The unit cells for most alkali metal halides are face-entered for both the cations and anions. In the case of NaCl the sodium ion is just a little too large for the chloride ions to touch each other along the face diagonal. The unit cell edge length is 564.0 pm. The sodium ion has a radius of 95.0 pm. What is the DIAMETER of the chloride ion in picometers?

how would i work out this question?

Jake--It seems to me that the unit cell edge of 564.0 pm is the radius of 1 Na ion + the diameter of the chloride ion + the radius of a second Na ion with the edge of each just touching. So wouldn't the diameter of the chloride ion just be 564.0 - 2*(radius Na ion) = ?? about 374 pm. Check my thinking.

You can check that out by looking at the diagonal of the face (where radius Cl + space + diameter Cl + space + radius Cl) DON'T touch. The diagonal will be sqrt(a^2+ a^2) = 797.6 pm. So if diameter is 374, then radius is 374/2 = 187. That means that 187 for the radius of the first Cl at the lower corner + 374 for the diameter of the Cl in the face + 187 for the Cl at the upper diagonal corner = 187 + 374 + 187 = 748 pm. That leaves 797.6 - 748 = 49.6 pm for the spaces between the Cl ions. We have two spaces which means the Cl atoms on the diagonal of the face are separated by 49.6/2 pm. Again, CHECK MY THINKING.

The purpose of that exercise on the diagonal was just to prove that the Cl ions were NOT touching and that there was, indeed, a little space between Cl ions. That gave some credibility, I thought, to the value of 374 for the diameter of the Cl ion.

To work out the diameter of the chloride ion in picometers, we can use the information provided about the unit cell edge length and the radius of the sodium ion. Here's a step-by-step process:

1. Determine the distance between the centers of neighboring sodium ions along the face diagonal.
Since the unit cells are face-centered, we can use the unit cell edge length to calculate this. The face diagonal of a face-centered unit cell is equal to the unit cell edge length multiplied by the square root of 2.
Face diagonal length = 564.0 pm * √2

2. Calculate the distance between the centers of neighboring sodium ions.
The distance between neighboring sodium ions is equal to the face diagonal length minus twice the radius of the sodium ion.
Distance between sodium ions = (Face diagonal length) - 2 * (radius of sodium ion)

3. Determine the distance between the centers of neighboring chloride ions.
Since the sodium ion is just a little too large for the chloride ions to touch each other along the face diagonal, the distance between neighboring chloride ions is equal to the distance between neighboring sodium ions.

4. Calculate the diameter of the chloride ion.
The diameter of the chloride ion is equal to the distance between neighboring chloride ions.

Now, let's substitute the values into the equations:

Face diagonal length = 564.0 pm * √2

Distance between sodium ions = (Face diagonal length) - 2 * (radius of sodium ion)

Distance between chloride ions = Distance between sodium ions

Diameter of chloride ion = Distance between chloride ions

To perform the calculations, you can use a scientific calculator or a programming language like Python. Ensure that all the measurements are in the same unit (picometers in this case) for consistent results.