Chapter 1, Problem 43RGQ

The smallest repeating unit of a crystal of common salt is a cube (called a unit cell) with an edge length of 0.563 nm.

1. (a) What is the volume of this cube in cubic nanometers? In cubic centimeters?
2. (b) The density of NaCl is 2.17 g/cm³. What is the mass of this smallest repeating unit (“unit cell”)?
3. (c) Each repeating unit is composed of four NaCl units. What is the mass of one NaCl formula unit?

Interpretation Introduction

Interpretation: The volume of this cube in cubic nanometers.

Concept Introduction:

Density: Is a characteristic property of a substance. The density of a substance is the relationship between the mass of the substance and how much space it takes up (volume).

  $\text{Density = }\frac{\text{Mass}}{\text{Volume}}$

Answer to Problem 43RGQ

The volume is $\text{0}{\text{.178 nm}}^3$ and in cubic centimeters is $\text{1}\text{.78 }\times {\text{10}}^{-22}{\text{ cm}}^3$

Explanation of Solution

  $\begin{array}{l}\underset{\_}{\text{Calculate the volume of unit cell:}}\\ \text{Volume of a unit cell is }{l}^3\\ \text{Volume = (0}{\text{.563)}}^3=0.179{\text{nm}}^3\\ \underset{\_}{\text{Conversion of units:}}\\ \text{Given: }\text{Volume = }0.179{\text{nm}}^3\\ {\text{Known: 1 nm = 10}}^{-7}\text{cm}\\ \text{Whereas,}\\ 0.179\text{nm }is\\ \text{ = }\frac{0.179\text{nm}\times {\text{10}}^{-7}\text{cm}}{\text{1 nm}}=0.179\times {\left({\text{10}}^{-7}\text{cm}\right)}^3=0.179\times {\text{10}}^{-21}{\text{cm}}^3\\ \text{ =}1.79\times {\text{10}}^{-22}{\text{cm}}^3.\end{array}$

The volume in nanometers is $\text{0}{\text{.178 nm}}^3$ and in cubic centimeters is $\text{1}\text{.78 }\times {\text{10}}^{-22}{\text{ cm}}^3$

Interpretation Introduction

Interpretation: The volume of this cube in cubic nanometers.

Concept Introduction:

Density: Is a characteristic property of a substance. The density of a substance is the relationship between the mass of the substance and how much space it takes up (volume).

  $\text{Density = }\frac{\text{Mass}}{\text{Volume}}$

Answer to Problem 43RGQ

Mass of the smallest repeating unit is $\text{3}{\text{.87 ×10}}^{\text{-22}}\text{ g}$

Explanation of Solution

  $\begin{array}{l}\underset{\_}{\text{Calculate the mass of the unit cell:}}\\ \text{Given: }\text{Density = 2}{\text{.17g/cm}}^{\text{3}}\text{ ; Volume = 1}{\text{.78×10}}^{\text{-22}}{\text{cm}}^{\text{3}}\\ \text{Known: Density = }\frac{\text{Mass}}{\text{Volume}}\\ \text{Whereas,}\\ \text{Mass=Density×Volume }\\ \text{ = 2}{\text{.17g/cm}}^{\text{3}}\text{× 1}{\text{.78×10}}^{\text{-22}}{\text{cm}}^{\text{3}}\\ \text{ =}\text{3}{\text{.87×10}}^{\text{-22}}\text{g}\text{.}\end{array}$

The obtained mass of the smallest repeating unit is $\text{3}{\text{.87 ×10}}^{\text{-22}}\text{ g}$

Interpretation Introduction

Interpretation: The volume of this cube in cubic nanometers.

Concept Introduction:

Density: Is a characteristic property of a substance. The density of a substance is the relationship between the mass of the substance and how much space it takes up (volume).

  $\text{Density = }\frac{\text{Mass}}{\text{Volume}}$

Answer to Problem 43RGQ

Mass of one formula unit is $\text{9}{\text{.68 ×10}}^{\text{-23}}\text{ g}$

Explanation of Solution

  $\begin{array}{l}\underset{\_}{\text{Calculate the mass of one NaCl unit cell:}}\\ \text{Mass of four unit cell is 3}{\text{.87×10}}^{\text{-22}}\text{g}\text{.}\\ \text{Therefore,}\\ \text{Mass of 1 unit cells =}\frac{\text{3}{\text{.87×10}}^{\text{-22}}\text{g}}{4}\\ \text{ = 9}{\text{.68 ×10}}^{\text{-23}}\text{g}.\end{array}$

The obtained mass of one formula unit is $\text{9}{\text{.68 ×10}}^{\text{-23}}\text{ g}$