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27 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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1

3 Laws of Thermodynamics


O)

If two systems are in equilibrium with a third, they are in equilibrium with each other.

1)

Conservation of energy

E
th

= W + Q

2)

The entropy of an isolated system never decreases. The entropy either increases until it reaches
equilibr
ium, or if it’s in equilibrium, it stays the same.




given two system w/

1

>

2
, heat wi
ll be spontaneously transferred from system 1 to 2.




heat cannot be completely converted into work.


Thermodynamic basics


Partition function:

Probability of being in a state w/energy





The
fundamental assumption
: a closed system is equally likely to be any of the quantum states
accessible to it.

g(N, U)


The multiplicity of a system

with N particles and energy

S = k
B


= k
B
logg(N, u)


Specific case: Hermonic oscillator: g(N, n) =

w
here N = #oscillators, n = quantum #

Specific case: N magnets with S
p

in excess Zs =
N




N


:

g(N, s) =

where
=
where B is the magnetic field and M is the magnetic moment


Kinds of energy
:










d


= du + pdV



N




Helmholtz Free Energy

(isothermal)


F =
u






=



logz


dF = du



d


+

dN









=


d




PdV +

dN

(isobaric)


Enthalpy





H =


+ pV



dH =

d


+ Vdp



dN


(isobaric, isothermal)

Gibbs Free Energy





G = F
+
pV = u+pV






dG =


d


+ Vdp +

dN










,








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2

Distributions


Fermi


Dirac : Average

occupancy of an orbital w/energy

, for fermions







Bose


Einstein: Average occupancy of an orbital w/energy

, bosons







Plank distr
ibution: Thermal average number of photons in a single mode a







Ideal gas


PV
=
n
RT

= Nk
B
T




= KE
avg

=

(

for each degree of freedom, note

that f


potential energy, each of those
degrees of freedom gets

as well by the Equipartition Theorem)




Heat capacity, constant volume :


(


= k
B
T)








Partition fun
ction of an atom in a box. Z
1

=




n
a

=


Partition function of N atoms in a box :


Entropy S = k
B


= k
B
N


Chemical potential :


Average occupancy of an orbital of energy





where


= e

/



Free energy:





Per atom in a
monatomic gas




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3


u
2



u
1



2




1

W

Q




O







O


O


O


O

O

Diatomic Gas:


2
-
D ideal gas



u = k
B
T,

C
v

= Nk
B

C
p

= 2Nk
B


Van DerWalls



attemps to modify the
ideal gas law to take into account interactions between atoms
or molecules






where a is a measure of the long
-
range attractive part (adds to internal pressure) of the interaction



and


b is a measure of the short
-
range re
pulsion (volume of molecules themselves)


Critical points : Pc =


Vc = 3Nb,




at this point, there is no separation between the vapor



and liquid phases (a horizontal point of inflection)













( K


K
1

p
290

Fig. 10.10)














(For a given
,


<

c
, V < V
1



liquid V > V
2



gas,


V
1

< V < V
2



both show that sum of volume of


liquid G gas = V)



Phase Diagram

Triple point : The
one

value of T and P for

which all three phases
can happily coexist. Happily.


Critical point: below this point
a phase change between liquid &
gas. Above this point
phase change (fluid


continuously
between high & low densit
y)



Diffusion


,


Reversible isothermal

Reversible isentropic

Irreversible extension into
vacuum

v
1

v
2




<

c



=

c



>

c


Solid

Liquid

Critical point

Triple point

T




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4



Maxwell velocity distribution


Fick’s Law




(diffusion constant)


Four
ier’s Law











Carnot cycle and Work in general

Work done on a system =
=

(area under pV curve)


Energy in: heat from resevoir R
H

(@

H
)


Energy out: heat to resevoir R
L

(@

L

<

H
)


For a reversible engine,


H

=


L
(if

H



L,
only work may be transferred)

e
fficiency:

1)

compress isothermally (Q

)





2)

compress isentropically (



)

3)

expand isothermally (Q

)

4)

expand isentropically (


)


(for
a heat pump, reverse or
der)


for the carnot cycle, efficiency is at a maximum



or



(engine)


(pump/refrigerator)


For an ideal gas, isothermal process


Q
H

=


= N

H

log




isentropic process




=



=
















V

P











V