(a)
To show that the force exerted on an ionic solid is
(a)
Answer to Problem 56AP
It is showed that the force exerted on an ionic solid is
Explanation of Solution
Write the expression for the total potential energy of an ionic solid.
Here,
The ionic bond formation will take place at equilibrium spacing where potential energy takes it minimum value.
Write the expression for the interatomic force.
Let
Write the expression for the equilibrium condition for the force
Here,
Conclusion:
Take the derivative of the equation (II) to get expression of force.
Use equation (II) in equation (III) to get value of
Rearrange above equation to get
Substitute
Therefore, it is showed that the force exerted on an ionic solid is
(b)
To show that the restoring force experienced by ion is
(b)
Answer to Problem 56AP
It is showed that the restoring force experienced by ion is
Explanation of Solution
Rewrite expression for the force from equation (V).
Assume that the ion in the solid is displaced a small distance
Conclusion:
Let
Substitute
Expand
Take the first two terms of the binomial expansion.
Substitute
Substitute
The above equation indicates that the force is proportional to displacement from equilibrium position and force acts in the opposite direction of displacement. Therefore above equation is equivalent to restoring force.
Write the above expression in generalized form.
Here,
Compare above two expressions to get
Therefore, it is showed that the restoring force experienced by ion is
(c)
The frequency of vibration of a
(c)
Answer to Problem 56AP
The frequency of vibration of a
Explanation of Solution
In figure
Then interatomic spacing in NaCl is half of the distance between sodium ion to sodium ion.
Therefore, calculate the spacing in
Write the expression for the ionic cohesive energy.
Here,
Use expression (V) to get stiffness constant.
Write the expression for frequency of vibration of sodium ion.
Here,
Conclusion:
The ionic cohesive energy for the crystal is
Substitute
Substitute
Substitute
Therefore, the frequency of vibration of a
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Chapter 43 Solutions
Physics for Scientists and Engineers With Modern Physics
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- Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning