Using the solution to Problem 6.13, 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 6.31) 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 | 1.5 | 7.0 × 10-6 | 8 |
Y | 2.0 | 1.0 × 10-5 | 9 |
Z | 3.5 | 4.0 × 10-6 | 7 |
(6.31)
Problem 6.13
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 τ as follows:
where A, B, and n are constants for the particular ion pair. Equation 6.31 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 τ, realizing that
2. Now take the derivative dF/dr.
3. Develop an expression for τ0, the equilibrium separation. Because τ0 corresponds to the value of τ at the minimum of the EN-versus-τ curve (Figure 2.10b), take the derivative dEN/dr, set it equal to zero, and solve for r, which corresponds to τ0.
4. Finally, substitute this expression for r0 into the relationship obtained by taking dF/dr.
Figure 2.10 (b) The dependence of repulsive, attractive, and net potential energies on interatomic separation for two isolated atoms.
We need at least 10 more requests to produce the solution.
0 / 10 have requested this problem solution
The more requests, the faster the answer.