Mid-Term Exam for Stat. Mech.

Spring 2013.

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N atoms are arranged regularly so as to form a perfect crystal. If one replaces n atoms among them ( 1 < nN ) from the lattice sites to interstices of the lattice, this becomes an imperfect crystal with n defects of the Frenkel type. The number N of interstitial sites into which an atom can enter is of the same order of magnitude as N. Let w be the energy necessary to remove an atom from a lattice site to an interstitial site. Show that, in the equilibroum state at temperature Tsuch that, the following relation is valid:

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Let ps be the probability that a system is in a state s with energy Es . Show that if the entropy is defined by

The value of the ps which make S a maximum under the condition that the mean energy of the system is E, follows the canonical distribution.

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Let the spatial distribution of particles with charge e be given by the number density . If the potential of an external field is , the total potential energy is

Assume that the entropy of this system is

and find the equation which and satisfy in the equilibrium state at temperature T.

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N monomeric units are arranged along a straight line to form a chain molecule. Each monomeric unit is assumed to be capable of being either in an  or  state. In the former state, the length is a and the energy is Ea. The corresponding values in the latter state ar b and Eb. Derive the relation between the length L of the chain molecule and the tension X applied between the both ends of the molecule. use the canonical ensemble at constant tension.

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Charge Q is uniformly distributed on a circular arc of radius a and angle 90. What is the magnitude and direction of the electric field at the center of the circle that contains the arc?

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The figure shows two solid spheres with uniformly distributed charge throughout their volumes. Each has radius R. Point P lies on a line connecting the centers of the spheres, at radial distance R/2 from the center of sphere 1. If the net electric field at P is zero, what is the ratio q2 / q1 of the total charges.

A mass point with mass m moves within the range and is reflected by walls at and l.

(a) Illustrate the trajectory of this mass point in the phase space ,

(b) find the volume of the phase space with energy smaller than E and

(c) show that is kept constant when the wall at is moved slowly.

(d) Going over to quantum mechanics,

N monomeric units are arranged along a straight line to form a chain molecule. Each monomeric unit is assumed to be capable of being either in an  or  state. In the former state, the length is a and the energy is Ea. The corresponding values in the latter state ar b and Eb. Derive, for a given temperature T, the relation between the length L of the chain molecule and the tension X applied to both ends of the molecule.

Hint: Use the canonical ensemble at constant tension.

Answer

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