Question

An element \(X\) (At. wt. \(=80 \mathrm{~g} / \mathrm{mol}\) ) having fcc structure, calculate no. of unit cells in \(8 \mathrm{gm}\) of \(X\) : (a) \(0.4 \times N_{A}\) (b) \(0.1 \times N_{A}\) (c) \(4 \times N_{A}\) (d) none of these

Short answer

None of these options are correct since the calculated number of unit cells is \(0.025 \times N_{A}\), which does not match any of the provided choices.

Step-by-step solution

Calculate the number of moles of element X in 8g

Using the molar mass of element X, calculate the number of moles present in 8g. The number of moles, n, is given by the formula: \( n = \frac{mass}{molar\ mass} \). For element X, \( n = \frac{8g}{80g/mol} = 0.1 \, mol \).

Determine the number of atoms in 0.1 mol of X

Avogadro's number, \(N_{A}\), is the number of atoms in one mole of any substance. The number of atoms in 0.1 moles is \(0.1 \times N_{A}\).

Calculate the number of unit cells in the FCC structure

In a face-centered cubic (fcc) structure, there are 4 atoms per unit cell. We already calculated the total number of atoms is \(0.1 \times N_{A}\). To find the number of unit cells, divide the total number of atoms by the number of atoms per unit cell: \(\frac{0.1 \times N_{A}}{4}\).

Simplify the expression to find the number of unit cells

Simplifying the expression \(\frac{0.1 \times N_{A}}{4}\), we get the number of unit cells as \(0.025 \times N_{A}\), which is not an exact match to the given options.

Key concepts

Avogadro's Number

Understanding Avogadro's number is essential when studying chemistry, particularly when delving into the mole concept. Avogadro's number, denoted as \(N_A\), is a constant that represents the number of particles—be they atoms, molecules, ions, or electrons—in one mole of a substance. It's an incredibly large number, approximately \(6.022 \times 10^{23}\).

When you have a specific mass of an element, you can calculate the number of moles by dividing the mass by the molar mass. From there, multiplying the moles by Avogadro's number gives you the total number of particles. For instance, in the exercise, figuring out the number of atoms in 8 grams of element X necessitated multiplying the number of moles (0.1 mol) by Avogadro's number, leading us to \(0.1 \times N_A\) atoms.

Molar Mass

Molar mass acts as a bridge connecting the mass of a substance to the number of particles it contains. Defined as the mass of one mole of a substance, it is expressed in units of grams per mole (g/mol). Each element has its own unique molar mass, allowing us to calculate how many moles are in a given sample.

To find the moles of element X from the exercise, the molar mass played a crucial role. With the given atomic weight of 80 g/mol for element X, we determine the moles present in an 8-gram sample through the formula \(n = \frac{mass}{molar\ mass}\). This straightforward calculation helps us advance toward understanding the composition of our sample on an atomic scale.

Face-Centered Cubic Structure

Delving into the world of crystal structures, the face-centered cubic (fcc) structure is a common arrangement of atoms in a solid. Each unit cell of this structure has atoms at each of the eight corners and one at the center of each face, making a total of 4 whole atoms per unit cell. This is because the corner atoms are shared by eight unit cells, and the face atoms are shared by two.

In the context of the exercise, knowing that each fcc unit cell contains 4 atoms allows us to calculate the number of unit cells in a sample. After determining the total number of atoms in our 8-gram sample, we divide that number by 4 to arrive at the number of unit cells. The face-centered cubic structure is fundamental to understanding material properties, as the arrangement of atoms affects how a material behaves, including its density and how it interacts with other substances.