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"
