KCl crystallizes in a structure like NaCl. The ionic radius of Na+ is 0.525 times the ionic radius of Cl^- and 0.714 times the ionic radius of K+. What is the ratio of the density of NaCl to that of KCl?
To find the ratio of the density of NaCl to that of KCl, we need to consider two things: the relative molar masses of NaCl and KCl, and their respective unit cell structures.
1. First, let's compare the molar masses of NaCl and KCl. The molar mass of NaCl can be calculated as follows:
Molar mass of NaCl = Molar mass of Na+ + Molar mass of Cl-
The atomic mass of Na is approximately 22.99 g/mol, and the atomic mass of Cl is approximately 35.45 g/mol.
Therefore, the molar mass of NaCl = 22.99 g/mol + 35.45 g/mol = 58.44 g/mol.
Similarly, the molar mass of KCl can be calculated as follows:
Molar mass of KCl = Molar mass of K+ + Molar mass of Cl-
The atomic mass of K is approximately 39.10 g/mol.
Therefore, the molar mass of KCl = 39.10 g/mol + 35.45 g/mol = 74.55 g/mol.
2. Next, let's consider the unit cell structure of NaCl and KCl. Both NaCl and KCl crystallize in a face-centered cubic (FCC) structure, meaning each ion in the crystal lattice is surrounded by six oppositely charged ions located at the corners of a cube.
In an FCC crystal structure, the coordination number (number of nearest neighbors) for each ion is 6. This means that each Na+ or K+ ion is surrounded by 6 Cl- ions, and each Cl- ion is surrounded by 6 Na+ or K+ ions.
The formula unit of NaCl consists of 1 Na+ ion and 1 Cl- ion, while the formula unit of KCl consists of 1 K+ ion and 1 Cl- ion. Since the coordination number is the same, the number of ions per unit cell in NaCl and KCl is the same.
3. Finally, let's calculate the density of NaCl and KCl using the formula:
Density = (Molar mass) / (Volume of the unit cell)
Since both NaCl and KCl have the same number of ions per unit cell, the volume of the unit cell will be the same for both compounds.
Therefore, the ratio of the density of NaCl to that of KCl is equal to the ratio of their molar masses:
Ratio of densities = (Molar mass of NaCl) / (Molar mass of KCl)
= 58.44 g/mol / 74.55 g/mol
≈ 0.784
So, the ratio of the density of NaCl to that of KCl is approximately 0.784.