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An Ionic Crystal. Figure P23.57 shows eight-point charges arranged at the corners of a cube with sides of length d. The values of the charges are +q and -q, as shown. This is a model of one cell of a cubic ionic crystal. In sodium chloride (NaCl), for instance, the positive ions are Na+ and the negative ions are CI. (a) Calculate the potential energy Uof this arrangement. (Take as zero the potential energy of the eight charges when they are infinitely far apart.) (b) In part (a), you should have found that U<0. Explain the relationship between this result and the observation that such ionic crystals exist in nature.

Short Answer

Expert verified

a) Potential energy of this arrangement is Vt=-1.46q2od

b) As the crystal is solid, thus due to attraction force our result is negative.

Step by step solution

01

Step 1:

In each square face, the diagonal of the square and radius2d and each square has two connection and the cube have a total of 12 connection and their potential is positive.

The cube has 4 diagonals so from the cube rules each diagonal is3dand as the charges are opposite so the potential is negative.

Taking charges at the distance (d) far from the each other and has 12 sides and since charges are not the same therefore potential is negative.

02

Step 2:

Let the voltage is positive and its radius is 2dand charges are (q)

So, potential it is given by

V1=12kq2d

Now taking voltage as negative and radius is2dand charges (q) from 4 connection

V2=-4kq3d

As the voltage is negative, the charge is (q) and radius (r)

V3=-12kqd

So, the total voltage is

vt=12kq2d-4kq3d-12kqd=-1.45q20d

Therefore, the Potential energy of this arrangement isrole="math" localid="1664271395597" Vt-1.45q20d

03

Step 3:

Hence, As the crystal is solid, thus due to attraction force our result is negative.

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