Density of an Ionic Crystal

[roeb]

Homework Statement

The distance between the Li+ and Cl- ions in a LiCl is .257 nm. Use this and the molecular mass of LiCl 42.4 g/mol to compute teh density of LiCl.

Homework Equations

The Attempt at a Solution

I am surprisingly having a lot of difficultly with this problem.
The LiCl crystal is face centered (as far as I can tell) so there is an entire LiCl and either a Li or Cl in the center (I think Cl, but I'm not sure).

If I follow the example in the book I can simply find the answer, but it doesn't make sense.

$$\rho$$ = m/V = 42.4 / (2 * 6.02 x 10^23 * r0^3)

This works but I don't understand it. Here is how I attempted to do it...

$$\rho$$ = m/V = (42.4 + 1/2 42.4) / (6.02 x 10^23 * r0^3)

I am assuming that since the problem doesn't give the information I can't look up whether or not Cl is in the middle or even the mass of Li and Cl separately. So I took 1/2 the mass of LiCl hoping that would average out. Unfortunately that gives an incorrect answer.

Does anyone have a way to explain why 1/2 is the correct factor in this case? It seems strange to me.

 

[roeb]

sorry, i seem to have confused fcc with bcc...

 

[thomate1]

The reason why 1/2 is the correct factor in this case is because in a face-centered cubic (FCC) crystal structure, there are 4 ions per unit cell. In this case, there are 2 Li+ ions and 2 Cl- ions in each unit cell. Therefore, when calculating the density, we need to take into account the total mass of all 4 ions in the unit cell, not just half of the LiCl molecule. This is why the correct equation is \rho = m/V = 42.4 / (2 * 6.02 x 10^23 * r0^3), where 2 represents the total number of ions in the unit cell.

In your attempt, you only considered half of the LiCl molecule, which is incorrect. Additionally, the FCC crystal structure does not have a single Li or Cl ion in the center, but rather a combination of both Li and Cl ions. Therefore, taking half of the mass of LiCl would not give an accurate result.

I hope this explanation helps clarify the concept.