Problem 141 - A Complete Resource Book in Chemistry for JEE Main

Question

Copper crystallizes in fcc with a unit cell length of \(361 \mathrm{pm}\). What is the radius of copper atom? [2009] (a) \(127 \mathrm{pm}\) (b) \(157 \mathrm{pm}\) (c) \(181 \mathrm{pm}\) (d) \(108 \mathrm{pm}\)

Step-by-Step Solution

Answer

The radius of a copper atom is 127 pm, option (a).

Step 1: Understanding the FCC Structure

Face-centered cubic (fcc) structure means that the atoms are located at each of the corners and at the centers of all the cube faces of the unit cell. There are 4 atoms per unit cell in an fcc lattice.

Step 2: Geometry of FCC Lattice

In an fcc lattice, the face diagonal is equal to four times the atomic radius, i.e., \(4r\), where \(r\) is the atomic radius of copper. The face diagonal also relates to the unit cell length \(a\) using the Pythagorean theorem, giving the formula \(\sqrt{2}a\). Thus, we have the equation: \(\sqrt{2}a = 4r\).

Step 3: Solving for Atomic Radius

Given \(a = 361 \mathrm{pm}\), substitute this into the equation \(\sqrt{2}a = 4r\). Calculate the value of \(r\) by rearranging the formula: \[r = \frac{\sqrt{2} \times 361}{4}\].

Step 4: Calculate and Simplify

Calculate \(\sqrt{2} \approx 1.414\). Substitute this value into the equation:\[r = \frac{1.414 \times 361}{4}\]. Compute the result to find \(r\):\[r \approx \frac{510.154}{4} \approx 127.54 \mathrm{pm}\]. Round \(r\) to the nearest whole number, which gives \(127 \mathrm{pm}\).

Key Concepts

Face-Centered Cubic (FCC) Structure

The face-centered cubic (fcc) structure is one of the simplest and most symmetrical types of crystal lattices. In an fcc structure, each unit cell is a cube with atoms positioned at each corner and at the center of each face of the cube.

The fcc structure is not just limited to copper; it is also found in other metals like aluminum and silver. The distinct feature of this arrangement helps in understanding the packing and density of materials. In each fcc unit cell, there are effectively four complete atoms because the corner atoms are shared among eight neighboring cells, while the face-centered atoms are shared with two.

• Corner atoms contribute 1/8th to the unit cell, resulting in a total of 1 atom from corners.
• Face-centered atoms contribute 1/2 each, adding up to 3 atoms.

Thus, an fcc lattice has a total of 4 atoms per unit cell, fostering a high packing efficiency, which is about 74%. This makes fcc metals particularly dense and strong.

Unit Cell Dimensions

The dimensions of a unit cell are crucial for understanding the spatial relationship and size of atoms within the crystal lattice. The unit cell length, often denoted as 'a', is the side length of the cube. This length is directly tied to the atomic radius when working with different structures like fcc.

In the context of the exercise, the copper's unit cell length is provided as 361 picometers (pm). For fcc structures, the face diagonal of the cube is directly linked to the atomic radius. Utilizing geometry, particularly the Pythagorean theorem, allows us to understand how dimensions relate to each other.

• The face diagonal is given by \[\sqrt{2}a = 4r\]
• This equation shows that the face diagonal, which connects corner atoms through the center of a face atom, equals four times the atomic radius.

With these relationships, understanding the geometry of the lattice becomes much more intuitive.

Crystal Lattice Geometry

The geometry of a crystal lattice is a three-dimensional structure where atoms are arranged in a repeating pattern. The symmetry and efficiency of this arrangement are essential for the material's properties. In face-centered cubic (fcc) lattices, geometry plays a key role in defining how atoms pack and interact.

FCC geometry ensures minimal wasted space, leading to high density and strong bonding interactions. The lattice's geometry includes:

• A cube's corners and face centers are key sites for atom placement, promoting uniformity and symmetry.
• Pythagorean relationships in the lattice, like \[\sqrt{2}a = 4r\], highlight how atomic radius connects with unit dimensions.

Through such geometric principles, we can reliably calculate properties like atomic radius and relate those to the material's overall stability and performance. Understanding these geometrical arrangements is crucial for students and professionals working in materials science, chemistry, or solid-state physics.