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

(a) \(2.57 \times 10^{21}\) unit cells.

Step 1: Calculate the Molar Mass of NaCl

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

Step 2: Determine the Number of Moles of NaCl

Calculate the number of moles in 1.00 g of NaCl using the formula: \ \( n = \frac{\text{mass}}{\text{molar mass}} = \frac{1.00}{58.5} \, \text{mol} \).

Step 3: Calculate the Number of NaCl Formula Units

Use Avogadro's number to find the number of formula units: \ \( N = n \times 6.022 \times 10^{23} \approx \frac{1.00}{58.5} \times 6.022 \times 10^{23} \approx 1.03 \times 10^{22} \).

Step 4: Determine the Number of Unit Cells

Since each unit cell of NaCl contains 4 formula units, divide the total number of formula units by 4: \ \( \text{Number of unit cells} = \frac{1.03 \times 10^{22}}{4} \approx 2.57 \times 10^{21} \).

Key Concepts

Molar Mass Calculation

When determining how many unit cells are present in a crystal, the first step is calculating the molar mass of the compound involved. Molar mass is the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). To find the molar mass of a compound like NaCl (sodium chloride), you add the atomic masses of all the atoms in its formula.
For NaCl, sodium (Na) has an atomic mass of 23 g/mol, and chlorine (Cl) has an atomic mass of 35.5 g/mol. By adding them, we get the molar mass of NaCl: 23 + 35.5 = 58.5 g/mol. This is a crucial step because it allows us to relate the mass of the substance we have, to the number of moles, using the formula:

• Number of moles (n) = mass of substance (in g) / molar mass (g/mol).

Avogadro's Number

After determining the moles of the compound, we employ Avogadro's number to convert those moles into formula units. Avogadro's number is a fundamental constant, approximately equal to \(6.022 \times 10^{23}\), representing the number of atoms, ions, or molecules in one mole of a substance.
In our exercise, we computed the number of moles of NaCl as \(\frac{1.00}{58.5}\) mol. Multiplying by Avogadro's number reveals the total number of formula units, since each mole contains \(6.022 \times 10^{23}\) formula units.
Applying this, we find the number of formula units:

• Number of formula units = Number of moles \( \times 6.022 \times 10^{23} \).

This calculation connects the microscopic world of atoms and molecules with the macroscopic masses we measure.

Formula Units in a Unit Cell

A unit cell is the smallest repetitive structure of a crystalline solid that retains the overall symmetry and properties of the entire crystal. For NaCl, each unit cell consists of a specific number of formula units. Understanding this is another step towards figuring out how many unit cells you have.
In sodium chloride's crystal structure, each unit cell contains 4 formula units of NaCl. Thus, to determine how many unit cells are present in a given mass of NaCl, divide the total number of formula units by the number of formula units per unit cell. This step bridges the quantity of substance at the molecular level to its structure in a solid form.
For instance, if you find \(1.03 \times 10^{22}\) formula units, dividing by 4 gives the total number of unit cells:

• Number of unit cells = \(\frac{1.03 \times 10^{22}}{4}\).

From these calculations, you gain insight into the smallest repeating unit that builds up a macroscopic sample of the crystal.