Take Home Exam
OPTI 6105/8105
3/22/2005
(60 pts Total)
Instructions:
- This is an open book, open note take home exam.
- Contact me if you need clarification on any question.
- Email: astratov@uncc.edu. If you cannot contact me, make any assumption you feel are necessary and work the problem. State your assumption clearly so I can follow what you did.
- The exam must be turned in by 5pm on Friday, 3/25/2004. Please place it under my door (Room 141 Burson).
THIS EXAM IS TO ASSESS YOUR UNDERSTANDING OF THE MATERIAL.
DO YOUR OWN WORK.
DO NOT WORK TOGETHER ON THIS EXAM.
If you find relevant information in a book, on the internet, etc.,
do not plagiarize (copy the text word for word).
YOUR ANSWERS MUST BE IN YOUR OWN WORDS.
VIOLATION OF THESE INSTRUCTIONS WILL BE CONSIDERED CHEATING AND WILL BE DELT WITH ACCORDING TO THE UNIVERSITY CODE OF ACADEMIC INTEGRITY (SEE SYLLABUS)
1. Fox, Ch.1: 1.15 (10 pts)
2. Fox, Ch.2: 2.2 (10 pts)
3. Fox, Ch.2: 2.7 (10 pts)
4. Fox, Ch.7: 7.5 (10 pts)
Additional problem (20 pts)
5. Along a crystal axis in a NaCl crystal, the potential energy that holds the ionic crystal together is well described by the equation,
where x is the separation distance of the Na+ and Cl– ions. The first term captures the net electrostatic attraction between the ions, and the second term provides a repulsion that keeps the ions from getting too close. The constant a is called theMadelung constant and it sets the strength of the attractive term. The Madelung constant for NaCl is 1.7476. The equilibrium separation between ions in NaCl is 0.281 nm and this occurs where the potential energy is a minimum. For most ionic solids, the value of n is between 8 and 10 and is approximately 8 for NaCl.
(a) Using the equilibrium condition, derive an expression that can be used to determine the value of the constant A.
(b) Determine the value of A. Use nanometer for the unit of length rather than meter.
(c) For small displacements from equilibrium, the restoring force is well described by Hooke’s Law. Carry out a Taylor series expansion of the potential function about the equilibrium point in this limit.
(d) Estimate the value of the effective spring constant for small changes in the NaCl bond length.