Chapter 10, Problem 105A

Interpretation Introduction

Interpretation: The value of Avogadro’s number from the given data of the X-ray diffraction method of sodium chloride needs to be determined.

Concept Introduction: For any crystal lattice, the density and Avogadro’s number are related as follows:

  $\rho =\frac{Z\times M}{N_A\times a^3}$

Here, $\rho$ is density, $Z$ is the number of atoms per unit cell, $M$ molar mass, $N_A$ is Avogadro’s number, and a is the distance between ions.

Explanation of Solution

According to the X-ray diffraction studies of sodium chloride, the distance between sodium and chloride ions is $2.819\times {10}^{-8}\text{ cm}$ . Also, the density of solid NaCl is $2.165{\text{ g/cm}}^3$ . Now, the molar mass of sodium chloride can be calculated from the atomic masses of sodium and chlorine as follows:

  $\begin{array}{c}M=22.989769\text{ g/mol}+35.4530\text{ g/mol}\\ =58.442769\text{ g/mol}\\ \approx 58.44\text{ g/mol}\end{array}$

Therefore, the molar mass of sodium chloride up to 4 significant figures is 58.44 g/mol. The relation between the distance between sodium and chloride ions, the density of sodium chloride, the molar mass of sodium chloride, and Avogadro’s number is represented as follows:

  $\rho =\frac{Z\times M}{N_A\times a^3}$

Here, $\rho$ is density, $Z$ is the number of atoms per unit cell, $M$ molar mass, $N_A$ is Avogadro’s number, and a is the distance between ions.

Rearrange the above equation to calculate Avogadro’s number,

  $N_A=\frac{Z\times M}{\rho \times a^3}$

Substitute the values,

  $\begin{array}{c}{N}_A=\frac{\left(4\right)\left(58.44\right)}{\left(2.165{\text{ g/cm}}^3\right){\left(2.819\times {10}^{-8}\text{ cm}\right)}^3}\\ =4.820\times {10}^{24}\end{array}$

Therefore, the value of Avogadro’s number will be $4.820\times {10}^{24}$ .