CHAPTER 3 ‐ CRYSTAL BINDING
3.2
(a) Determine the distance of separation between a positive ion and a negative ion, each
carrying a charge equal to that on an electron, if their mutual force of attraction is
(b) What is the coulomb potential energy of this ion pair? Give your answer in joules, calories,
electron volts, and joules per mole of ion pairs. See Appendix D.
Solution:
(a) Given:
and the relation:
Solving this latter for
and substituting the values for the given parameters into the equation
gives:
and
(b) The coulomb potential energy,
is given by the equation:
parameters are defined in part a of this problem. Thus,
; where all of the
equals:
or in terms of calories:
or on electron volts:
and in J/mole:
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21
3.3
(a) Make the actual calculation for the Madelung, or attractive energy term, of the Born
equation for the NaCl lattice using the cgs system. Give your answer in k‐cal per mol and
compare it with the value in Table 3.2.
(b) Also give your answer in joules per mol.
Solution:
and
Given:
statcoulomb, and the Madelung energy
Upon substituting r and e in this equation one obtains:
The Madelung energy per mol
can be obtained by multiplying
by Avogardro’s number
Thus:
3.4
Using cgs units, compute the repulsive energy term for the NaCl lattice. Assume that the Born
exponent is 8.00. B can be determined by taking the derivative of the Born equation with
respect to r , and assuming that at
the forces on the ions are zero. (Coulomb equals repulsive
force.)
Solution:
The Born equation for NaCl is:
Where U is the lattice potential energy, N is Avogardo’s number, e is the electron charge, r is the
distance between centers of an adjacent pair of negative and positive ions, n is a large exponent
of the order of 9, and A and B are constants. The first term of the Born equation, for the NaCl
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22
crystal, was determined in Problem 3.3. This problem involves the evaluation of the second term.
To do this, one needs to evaluate the constant B. This can be done by taking the derivative of
the lattice potential energy, U, at
derivative of U with respect to r at
. At this position the derivative should be zero. The
is thus:
f the above relation is solved for the constant B one obtains:
The right hand side of this latter equation should now be substituted for B in the repulsive
energy term as shown below.
At this point, attention is called to the fact that the Madelung or attractive energy term equals
. Thus at
the repulsive energy is 1/n times the attractive energy. Since the Madelung
energy was 205 kcal per mole the repulsive energy is 205/8 or 25.6 kcal per mole. Note that in
Table 3.2 the repulsive energy for NaCl is listed as 23.5 kcal per mole.
3.6
Determine the magnitude and direction of the force between a dipole and an electron if
, and the dipole charges are the same as that on an
electron.
Solution:
Here the parameters are
Under these conditions the radial component of the force is:
The tangential and radial components of the force differ only by a factor of 2 and the difference
between
these latter are equal so that
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23
The drawing below shows the orientation relationships involved in computing the force on the
electron due to the dipole.
r
F 3
0
Fr
Fq
°
a
45°
– +
Dipole
3.7
The zero point energy of the solid neon crystal is reported to be 590 J/mol. On the basis of this
information, estimate the maximum lattice vibrational frequency,
, of the neon lattice. Use
the mks system of units in solving this problem.
Solution:
According to the quantum theory, Eq. 3.31 gives the vibrational energy of a crystal at absolute
zero. This equation is:
where
is the vibrational energy per mole at 0 K, N is Avogadro’s number (6.03 x
Plank constant
and
is the maximum vibrational frequency. Assuming that
and solving for the above equation for
3.8
), h is
results in the following:
Empirical specific heat data imply that the Debye temperature of pure iron is close to 450 K. The
Dab temperature also has been proposed as the temperature at which the energy of the highest
vibrational mode of the lattice,
, equals the thermal energy or kT, i.e.
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On this basis, compute a value for iron of
using mks units
Solution:
The basic equation is
or:
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25