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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
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