Chemistry · Ch 1 — Solid State

Relationship between molar mass, density of the substance and unit cell edge length

1.5.2

Relationship between molar mass, density of the substance and unit cell edge length

The relationship between the density of a substance, its molar mass and its unit cell edge length is deduced in the following steps. If the edge length of the cubic unit cell is $a$ , the volume of the unit cell is $a^3$ . If the mass of one particle is $m$ and the unit cell contains $n$ particles, then

\text{mass of unit cell} = m \times n \qquad \text{...(1.1)}

The density of the unit cell is the same as the density $\rho$ of the substance:

\rho = \frac{\text{mass of unit cell}}{\text{volume of unit cell}} = \frac{m\,n}{a^3} \qquad \text{...(1.2)}

The molar mass $M$ of the substance is the mass of one particle multiplied by the number of particles per mole (Avogadro's number $N_A$ ), that is $M = m\,N_A$ , which rearranges to

m = \frac{M}{N_A} \qquad \text{...(1.3)}

Substituting Eq. (1.3) into Eq. (1.2) gives the working relation used throughout this chapter:

$\rho = \frac{n\,M}{a^3\,N_A} \qquad \text{...(1.4)}$ …

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