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 \(\mathrm{Na}=\) \(23, \mathrm{Cl}=35.5]\) [2003] (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 in the 1.00 g NaCl crystal is \(2.57 \times 10^{21} \).

Step 1: Calculate Molar Mass of NaCl

First, find the molar mass of NaCl by adding the atomic masses of Na and Cl. The molar mass of NaCl is: \ \( M_{\text{NaCl}} = 23 + 35.5 = 58.5 \text{ g/mol} \).

Step 2: Calculate Moles of NaCl

Find the number of moles in 1 g of NaCl using its molar mass. Use the formula: \ \( \text{moles of NaCl} = \frac{\text{mass}}{\text{molar mass}} = \frac{1.00}{58.5} \approx 0.0171 \text{ moles} \).

Step 3: Determine Number of NaCl Formula Units

Use Avogadro's number to find the number of formula units in the given moles of NaCl. Avogadro's number is \( 6.022 \times 10^{23} \text{ units/mol} \). Therefore: \ \( \text{Number of NaCl units} = 0.0171 \times 6.022 \times 10^{23} \approx 1.03 \times 10^{22} \).

Step 4: Calculate Number of Unit Cells

In a NaCl crystal, each unit cell contains four NaCl formula units. Thus, the number of unit cells is: \ \( \text{Number of unit cells} = \frac{1.03 \times 10^{22}}{4} \approx 2.57 \times 10^{21} \).

Key Concepts

Understanding Unit Cells in Crystals

A unit cell is the smallest repeating structure of a crystal. It acts similar to a building block for the crystal lattice. In the context of a NaCl crystal, the unit cell is shaped in a way that includes a specific arrangement of sodium and chloride ions. Each unit cell in NaCl contains four NaCl formula units. This means there are four pairs of sodium and chloride ions in these basic structural units.
Understanding unit cells helps in the study of crystal properties and behavior. It allows precise calculations related to the structure's density and mass, and why certain quantities of material are as they are. When complete lattices form, these unit cells repeat in three dimensions, creating the clear and structured solids we observe.

• Smallest repeating structure in a crystal
• Contains specific number of formula units
• Essential for structural calculations

Joins with others to form the larger crystal.

Exploring Molar Mass Calculation

Molar mass is a crucial concept in chemistry, defining the mass of a given substance which one mole of its molecules or atoms comprises. For a compound like NaCl, it is essential to determine the combined atomic masses of its constituent elements, sodium and chloride, to find this value.
The atomic mass of sodium (Na) is 23 g/mol and for chlorine (Cl) is 35.5 g/mol. Therefore, the molar mass of NaCl is calculated by addition: \[ M_{\text{NaCl}} = 23 + 35.5 = 58.5 \text{ g/mol} \]
This calculation is vital for determining how much of the substance is present, particularly in chemical reactions or when converting between mass and moles. Immediately knowing the molar mass aids in conversions between mass of a substance and the number of moles, acting as a bridge between atom-scale and macro-scale quantities. This helps in determining how many molecules or atoms are in a given mass.

• Defines mass of one mole of a substance
• Calculated by adding atomic masses of elements
• Essential for converting between mass and moles

Grasping the Concept of Avogadro's Number

Avogadro's number is fundamental in chemistry, allowing scientists to transition between the mole of a substance and the actual number of entities such as atoms, molecules, or ions. Avogadro's number is an impressive \(6.022 \times 10^{23} \) units per mole.
This large number represents the count of particles found in a single mole of a substance. It's akin to a chemistry constant that links mass and particles, making it indispensable in stoichiometric calculations. By using Avogadro's number, chemistry students and scientists can calculate the exact number of molecules in a sample, enabling precise chemical computations.

• Represents number of units in one mole
• Crucial for calculating particle counts
• Links mass quantities with molecular or atomic scales

It tells us how many individual units of a chemical structure are present in a substance's molecular providing accuracy for experimentation and analysis.