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Preliminary Examination: Thermodynamics and Statistical Mechanics

Department of Physics and AstronomyUniversity of New Mexico

Spring 2009

Instructions:

The exam consists of 10 short-answer problems (10 points each).Where possible, show all work; partial credit will be given if merited.Personal notes on two sides of an 811 page are allowed.Total time: 3 hours.

Unless otherwise noted, commonly used symbols are dened as follows:

P : Pressure

T : Absolute Temperature

k : Boltzmanns constant

~ : Plancks constant divided by 2

: (kT)1

S : Entropy

V : Volume

g : 9:8 m/s2

Useful Relations:Z1

1

dxex2

=

r

; Re > 0Z

1

0

dxxex = (+ 1)

+1

(+ 1) = ()NX

m=0

xm =1 x(N+1)

1 x; jxj < 1:

ln N! Nln N N

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The partition function for a gas of N noninteracting indistinguishable freeparticles of mass m in a vessel of xed volume V and temperature T is given by

Z0 =1

N!

1

(2~3)

Zdpxdpydpz

Zdxdydz exp

p2

2m

N=

VN

N!

mkT

2~2

3N=2:

Suppose now that the vessel is a tall vertical column with area A and height h,so that V = Ah: For the following three problems, consider the case that h islarge enough that gravitational eects are important.

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P1. Show that when gravity is included, the partition function for thenoninteracting gas,

Z(z1 ; z2 ) =

exp(mgz1) exp(mgz2)

mgh

NZ0;

is modied so that Z0 is multiplied by an additional factor having to do withthe gravitational potential energy. Here z1 is the elevation of the bottom of thecolumn, and z2 = z1 + h is the location of the top of the column.

P2. Starting with Z(z1 ; z2 ) above, derive expressions for the gas pressure atthe bottom and the top of the column respectively. If the gas under considera-tion is Xe; and the pressure at the bottom of the column is 1.00 atm, what is thepressure at the top of the column at an elevation h = 3000 m, assuming idealgas behavior at room temperature? The molecular weight of Xe is 54 g/mole.

P3. Derive an expression for the heat capacity at constant volume in thetwo limiting cases; h > kT=mg:

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A solenoid of length `, and radius r; with a winding density (number of turns

per unit length) ; carries a current I: The core of the solenoid is comprised of asolid having a permeability that is greater than the permeability of free space0. The free energy at constant I and T is given by

F=

1

22I2 + T ln TT0

V

where the constant is the heat capacity per unit volume of the core, T0 is aconstant, and V = r2` is the volume.

P4. The core can be withdrawn from the solenoid by sliding it along thecylindrical axis. What is the minimum work that must be performed to com-pletely withdraw the core at constant I and T?

P5. The permeability = 0 (1 + ) is determined by a linear susceptibility

=C

T

that is inversely proportional to temperature with a Curie constant C: Showthat the entropy of the solenoid/core system is given by

S = V

"

1 + ln TT0

(1202I2

T (core within solenoid)0 (core withdrawn)

)#

P6. Suppose that the core is withdrawn adiabatically and reversibly, whilethe current I is held constant. If the initial temperature of the core is T1; whatis the nal temperature T2 after it has been withdrawn?

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