Page 77 - Materials Science and Engineering An Introduction

Questions and Problems • 49

1. Differentiate E N with respect to r, and then set (ii) Solve for C in terms of D, r, and r 0 .
the resulting expression equal to zero, because the (iii) Determine the expression for E 0 by substitu-
curve of E N versus r is a minimum at E 0 . tion for C in Equation 2.18.
2. Solve for r in terms of A, B, and n, which yields (b) Derive another expression for E 0 in terms of
r 0 , the equilibrium interionic spacing.
r 0 , C, and r using a procedure analogous to the
3. Determine the expression for E 0 by substitut- one outlined in part (a).
ing r 0 into Equation 2.17.
2.19 For an Na —Cl ion pair, attractive and repul- Primary Interatomic Bonds

sive energies E A and E R , respectively, depend on 2.22 (a) Briefly cite the main differences among
the distance between the ions r, according to ionic, covalent, and metallic bonding.
(b) State the Pauli exclusion principle.
1.436
E A = - Make a plot of bonding energy versus melting
r 2.23
temperature for the metals listed in Table 2.3.
7.32 * 10 -6
E R = 8 Using this plot, approximate the bonding energy
r for molybdenum, which has a melting tempera-
For these expressions, energies are expressed ture of 2617 C.
in electron volts per Na —Cl pair, and r is the

distance in nanometers. The net energy E N is just Secondary Bonding or van der Waals Bonding
the sum of the preceding two expressions. 2.24 Explain why hydrogen fluoride (HF) has a
(a) Superimpose on a single plot E N , E R , and E A higher boiling temperature than hydrogen chlo-
versus r up to 1.0 nm. ride (HCl) (19.4 C vs. 85 C), even though HF
has a lower molecular weight.
(b) On the basis of this plot, determine (i) the
between the Na and Cl

equilibrium spacing r 0
ions, and (ii) the magnitude of the bonding energy Mixed Bonding
E 0 between the two ions. 2.25 Compute %IC of the interatomic bond for each
of the following compounds: MgO, GaP, CsF,
(c) Mathematically determine the r 0 and E 0 val- CdS, and FeO.
ues using the solutions to Problem 2.18, and com-
pare these with the graphical results from part (b). 2.26 (a) Calculate %IC of the interatomic bonds for
the intermetallic compound Al 6 Mn.
2.20 Consider a hypothetical X —Y ion pair for which

the equilibrium interionic spacing and bonding en- (b) On the basis of this result, what type of in-
ergy values are 0.38 nm and 5.37 eV, respectively. teratomic bonding would you expect to be found
If it is known that n in Equation 2.17 has a value of 8, in Al 6 Mn?
using the results of Problem 2.18, determine explicit
expressions for attractive and repulsive energies E A Bonding Type–Material Classification Correlations
and E R of Equations 2.9 and 2.11. 2.27 What type(s) of bonding would be expected for
2.21 The net potential energy E N between two adjacent each of the following materials: solid xenon, cal-
ions is sometimes represented by the expression cium fluoride (CaF 2 ), bronze, cadmium telluride
(CdTe), rubber, and tungsten?
C r
E N = - + D expa - b (2.18)
r r Spreadsheet Problems
in which r is the interionic separation and C, D, 2.1SS Generate a spreadsheet that allows the user to
and r are constants whose values depend on the input values of A, B, and n (Equation 2.17) and
specific material. then does the following:
(a) Derive an expression for the bonding energy (a) Plots on a graph of potential energy versus
in terms of the equilibrium interionic separa- interatomic separation for two atoms/ions, curves
E 0
tion r 0 and the constants D and r using the follow- for attractive (E A ), repulsive (E R ), and net (E N )
ing procedure: energies.
(i) Differentiate E N with respect to r, and set the (b) Determines the equilibrium spacing (r 0 ) and
resulting expression equal to zero. the bonding energy (E 0 ).