Refer to Figure 13-28 . (a) If the unit cell edge is...

Refer to Figure $13-28 .$ (a) If the unit cell edge is represented as $a$, what is the distance (center to center) from $\mathrm{Na}^{+}$ to its nearest neighbor? (b) How many equidistant nearest neighbors does each $\mathrm{Na}^{1}$ ion have? What are the identities of these nearest neighbors? (c) What is the distance (center to center), in terms of $a$, from a $\mathrm{Na}^{+}$ ion to the nearest $\mathrm{Na}^{+}$ ion? (d) How many equidistant nearest neighbors does each Cl $^{-}$ ion have? What are their identities?

Refer to Figure $13-28 .$ (a) If the unit cell edge is represented as $a$, what is the distance (center to center) from $\mathrm{Na}^{+}$ to its nearest neighbor? (b) How many equidistant nearest neighbors does each $\mathrm{Na}^{1}$ ion have? What are the identities of these nearest neighbors?
(c) What is the distance (center to center), in terms of $a$, from a $\mathrm{Na}^{+}$ ion to the nearest $\mathrm{Na}^{+}$ ion? (d) How many equidistant nearest neighbors does each Cl $^{-}$ ion have? What are their identities?

Key Concepts

Nearest Neighbor Distance Calculation in Lattices

Calculating the distance between nearest neighbor ions involves analyzing the geometric relationships within the unit cell. This includes determining distances along edges, face diagonals, or body diagonals based on the known cell edge length. Such calculations are essential in relating the crystal’s macroscopic parameters to the microscopic interactions that dictate properties like ionic bonding and lattice energy.

Ionic Crystal Structure and Sublattices

Many ionic compounds crystallize in structures where the cations and anions form two interpenetrating lattices, such as in the sodium chloride structure. Each ion type occupies one of the sublattices, and the relative positioning of these sublattices determines both which ions are nearest neighbors and the geometrical distances between ions of the same or opposite types.

Unit Cell Geometry

This concept involves understanding the basic building block of a crystal lattice, the unit cell, including its edge length and spatial arrangement. In calculations of interionic distances, the unit cell’s dimensions provide the framework to relate macroscopic lattice parameters to microscopic distances between ions positioned at specific fractional coordinates of the cell.

Coordination Number

The coordination number defines the number of nearest neighbor ions surrounding a given ion in a crystal structure. It is crucial for understanding the stability and bonding characteristics of ionic crystals, as it reflects how the arrangement of oppositely charged ions maximizes attractive interactions while minimizing repulsion.

Transcript

00:01 So for a, we want to know what is the distance from sodium to its nearest neighbor? so in an ionic crystal lattice, the optionally charged ions are arranged next to each other.

00:14 The nearest neighbor of sodium ion is going to be the chlorine ion.

00:21 So we can see that it would be one half of a because based off of this drawing, the distance from the center of one ion to the neighboring ion is one half of the edge of unit cells.

00:38 Which is a cube.

00:40 So then we want to know how many equidistant nearest neighbors does sodium ion have.

00:45 So the equidistant neighbors are going to be the oppositely charged ions.

00:52 In this case, sodium has six neighbors and they're all chlorine.

00:58 So the reason is so the little atoms are the sodium and then we have one, two, three, four of them.

01:07 Then we have to take into account that if there's another unit cell attached to this one, we're going to actually have a chlorine that is from the other unit cell touching the sodium.

01:29 And there will be two from the other unit cell that will be touching it.

01:35 So you can imagine that the unit cells are like blocks stacked on top of one another.

01:47 And if we stack the next unicel on top of this one, we should get six chlorine atoms next to the sodium.

01:54 So now for the next one, we have to find the distance in terms of a from central to the center of sodium to the nearest sodium ion.

02:06 So if we look at only the sodium ions in the unit cell, they're in an fcc arrangement.

02:13 So i'm going to just draw this unicel with just the sodium ion.

02:17 Then we close up on one face of this unit cell, this is what we'll see.

02:23 We can see that we have a line going through the center of the ions in the corners.

02:31 And let's say that r is equal to the radius of one ion...

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