---
title: "Density from lattice constant"
subject: "Material Science"
chapter: 2
key: "semester_3/material_science:copper-and-tungsten-density"
source: https://www.bctengineer.com/q/semester_3/material_science/copper-and-tungsten-density
site: BCT Engineering
---

# Density from lattice constant

Material Science · Chapter 2: Atomic Structure, arrangement of atoms

- Asked in: year not given, 5 marks
- Video: [ALEKS - Finding Density from an FCC or BCC Lattice Constant](https://www.youtube.com/watch?v=apXNryZTsQc) — Tony St John
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/material_science/copper-and-tungsten-density
- Whole chapter: https://www.bctengineer.com/md/semester_3/material_science/chapter-2

## Question

(a) Copper has an FCC structure with lattice constant 0.3615 nm and atomic mass 63.54 g/mol. Calculate its atomic radius and theoretical density. (b) Tungsten is BCC with a = 0.3165 nm and atomic mass 183.85 g/mol. Calculate its theoretical density. Take Avogadro's number as $6.022 \times 10^{23}$ atoms/mol.

## Short answer

Using ρ = nA/(a³Nₐ): FCC Cu (n=4) atomic radius R = (√2/4)×0.3615 nm = 0.128 nm, density 8.93 g/cm³. BCC W (n=2) density 19.26 g/cm³.

## Answer

**Formula:** theoretical density

$$
\rho = \frac{n A}{V_c N_A}
$$

where $n$ = atoms per unit cell, $A$ = atomic mass, $V_c = a^3$, $N_A$ = Avogadro's number.

#### (a) Copper (FCC, $n = 4$)
Atomic radius, from $4R = \sqrt{2}\,a$:

$$
R = \frac{\sqrt{2}\times 0.3615}{4} = 0.1278\ \text{nm}
$$

Cell volume: $a = 0.3615 \times 10^{-7}$ cm, so $V_c = (0.3615\times10^{-7})^3 = 4.724\times10^{-23}\ \text{cm}^3$.

$$
\rho = \frac{4 \times 63.54}{(4.724\times10^{-23})(6.022\times10^{23})} = \frac{254.16}{28.45} = 8.93\ \text{g/cm}^3
$$

#### (b) Tungsten (BCC, $n = 2$)
$a = 0.3165\times10^{-7}$ cm, so $V_c = 3.171\times10^{-23}\ \text{cm}^3$.

$$
\rho = \frac{2 \times 183.85}{(3.171\times10^{-23})(6.022\times10^{23})} = \frac{367.7}{19.10} = 19.26\ \text{g/cm}^3
$$

The copper value agrees with the handbook value of 8.94 g/cm$^3$ and tungsten with 19.25 g/cm$^3$.

**Answer: (a) $R = 0.128$ nm, $\rho_{Cu} = 8.93$ g/cm³ (8930 kg/m³); (b) $\rho_{W} = 19.26$ g/cm³ (19 260 kg/m³).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which formula gives theoretical density?
   A. n A / (a^3 N_A)
   B. A / (n a^3 N_A)
   C. n N_A / (A a^3)
   D. n a^3 / (A N_A)
   Answer: A. rho = n A / (Vc N_A), with Vc = a^3. (Key point: Density formula)

2. What values of n are used for Cu (FCC) and W (BCC)?
   A. 2 and 4
   B. 4 and 2
   C. 4 and 4
   D. 1 and 2
   Answer: B. FCC has 4 atoms per cell and BCC has 2. (Key point: Atoms per cell for Cu and W)

3. Which relation and result give copper's atomic radius?
   A. 4R = sqrt(3)a, R = 0.156 nm
   B. 2R = a, R = 0.181 nm
   C. 4R = sqrt(2)a, R = 0.128 nm
   D. 4R = sqrt(2)a, R = 0.256 nm
   Answer: C. FCC atoms touch along the face diagonal: R = sqrt(2)(0.3615)/4 = 0.128 nm. (Key point: Copper radius)

4. Why is a converted to cm before computing density?
   A. To get density in g/cm^3 consistent with g/mol
   B. To get the radius in nm
   C. Because N_A is in nm
   D. To change n
   Answer: A. A is in g/mol, so volume in cm^3 gives density in g/cm^3. (Key point: Unit conversion)

## Related in Material Science

- [BCC, FCC and HCP crystal structures](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Abcc-fcc-hcp-structures) — Material Science, chapter 2
- [Planar and linear density of BCC iron](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Aplanar-and-linear-density-bcc-iron) — Material Science, chapter 2
- [Critical nucleus radius for copper](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Acritical-nucleus-radius-copper) — Material Science, chapter 5
- [Volume change in iron allotropy](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Avolume-change-bcc-to-fcc-iron) — Material Science, chapter 6

## Related in other subjects

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- [Copper FCC: atoms, packing, concentration and density](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-atoms-density) — Electrical Engineering Material, chapter 1
- [Copper FCC: packing, density and Fermi energy](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-fermi-energy) — Electrical Engineering Material, chapter 1
- [Lattice constant of a simple cubic structure](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asimple-cubic-lattice-constant) — Electrical Engineering Material, chapter 1