Using the solution to Problem 7.13, rank the magnitudes of the moduli of elasticity for the following hypothetical X,Y, and Z materials from the greatest to the least. The appropriate A, B, and n parameters (Equation 7.30) for these three materials are shown in the following table; they yield EN in units of electron volts and r in nanometers:
Material | A | B | n |
X | 2.5 | 2.0 × 10−5 | 8 |
Y | 2.3 | 8.0 × 10−6 | 10.5 |
Z | 3.0 | 1.5 × 10−5 | 9 |
Problem 7.13
In Section 2.6 it was noted that the net bonding energy EN between two isolated positive and negative ions is a function of interionic distance r as follows:
where A, B, and n are constants for the particular ion pair. Equation 7.30 is also valid for the bonding energy between adjacent ions in solid materials. The modulus of elasticity E is proportional to the slope of the interionic force–separation curve at the equilibrium interionic separation; that is,
Derive an expression for the dependence of the modulus of elasticity on these A, B, and n parameters (for the two-ion system), using the following procedure:
1. Establish a relationship for the force F as a function of r, realizing that
2. Now take the derivative dF/dr.
3. Develop an expression for r0, the equilibrium separation. Because r0 corresponds to the value of r at the minimum of the EN-versus-r curve (Figure 2.8b), take the derivative dEN/dr, set it equal to zero, and solve for r, which corresponds to r0.
4. Finally, substitute this expression for r0 into the relationship obtained by taking dF/dr.
Figure 2.8b The dependence of repulsive, attractive, and net potential energies on interatomic separation for two isolated atoms.
(7.30)
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