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Good Day! There are 4 thermodynamics questions that I need help-(Answered)

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Good Day! There are 4 thermodynamics questions that I need help with. I'll need detailed solutions. Thank you!


PHYS 2002

 

Statistical Mechanics and Thermodynamics

 

Assignment 1b, Semester 1, 2016

 


 

Curtin University

 

Faculty of Science and Engineering

 

Department of Physics and Astronomy

 


 

Page 1 of 3

 


 

PHYS2002: Statistical Mechanics and Thermodynamics

 


 

1.

 


 

Assignment 1b, Semester 1, 2016

 


 

(Total 6 marks)

 


 

a. (6 marks) Beginning from the four thermodynamic potentials U , H, F , and G,

 

derive the four Maxwell relations for an ideal gas.

 


 

2.

 


 

(Total 12 marks)

 


 

a. (3 marks)

 

theorem,

 


 

Describe what is meant by entropy, and beginning from Clausius?s

 

I

 

d?Q

 

? 0,

 

T

 


 

show that dS ? 0.

 

b. (6 marks) Consider a container of volume V separated into two equal halves by

 

a movable piston. The two halves are in thermal equilibrium at a temperature T . One half

 

contains nA moles of gas A at pressure p and the other half contains nB moles of gas B at

 

pressure 3p. Both gases are ideal, and the container is thermally isolated from its surroundings.

 

The piston is initially held in place, and then released to allow the system to come to pressure

 

equilibrium. Determine an expression for the change in entropy of the system as a whole, in

 

terms of nA and the gas constant R.

 

c. (2 marks) Determine the additional entropy change that would be produced if the

 

piston were completely removed, again in terms of nA and R.

 

d. (1 mark) Would either of these answers be different if both halves were initially

 

filled with gas A, at the original temperature, pressures and volumes?

 


 

3.

 


 

(Total 12 marks) Paramagnetic salts obey Curie?s law, such that ? = ?T ?1 , where ? is

 

the magnetic susceptibility, defined as

 

? = limH?0

 


 

M

 

,

 

H

 


 

where M is the magnetization and H is the magnetic field strength, such that the magnetic

 

induction B = ?0 (H + M ), wher ?0 is the permeability of free space. Changes in the Gibbs

 

free energy for such a salt may be written as dG = ?SdT ? ?0 M V dH.

 

Consider a paramagnetic salt whose heat capacity at constant magnetic field strength, CH ,

 

scales as (? + ?H 2 )V T ?2 , where ? and ? are constants (with ? = ?T ). This salt is at an initial

 

temperature T1 and is placed in a magnetic field of strength H1 . The strength of the field is

 

reduced to zero via an adiabatic process.

 

a. (4 marks) Derive the Maxwell relation arising from the Gibbs free energy for this

 

system, and thereby show that

 



 



 

 

 

?T

 

?0 V HT ??

 

=?

 

.

 

?H S

 

CH

 

?T H

 

Page 2 of 3

 


 

PHYS2002: Statistical Mechanics and Thermodynamics

 


 

Assignment 1b, Semester 1, 2016

 


 

b. (4 marks) Find the final temperature of the salt following the adiabatic demagnetization.

 

c. (4 marks) Describe, on a microscopic level, how the process of adiabatic demagnetization leads to a reduction in temperature. You may wish to refer to Gibbs?s expression for

 

the entropy. Discuss whether or not this process might be used to cool a salt all the way to 0 K,

 

and why.

 


 

4.

 


 

(Total 20 marks)

 


 

a. (5 marks) Along a phase boundary the chemical potential of the two phases is

 

equal, such that ?1 (p, T ) = ?2 (p, T ). Starting from this equality, derive the Clausius-Clapeyron

 

equation for the gradient of the phase boundary in the p ? T plane, namely

 

L

 

dp

 

=

 

,

 

dT

 

T (V2 ? V1 )

 

where L is the latent heat, T is the temperature, and V is the volume of a given phase.

 

b. (4 marks) Show that along the phase equilibrium line between liquid and vapour,

 

the rate of change of latent heat with temperature, dL/dT , is given by

 

L

 

dp

 

dL

 

= + (Cp,v ? Cp,l ) ? (Vv ?p,v ? Vl ?p,l )T

 

dT

 

T

 

dT

 

where subscripts v and l mark the vapour and liquid phases, respectively, and ?p is the isobaric

 

expansivity,

 



 



 

?V

 

?p =

 

.

 

?T p

 

c. (4 marks) Assuming that Vv >> Vl , and using the Clausius-Clapeyron equation and

 

the ideal gas equation, show that

 

dL

 

= (Cp,v ? Cp,l ),

 

dT

 

and hence derive an expression for L.

 

d. (4 marks) The saturated vapour of an incompressible liquid is expanded adiabatically. By considering S = S(T, p), show that for such an adiabatic expansion, we can write

 

dp

 

Cp,v

 

 .

 

=

 

v

 

dT

 

T ?V

 

?T p

 

e. (3 marks) During this expansion, some liquid will condense out if dp/dT for the

 

expansion is shallower than the gradient of the phase boundary predicted in part (b). Show that

 

the condition for liquid to condense out is then

 

 

 

d L

 

Cp,l + T

 

< 0.

 

dT T

 

Page 3 of 3

 


 

 

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