2 × 10⁸ atoms of carbon are arranged side by side. Calculate the radius of carbon atom if the length of this a

Exercises · 2.36

2 \times 10^{8}

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Concept understanding — Atomic Packing Scale

Atomic Packing Factor (APF) — First Principles

Imagine you're packing oranges into a crate. No matter how carefully you arrange them, there will always be some empty space between the spheres. The Atomic Packing Factor (APF) is simply the fraction of that crate's volume that is actually filled by the oranges — except here, the "oranges" are atoms, and the "crate" is the unit cell of a crystal.

The Intuition

Atoms in a solid are not tiny cubes that stack perfectly. They're spheres. And spheres, when packed together, always leave gaps. The APF tells you how efficiently a given crystal structure uses space. A higher APF means atoms are packed more tightly together — less wasted volume.

For example, if you stack cannonballs in a pyramid, they pack more densely than if you just pile them randomly. Different crystal structures (like simple cubic, body-centered cubic, face-centered cubic) have different packing efficiencies. The APF is the number that quantifies this.

The Precise Definition

\text{APF} = \frac{\text{Volume of atoms in a unit cell}}{\text{Volume of the unit cell}}

That's it. A ratio between 0 and 1 (or 0% to 100%). For most metals, APF values range from about 0.52 to 0.74.

How to Calculate It — Step by Step

You need three things:

Number of atoms per unit cell — Count carefully. Atoms at corners are shared by 8 cells, atoms on faces by 2 cells, atoms at edges by 4 cells, and atoms fully inside belong entirely to that cell.
Radius of the atom — Usually given as $R$ .
Volume of the unit cell — Depends on the crystal structure. For a cube of side $a$ , it's $a^3$ . But $a$ itself depends on $R$ through the geometry of how atoms touch.

Watch out

The most common mistake: forgetting that atoms are spheres, not cubes. The volume of one atom is $\frac{4}{3}\pi R^3$ , not $R^3$ or $(2R)^3$ .

Worked Example: Simple Cubic (SC)

In a simple cubic cell:

8 corner atoms, each shared by 8 cells → $8 \times \frac{1}{8} = 1$ atom per cell
Atoms touch along the cube edge: $a = 2R$
Volume of cell: $a^3 = (2R)^3 = 8R^3$
Volume of one atom: $\frac{4}{3}\pi R^3$

\text{APF} = \frac{\frac{4}{3}\pi R^3}{8R^3} = \frac{\pi}{6} \approx 0.524

So only about 52.4% of the space is filled. Nearly half is empty — that's why simple cubic is rare in real metals.

Common APF Values

Crystal Structure	APF	Atoms per cell
Simple Cubic (SC)	0.524	1
Body-Centered Cubic (BCC)	0.680	2
Face-Centered Cubic (FCC)	0.740	4
Hexagonal Close-Packed (HCP)	0.740	6

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Dividing the total length by the number of atoms gives the diameter of one carbon atom; halving that yields the radius: $6 \times 10^{-9}$ cm or 60 pm.

Understanding the Atomic Packing Scale

When atoms are arranged side by side in a straight line, each atom contributes its full diameter to the total length. Think of it like beads on a string: if you have $N$ beads each of diameter $d$ , the string spans a length $L = N \times d$ . This simple linear arrangement lets us work backwards from a macroscopic measurement to the atomic scale.

Carbon atoms, like all atoms, are roughly spherical. When placed side by side, they touch along their diameters. So the total length is just the number of atoms multiplied by one diameter per atom.

Step-by-Step Calculation

1. Identify what we know

We have:

Number of carbon atoms: $N = 2 \times 10^8$
Total length of the arrangement: $L = 2.4 \text{ cm}$
Goal: find the radius $r$ of one carbon atom

2. Relate total length to atomic diameter

Since atoms are arranged side by side (touching), the total length equals the number of atoms times the diameter of one atom:

L = N \times d

where $d$ is the diameter of a carbon atom.

3. Solve for the diameter

Rearranging:

d = \frac{L}{N} = \frac{2.4 \text{ cm}}{2 \times 10^8}

d = \frac{2.4}{2 \times 10^8} = \frac{1.2}{10^8} = 1.2 \times 10^{-8} \text{ cm}

4. Convert diameter to radius

The radius is half the diameter: …

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Q.2×1082 \times 10^{8}2×108 atoms of carbon are arranged side by side. Calculate the radius of carbon atom if the length of this arrangement is 2.4 cm.

Understanding the Atomic Packing Scale

Step-by-Step Calculation

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Q. $2 \times 10^{8}$ atoms of carbon are arranged side by side. Calculate the radius of carbon atom if the length of this arrangement is 2.4 cm.

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