Since KF has the NaCl structure, I will use images of #"NaCl"# for reference.
The #"KF"# unit cell
The #"NaCl"# unit cell is a face-centred cube.
It is called a face centred cubic (fcc) cell because the chloride ions (green) are centred on the faces of a cube.
A #"KF"# unit cell also has an fcc structure. Here is an expanded view of a #"KF"#
unit cell.
Let's say the #"K"^"+"# ions are purple and the #"F"^"-"# ions are red.
Step 1. Calculate the numbers of #"K"^+# and #"F"^"-"# ions in a unit cell
#"Number of K"^"+" "ions" = 12 color(red)(cancel(color(black)("edges"))) × (1/4 color(white)(l)"ion")/(1 color(red)(cancel(color(black)("edge")))) + 1 color(red)(cancel(color(black)("centre")))× "1 ion"/(1 color(red)(cancel(color(black)("centre")))) = "3 ions + 1 ion" = "4 ions"#
#"Number of F"^"-" "ions" = 8 color(red)(cancel(color(black)("corners"))) × (1/8 color(white)(l)"ion")/(1 color(red)(cancel(color(black)("edge")))) + 6 color(red)(cancel(color(black)("faces")))× "1/2 ion"/(1 color(red)(cancel(color(black)("face")))) = "1 ion + 3 ions" = "4 ions"#
There are four #"K"^+# and four #"F"^"-"# ions or four #"KF"# formula units (FU) in a unit cell of potassium fluoride.
Step 2. Calculate the mass of a unit cell
#"Mass" = 4 color(red)(cancel(color(black)("FU"))) × (1 color(red)(cancel(color(black)("mol"))))/(6.022 × 10^23 color(red)(cancel(color(black)("FU")))) × "58.10 g"/(1 color(red)(cancel(color(black)("mol")))) = 3.86 ×10^"-22"color(white)(l) "g"#
Step 3. Calculate the volume of a unit cell
The density of a unit cell is the same as the bulk density of the solid.
∴ # V= 3.86 ×10^"-22"color(red)(cancel(color(black)("g"))) × "1 cm"^3/(2.48 color(red)(cancel(color(black)("g")))) = 1.56 × 10^"-22"color(white)(l) "cm"^3#
Step 4. Calculate the edge length of the unit cell
#V = l^3#
∴ #l = root(3)(V) = root(3)(1.56 × 10^"-22"color(white)(l) "cm"^3) = 5.38 × 10^"-8" color(white)(l)"cm" = "538 pm"#
Step 5. Calculate the interionic distance
If you return to the diagram of an #"NaCl"# unit cell, you will see that the edge length is twice the interionic distance.
∴ #"Interionic distance" = "538 pm"/2 = "269 pm"#
