.\[ Cu = 63.55, \ S = 32.07, \ O = 16.00 \times 4 (for \ SO_4) + 16.00 \times 2 (for \ 2H_2O), \ H = 1.01 \times 4 (for \ 2H_2O) \] \[ \text{Molar mass} = 63.55 + 32.07 + (16.00 \times 6) + (1.01 \times 4) = 81.08 \] Hence, the molar mass of CuSO₄ · 2H₂O is \[ 177.15 \text{ g/mol} \].

Step 2: Convert Molecules to Moles

Use Avogadro's number \(6.022 \times 10^{23}\) to find how many moles are present in \(1 \times 10^{22}\) molecules. \[ \text{Moles} = \frac{1 \times 10^{22}}{6.022 \times 10^{23}} \] Calculating this gives approximately \(1.66 \times 10^{-2}\) moles.

Step 3: Calculate the Mass

Multiply the number of moles by the molar mass to find the weight of the molecules. \[ \text{Mass} = 1.66 \times 10^{-2} \text{ moles} \times 177.15 \text{ g/mol} \] This calculates to be approximately \(29.41 \text{ g}\).

Key Concepts

Avogadro's number

Avogadro's number is a fundamental constant in chemistry that helps us understand the scale of atoms and molecules. Named after the early 19th-century scientist Amedeo Avogadro, it is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance.
This number is approximately equal to \( 6.022 \times 10^{23} \).

Here's why Avogadro's number is important:

• It allows chemists to "count" molecules by weighing them, which is impractical to do directly given their tiny size.
  
• It provides a common reference point in chemical calculations, such as determining the number of particles in a given amount of substance.

In the context of the exercise, dividing the given number of molecules by Avogadro's number allowed us to convert the count of molecules into moles, a more workable unit in chemistry.
This conversion is essential for further calculations, like deriving mass from moles.

Molecule Weight

Molecule weight is another crucial concept in chemistry, as it refers to the total weight of a molecule calculated from the atomic weights of its constituent atoms.
This weight is often represented in terms of "molar mass," which is the mass of one mole of a given compound.

In the exercise, we calculated the molar mass of \(\text{CuSO}_4 \cdot 2 \text{H}_2\text{O}\):

• The atomic weights of each element (Copper, Sulfur, Oxygen, Hydrogen) were added together by considering their counts in the compound.
• This sum resulted in the molecule having a molar mass of \(177.15 \text{ g/mol}\).

Understanding the molar mass is imperative for converting between moles and grams.
In chemical reactions and laboratory work, knowing the molar mass allows chemists to measure out precise quantities of a substance based on how much it weighs.

Chemical Formula Interpretation

Chemical formula interpretation involves breaking down the symbols and numbers that make up a chemical compound's formula.
This interpretation is vital for understanding the composition of a compound and performing accurate calculations.

A chemical formula like \(\text{CuSO}_4 \cdot 2 \text{H}_2\text{O}\) tells us the elements involved and their respective quantities:

• - "\(\text{Cu}\)" is copper, of which there is one atom.
  - "\(\text{SO}_4\)" is a sulfate group, consisting of one sulfur atom and four oxygen atoms.
  - "\(\cdot 2 \text{H}_2\text{O}\)" indicates the presence of two water molecules, each containing two hydrogens and one oxygen.

This means interpreting the formula requires an understanding of how many of each atom type is present, enabling the calculation of molar mass as seen in the solution.
Grasping this concept allows students to better solve exercises involving mass calculations and molecular interpretations.