Problem 136 - A Complete Resource Book in Chemistry for JEE Main

Question

How many unit cells are present in a cube shaped ideal crystal of \(\mathrm{NaCl}\) of mass \(1.00 \mathrm{~g} ?\) [Atomic mass of \(\mathrm{Na}=\) \(23, \mathrm{Cl}=35.5]\) (a) \(2.57 \times 10^{21}\) (b) \(6.14 \times 10^{21}\) (c) \(3.28 \times 10^{21}\) (d) \(1.71 \times 10^{21}\)

Step-by-Step Solution

Answer

The number of unit cells is approximately \(2.57 \times 10^{21}\), option (a).

Step 1: Calculate Molar Mass of NaCl

First, determine the molar mass of NaCl by adding the atomic masses of sodium and chlorine. The atomic mass of Sodium (Na) is 23 g/mol, and Chlorine (Cl) is 35.5 g/mol. Therefore, the molar mass of NaCl is calculated as \(23 + 35.5 = 58.5\) g/mol.

Step 2: Calculate Moles of NaCl

To find out how many moles of NaCl are present in 1 gram, divide the mass of the sample by the molar mass. Thus, the moles of NaCl are \( \frac{1}{58.5} \approx 0.0171\) moles.

Step 3: Calculate Number of Formula Units

Use Avogadro's number to determine the number of formula units in the moles calculated. Avogadro’s number is \(6.022 \times 10^{23}\) formula units per mole. Therefore, multiply the moles of NaCl by Avogadro's number: \(0.0171 \times 6.022 \times 10^{23} \approx 1.03 \times 10^{22}\) formula units.

Step 4: Calculate Number of Unit Cells

Each unit cell of NaCl contains 4 formula units because NaCl crystallizes in a face-centered cubic structure, where each unit cell contains 4 formula units. Thus, divide the total number of formula units by 4: \(\frac{1.03 \times 10^{22}}{4} \approx 2.57 \times 10^{21}\) unit cells.

Key Concepts

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is a type of crystalline structure that is commonly found in metals and some ionic compounds, like sodium chloride (NaCl). In an FCC lattice, the atoms are arranged in a cube, and every cube has an atom at each of its corners and one at the center of each face of the cube. This arrangement is known for its high packing efficiency. For ionic compounds like NaCl, this means an efficient way of arranging ions in a solid.
NaCl's FCC structure results in each unit cell containing four formula units. This is because each corner atom is shared among eight neighboring cubes, and each face-centered atom is shared between two cubes. In total, in an FCC unit cell:

• 8 corner atoms contribute 1/8th of an atom each, totaling 1 atom.
• 6 face-centered atoms contribute 1/2 of an atom each, totaling 3 atoms.

This totals to 4 atoms per unit cell in the NaCl face-centered cubic structure. This compact and efficient structure is why NaCl is such a stable ionic compound.

Molar Mass Calculation

Molar mass is essential for converting between grams and moles in any chemical calculations. It is determined by adding together the atomic masses of the elements in a chemical formula.
For sodium chloride (NaCl), the molar mass is calculated by taking the atomic mass of sodium (Na), which is 23 grams per mole, and the atomic mass of chlorine (Cl), which is 35.5 grams per mole. By adding these two together:

• The molar mass of Na is 23 g/mol.
• The molar mass of Cl is 35.5 g/mol.
• So, NaCl's molar mass is 23 + 35.5 = 58.5 g/mol.

Knowing the molar mass allows us to determine how many moles are present in a sample by dividing the sample’s mass by the molar mass. This conversion is crucial in solving problems involving chemical equations and stoichiometry.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, defined as the number of particles (atoms, ions, or molecules) in one mole of a substance. It is approximately 6.022 x 10^23 particles per mole. This number helps in bridging the gap between the macroscopic world that we can measure and the microscopic world of atoms and molecules.
When you have the number of moles of a substance, you can use Avogadro's number to find the exact number of particles present. For example, if we have approximately 0.0171 moles of NaCl, multiplying by Avogadro's number gives the total number of formula units:

• Number of formula units = 0.0171 moles × 6.022 x 10^23 formula units/mole = approximately 1.03 x 10^22 formula units.

This conversion helps in understanding how many smallest units (atoms, ions, or molecules) are interacting in any chemical process. It's a bridge that links mass with number of particles, indispensable for chemists analyzing reactions and creating compounds.