The Thermodynamics in Planck's Law

709
came in direct conflict, however, with Einstein's Photon Hypothesis explanation of the
Photoelectric Effect which establishes the particle nature of light.
Reconciling these logically
antithetical views has been a major challenge for physicists. The double-slit experiment
embodies this quintessential mystery of Quantum Mechanics. Fig. 6.
There are many variations and strained explanations of this simple experiment and new
methods to prove or disprove its implications to Physics. But the 1989 Tonomura 'single
electron emissions' experiment provides the clearest expression of this wave-particle
enigma. In this experiment single emissions of electrons go through a simulated double-slit
barrier and are recorded at a detection screen as 'points of light' that over time randomly fill
in an interference pattern. The picture frames in Fig. 6 illustrate these experimental results.
We will use these results in explaining the
double-slit experiment.
12.1 Plausible explanation of the double-slit experiment
The basic logical components of this double-slit experiment are the 'emission of an electron at
the source' and the subsequent 'detection of an electron at the screen'. It is commonly
assumed that these two events are directly connected. The electron emitted at the source is
assumed to be the same electron as the electron detected at the screen. We take the view that
this may not be so. Though the two events (emission and detection) are related, they may
not be directly connected. That is to say, there may not be a 'trajectory' that directly connects
the electron emitted with the electron detected. And though many explanations in Quantum
Mechanics do not seek to trace out a trajectory, nonetheless in these interpretations the
detected electron is tacitly assumed to be the same as the emitted electron. This we believe is
the source of the dilemma. We further adapt the view that while energy propagates

undetectable as a whole. However, when at a point on the screen
local equilibrium occurs, we
get a 'light burst' that in effect discharges the screen of an amount of energy equal to the
energy burst that illuminated the screen. These points of discharge will be more likely to
occur at those areas on the screen where the illumination is greatest. Over time we would
get these dots of light filling the screen in the interference pattern.
We have a 'reciprocal relation' between 'energy' and 'time'. Thus, 'lowering energy intensity'
while 'increasing time duration' is equivalent to 'increasing energy intensity' and 'lowering
time duration'. But the resulting phenomenon is the same: the interference pattern we observe.
This explanation of the
double-slit experiment is logically consistent with the 'probability
distribution' interpretation of Quantum Mechanics. The view we have of energy
propagating continuously as a wave while manifesting locally in discrete units (
equal size
sips)
when local equilibrium occurs, helps resolve the wave-particle dilemma.
12.2 Explanation summary
The argument presented above rests on the following ideas. These are consistent with all our
results presented in this Chapter.
1.
The 'electron emitted' is not be the same as the 'electron detected'.
2.
Energy 'propagates continuously' but 'interacts discretely' when equilibrium occurs
3.
We have 'accumulation of energy' before 'manifestation of energy'.
Our thinking and reasoning are also guided by the following attitude of
physical realism:
a.
Changing our detection devices while keeping the experimental setup the same can
reveal something 'more' of the examined phenomenon but not something

quantity eta that naturally appears in
our derivation as
prime physis. Planck's constant h is such a quantity. Energy can be defined
as the time-rate of
eta while momentum as the space-rate of eta. Other physical quantities
can likewise be defined in terms of
eta. Laws of Physics can and must be mathematically
derived and not physically posited as Universal Laws chiseled into cosmic dust by the hand
of God.
We postulated the
Identity of Eta Principle, derived the Conservation of Energy
and Momentum, derived Newton's Second Law of Motion, established the intimate
connection between entropy and time, interpreted Schoedinger's equation and suggested
that the
wave-function ψ is in fact prime physis η. We showed that The Second Law of
Thermodynamics pertains to
time (and not entropy, which can be both positive and
negative) and should be reworded to state that
'all physical processes take some positive duration
of time to occur'
. We also showed the unexpected mathematical equivalence between Planck's
Law and Boltzmann's Entropy Equation
and proved that "if the speed of light is a constant, then
light is a wave".
14. Appendix: Mathematical derivations
The proofs to many of the derivations below are too simple and are omitted for brevity. But
the propositions are listed for purposes of reference and completeness of exposition.
Notation. We will consistently use the following notation throughout this APPENDIX:

()Et is a real-valued function of the real-variable t

0
()
rt
Et Ee
if and only if
t
DE rE


Characterization 1:
0
()
rt
Et Ee if and only if EPr


Proof: Assume that
0
()
rt
Et Ee
. We have that




00
rt rs
EEt Es Ee Ee   
,

712
Theorem 1:
0
()
rt
Et Ee if and only if
1
rt
Pr
e


is invariant with respect to t
Proof: Assume that
0
()
rt
Et Ee
. Then we have, for fixed s,

() ()
00
0
()
11
t
rs
rt s rt s
ru rt rs
s



is constant
with respect to t, for fixed s.
Therefore,

2
() 1
0
1
1
rt rt
t
rt
rt
rE t e rP re
Pr
D
e
e











Et Ese Ee

. q.e.d
From the above, we have
Characterization 2:
0
()
rt
Et Ee if and only if
()
()
1
rt s
Pr
Es
e




Clearly by definition of
av
E
,
av
Pr
rt
E
 . We can write
1

Et Ee if and only if
()
1
av
Pr E
Pr
Es
e


.
But if
()
1
av
Pr E
Pr
Es
e


, then by Characterization 2a ,
0
()
rt
Et Ee . Then, by Characterization 1,
we must have that EPr

 . And so we can write equivalently
()

As we've seen above, it is always true that
av
Pr
rt
E


. But for exponential functions ()Et we
also have that EPr . So, for exponential functions we have the following.
Characterization 4:
0
()
rt
Et Ee if and only if
av
E
rt
E



14.2 Part II: Integrable functions
We next consider that ()Et is any function. In this case, we have the following.

The Thermodynamics in Planck's Law

713
Theorem 2: a) For any differentiable function ()Et ,
lim ( )
1

av
EE
E
e




and
0
0
1
rt
Pr
e



as ts , we apply L’Hopital’s Rule.
2
()
lim lim
()
1
t
EE
ts ts
EE
tt
DEt










()Es


since 0E and
()EEs as ts .
Likewise, we have
()
lim lim ( )
1
rt rt
ts ts
Pr E s r
Es
eer





. q.e.d.
Corollary A:




. Since
1
av
EE
E
e



is constant with
respect to t, we have
()
1
av
EE
E
Es
e




. Conversely, if
()
1
av
EE

()Et ,
lim ( )
1
av
Pr E
ts
Pr
Es
e




As a direct consequence of the above, we have the following interesting and important result:
Corollary B:
()
1
av
EE
E
Es
e




and
()
1
av

ts




Proof: We let tts  and
1
()
t
s
EEudu
ts



.
Differentiating with respect to t we have


() ()
t
tsDEt EEt  .

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

714
Rewriting, we have
()
()
t

av
EE
E
Es
e





Proof: Assume that
0
()
rt
Et Ee
. From,
00
0
()
11
t
rs
ru rt rs r t r t
s
EEe
Es
PEedu ee e e
rr r



further be written as
()
1
av
EE
E
Es
e




.
Conversely, consider next a function
()Es satisfying
()
1
E
Es
e




, where
() ()
1
()
t
s

EEtEsEsEt
e
Es Es Es


 
.
Differentiating with respect to
s, we get
2
() () ()
()
()
ss
s
Et DEs DEs
eD e
Es
Es



 

and so,
()
()
s
s
DEs

E




(A3)
and combining (A1), (A2), and (A3) we have


22
() ()
()
() ()
()
s
ss
E
DEs E E Es
EEs
DEs DEs
E
t
Es E t
EE





 


and so,
()
1
()
s
DEs
E
Es t E




.
Using (A1), this can be written as

s
D
t




, or as
s
Dt





and therefore
0
()
rs
Es Ee
. q.e.d.
15. Acknowledgement
I am indebted to Segun Chanillo, Prof. of Mathematics, Rutgers University for his
encouragement, when all others thought my efforts were futile. Also, I am deeply grateful to
Hayrani Oz, Prof. of Aerospace Engineering, Ohio State University, who discovered my
posts on the web and was the first to recognize the significance of my results in Physics.
Special thanks also to Miguel Bayona of The Lawrenceville School for his friendship and
help with the graphics in this chapter. And Alexander Morisse who is my best and severest
critic of the Physics in these results.
16. References
Frank, Adam (2010), Who Wrote the Book of Physics? Discover Magazine (April 2010)
Keesing, Richard (2001). Einstein, Millikan and the Photoelectric Effect, Open University
Physics Society Newsletter, Winter 2001/2002 Vol 1 Issue 4

Öz, H., Algebraic Evolutionary Energy Method for Dynamics and Control, in: Computational
Nonlinear Aeroelasticity for Multidisciplinary Analysis and Design, AFRL, VA-WP-TR-
2002 -XXXX, 2002, pp. 96-162.
Öz, H., Evolutionary Energy Method (EEM): An Aerothermoservoelectroelastic
Application,: Variational and Extremum Principles in Macroscopic Systems, Elsevier,
2005, pp. 641-670.
Öz , H., The Law Of Evolutionary Enerxaction and Evolutionary Enerxaction Dynamics , Seminar
presented at Cambridge University, England, March 27, 2008,

Öz , Hayrani; John K. Ramsey, Time modes and nonlinear systems, Journal of Sound and
Vibration, 329 (2010) 2565–2602, doi:10.1016/j.jsv.2009.12.021

/>broglie-waves/ql47o1qdr604/18#
Ragazas, C. (2011) “If the Speed of Light is a Constant, Then Light is a Wave”, knol
/>constant/ql47o1qdr604/19#
Tonomura (1989)
Wikipedia, (n.d.)
0
Statistical Thermodynamics
Anatol Malijevský
Department of Physical Chemistry, Institute of Chemical Technology, Prague
Czech Republic
1. Introduction
This chapter deals with the statistical thermodynamics (statistical mechanics) a modern
alternative of the classical (phenomenological) thermodynamics. Its aim is to determine
thermodynamic properties of matter from forces acting among molecules. Roots of the
discipline are in kinetic theory of gases and are connected with the names Maxwelland
Boltzmann. Father of the statistical thermodynamics is Gibbs who introduced its concepts
such as the statistical ensemble and others, that have been used up to present.
Nothing can express an importance of the statistical thermodynamics better than the words
of Richard Feynman Feynman et al. (2006), the Nobel Prize winner in physics: If, in some
cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next
generations of creatures, what statement would contain the most information in the fewest words? I
believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that All things
are made of atoms – little particles that move around in perpetual motion, attracting each
other when they are a little distance apart, but repelling upon being squeezed into one
another.
In that one sentence, you will see, there is an enormous amount of information about the
world, if just a little imagination and thinking are applied.
The chapter is organized as follows. Next section contains axioms of the phenomenological
thermodynamics. Basic concepts and axioms of the statistical thermodynamics and relations
between the partition function and thermodynamic quantities are in Section 3. Section 4 deals

• The first law of thermodynamics
There is a function of state called internal energy U. For its total differential dU we write
dU
= ¯dW + ¯dQ , (1)
where the symbols ¯dQ and ¯dW are not total differentials but represent infinitesimal values
of heat Q and work W supplied to the system.
• The second law of thermodynamics
There is a function of state called entropy S. For its total differential dS we write
dS
=
¯dQ
T
,
[reversible process] , (2)
dS
>
¯dQ
T
,
[irreversible process] . (3)
• The third law of thermodynamics
At temperature of 0 K, entropy of a pure substance in its most stable crystalline form is
zero
lim
T→0
S = 0 . (4)
This postulate supplements the second law of thermodynamics by defining a natural
referential value of entropy. The third law of thermodynamics implies that temperature
of 0 K cannot be attained by any process with a finite number of steps.
Phenomenological thermodynamics using its axioms radically reduces an amount of

The time average
X
τ
of a thermodynamic quantity X is given by
X
τ
=
1
τ

τ
0
X(t) dt , (5)
where X
(t) is a value of X at time t and, τ is a time interval of a measurement.
• Ensemble average of thermodynamic quantity
The ensemble average
X
s
of a thermodynamic quantity X is given by
X
s
=
∑
i
P
i
X
i
, (6)

From Eq.(8) relations between the probability and energy can be derived:
Probability in the microcanonical ensemble
All the microscopic states in the microcanonical ensemble have the same energy. Therefore,
P
i
=
1
W
for i
= 1, 2, . . . , W , (9)
where W is a number of microscopical states (the statistical weight) of the microcanonical
ensemble.
Probability in the canonical ensemble
In the canonical ensemble it holds
P
i
=
exp(−βE
i
)
Q
, (10)
where β
=
1
k
B
T
, k
B

B
ln Q + k
B
T

∂ ln Q
∂T

V
. (14)
C
V
=

∂U
∂T

V
= k
B
T
2
∂
2
ln Q
∂T
2
+ 2k
B
T


V
+ Vk
B
T

∂ ln Q
∂V

T
, (17)
G
= A + pV = −k
B
T ln Q + Vk
B
T

∂ ln Q
∂V

T
, (18)
C
p
=

∂H
∂T


∑
i
P
i
ln P
i
. (20)
For the microcanonical ensemble a similar relation holds
S
= k
B
ln W , (21)
where W is a number of accessible states. This equation (with log instead of ln) is written in
the grave of Ludwig Boltzmann in Central Cemetery in Vienna, Austria.
4. Ideal gas
The ideal gas is in statistical thermodynamics modelled by a assembly of particles that do not
mutually interact. Then the energy of i-th quantum state of system, E
i
, is a sum of energies of
individual particles
E
i
=
N
∑
i=1

i,j
. (22)
In this way a problem of a determination of the partition function of system is dramatically

= 
0
+ 
trans
+ 
rot
+ 
vib
+ 
el
, (25)
where 
0
is the zero point energy. The partition function of system then becomes a product
Q
=
exp(−Nβ
0
)
N!
q
trans
q
rot
q
vib
q
el
. (26)
Consequently all thermodynamic quantities of the ideal gas become sums of the

tr
+ A
rot
+ A
vib
+ A
el
, (27)
where U
0
= N
0
and A
tr
, A
rot
, A
vib
, A
el
are the translational, rotational, vibrational,
electronic contributions to the Helmholtz free energy, respectively.
721
Statistical Thermodynamics
6 Will-be-set-by-IN-TECH
4.1 Translational contributions
Translational motions of a molecule are modelled by a particle in a box. For its energy a
solution of the Schrödinger equation gives

tr

, n
z
are the quantum numbers of translation. The partition function
of translation is
q
tr
=

2πmk
B
T
h
2

3/2
V . (29)
Translational contribution to the Helmholtz energy is
A
tr
= −RT ln q
tr
= −RT ln

λ
−3
V

, (30)
where R
= Nk

∂A
tr
∂V

T
=
RT
V
, (32)
U
tr
= A
tr
+ TS
tr
=
3
2
RT , (33)
H
tr
= U
tr
+ p
tr
V =
5
2
RT , (34)
G

tr
∂T

p
=
5
2
R . (37)
4.2 Rotational contributions
Rotations of molecule are modelled by the rigid rotator. For linear molecules there are two
independent axes of rotation, for non-linear molecules there are three.
4.2.1 Linear molecules
For the partiton function of rotation it holds
q
rot
=
8π Ik
B
T
σ h
2
, (38)
where σ is the symmetry number of molecule and I its moment of inertia
I
=
n
∑
1
m
i

= R ln

8π
2
Ik
B
T
σh
2

+ R , (40)
p
rot
= 0 , (41)
U
rot
= RT , (42)
H
rot
= U
rot ,
(43)
G
rot
= F
rot
, (44)
C
V,rot
= R , (45)

1/2
, (47)
where I
A
, I
B
and I
C
the principal moments of inertia. Contributions to the thermodynamic
quantities are
A
rot
= −RT ln q
rot
= −RT ln

1
σ

8π
2
k
B
T
h
2

3/2
(πI
A

)
1/2

+
3
2
R , (49)
p
rot
= 0 , (50)
U
rot
=
3
2
RT , (51)
H
rot
= U
rot
, (52)
G
rot
= A
rot
, (53)
C
V,rot
=
3

quantities are
A
vib
= −RT ln q
vib
= RT ln

1 − e
−x

, (57)
S
vib
= R
xe
−x
1 − e
−x
− R ln

1 − e
−x

, (58)
p
vib
= 0 , (59)
U
vib
= RT

V,vib
, (64)
where x
=
hν
0
k
B
T
.
4.3.2 Polyatomic molecules
In n-atomic molecule there is f fundamental harmonic frequencies ν
i
where
f
=
⎧
⎨
⎩
3n
−5 linear molecule
3n
−6 non-linear molecule
The partition function of vibration is
q
vib
=
f
∏
i=1

i
e
−x
i
1 − e
−x
i
− R
f
∑
i=1
ln

1 − e
−x
i

, (67)
p
vib
= 0 , (68)
U
vib
= RT
f
∑
i=1
x
i
e

i
)
2
, (72)
C
p,vib
= C
V,vib
, (73)
724
Thermodynamics – Interaction Studies – Solids, Liquids and Gases
Statistical Thermodynamics 9
where x
i
=
hν
i
k
B
T
.
4.4 Electronic contributions
The electronic partition function reads
q
el
=
∞
∑
=0
g

1
!
q
N
2
2
N
2
!
(75)
where q
1
and q
2
are the partition functions of molecules 1 and 2, respectively. Let us denote
X
m,i
the molar thermodynamic quantity of pure component i, i = 1, 2 and x
i
=
N
i
N
1
+N
2
its mole
fraction. Then
A
= RT

2
)
+
x
1
S
m,1
+ x
2
S
m,2
, (77)
G
= RT
(
x
1
ln x
1
+ x
2
ln x
2
)
+
x
1
G
m,1
+ x

2
U
m,2
, (80)
H
= x
1
H
m,1
+ x
2
H
m,2
, (81)
C
V
= x
1
C
Vm,1
+ x
2
C
Vm,2
, (82)
C
p
= x
1
C

An older and simpler Einstein model is based on the following postulates
1. Vibrations of molecules are independent:
Q
vib
= q
N
vib
, (85)
where q
vib
is the vibrational partition function of molecule.
2. Vibrations are isotropic:
q
vib
= q
x
q
y
q
z
= q
3
x
. (86)
3. Vibrations are harmonical
q
x
=
∞
∑

/(2T)
1 − e
−Θ
E
/T

3N
, (88)
where
Θ
E
=
hν
k
B
is the Einstein characteristic temperature.
For the isochoric heat capacity it follows
C
V
= 3Nk
B

Θ
E
T

2
e
−Θ
E

T
Θ
D

3

Θ
D
/T
0
x
2
ln(1 −e
−x
) dx , (90)
where
Θ
D
=
hν
max
k
B
is the Debye characteristic temperature with ν
max
being the highest frequency of crystal.
For the isochoric heat capacity it follows
C
V
= 3R

C
V
= 36R

T
Θ
D

3

∞
0
x
3
e
x
−1
dx
= aT
3
,
while the Einstein model incorrectly gives
C
V
= 3R

Θ
E
T


The partition function of the real gas or liquid may be written in a form
Q
=
1
N!
exp
(−Nβ
0
)q
N
int

2πmk
B
T
h
2

3
2
N
Z . (92)
where q
int
= q
rot
q
vib
q
el


r
2
d

r
N
, (93)
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Statistical Thermodynamics
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where symbol

(V)
···d

r
i
=

L
0

L
0

L
0
···dx
i

i
. However, we will use the above simplified notation.
The interaction potential energy u
N
of system may be written as an expansion in two-body,
three-body, e.t.c contributions
u
N
(

r
1
,

r
2
, ,

r
N
)=
∑
i<j
u
2
(

r
i
,

(

r
i
,

r
j
) , (95)
where u
2
is the pair intermolecular potential. The three-body potential u
3
is used rarely at
very accurate calculations, and u
4
and higher order contributions are omitted as a rule.
6.2 The pair intermolecular potential
The pair potential depends of a distance between centers of two molecules r and on their
mutual orientation

ω. For simplicity we will omit the angular dependence of the pair potential
(it is true for the spherically symmetric molecules) in further text, and write
u
2
(

r
i
,

(r)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∞ r
< σ
−σ< r < λσ
0 r
> λσ
(97)
Here σ is a hard-sphere diameter,  a depth of the attractive well, and the attraction region
ranges from σ to λσ.
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Thermodynamics – Interaction Studies – Solids, Liquids and Gases
Statistical Thermodynamics 13
6.2.3 Lennard-Jones potential
This well known pair intermolecular potential realistically describes a dependence of pair
potential energy on distance
u
(r)=4


σ
r

12

hard ellipsoids, and so on.
Examples of soft pair potentials are Lennard-Jones multiatomics, molecules whose atoms
interact according to the Lennard-Jones potential (98).
Another example is the Stockmayer potential, the Lennard-Jones potential with an indebted
dipole moment
u
(r, θ
1
, θ
2
, φ)=4


σ
r

12
−

σ
r

6

−
μ
2
r
3
[

(
3 cos θ
1
cos θ
2
cos θ
3
+ 1
)
, (101)
where ν is a strength parameter. It is a first term (DDD, dipole-dipole-dipole) in the multipole
expansion. Analytical formulae and corresponding strength parameters are known for higher
order terms (DDQ, dipole-dipole-quadrupole, DQQ, dipole-quadrupole-quadrupole, ) as
well.
More accurate three-body potentials can be obtained using quantum chemical ab initio
calculations Malijevský et al. (2007).
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7. The virial equation of state
The virial equation of state in the statistical thermodynamics is an expansion of the
compressibility factor z
=
pV
RT
in powers of density ρ =
N
V
z = 1 + B
2


e
−βu(r)
−1

r
2
dr , (103)
where
f
(r)=exp[−βu(r)] −1
is the Mayer function. For linear molecules we have
B
= −
1
4

∞
0

π
0

π
0

2π
0

e

d

ω
1
d

ω
2

∞
0

ω
1

ω
2

e
−βu(r,

ω
1
,

ω
2
)
−1



r
0

r+s
|r−s|

e
−βu(r)
−1

e
−βu(s)
−1

e
−βu(t)
−1

rstdr ds dt , (107)
and
C
nadd
=
8
3
π
2

∞

dimensionalities are up to 21 Malijevský & Kolafa (2008) in a simplest case of spherically
symmetric molecules. For hard spheres the virial coefficients are known up to ten, which is at
the edge of a present computer technology Labík et al. (2005).
7.4 Virial coefficients of mixtures
For binary mixture of components 1 and 2 the second virial coefficient reads
B
2
= x
2
1
B
2
(11)+2x
1
x
2
B
2
(12)+x
2
2
B
2
(22) , (109)
where x
i
are the mole fractions, B
2
(ii) the second virial coefficients of pure components
and B

(222) . (110)
Extensions of these equations on multicomponent mixtures and higher virial coefficients is
straightforward.
8. Dense gas and liquid
Determination of thermodynamic properties from intermolecular interactions is much more
difficult for dense fluids (for gases at high densities and for liquids) than for rare gases and
solids. This fact can be explained using a definition of the Helmholtz free energy
A
= U − TS. (111)
Free energy has a minimum in equilibrium at constant temperature and volume. At high
temperatures and low densities the term TS dominates because not only temperature but also
entropy is high. A minimum in A corresponds to a maximum in S and system, thus, is in the
gas phase. Ideal gas properties may be calculated from a behavior of individual molecules
only. At somewhat higher densities thermodynamic quantities can be expanded from their
ideal-gas values using the virial expansion.
At low temperatures the energy term in equation (111) dominates because not only
temperature but also entropy is small. For solids we may start from a concept of the ideal
crystal.
No such simple molecular model as the ideal gas or the ideal crystal is known for liquid and
dense gas. Theoretical studies of liquid properties are difficult and uncompleted up to now.
8.1 Internal structure of fluid
There is no internal structure of molecules in the ideal gas. There is a long-range order in
the crystal. The fluid is between of the two extremal cases: it has a local order at short
intermolecular distances (as crystal) and a long-range disorder (as gas).
The fundamental quantity describing the internal structure of fluid is the pair distribution
function g
(r)
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, ,

r
N
)
d

r
3
d

r
N

(V)
···

(V)
e
−βu
N
(

r
1
,

r
2
, ,

dr , (114)
the energy equation
U
RT
=
U
0
RT
+ 2πρβ

∞
0
u(r)g(r)r
2
dr , (115)
where U
0
internal energy if the ideal gas, and the compressibility equation
β

∂p
∂ρ

β
=

1
+ 4πρ

∞

=
A
0
RT
+ 2πρβ

∞
0
u
p
(r)g
0
(r) r
2
dr , (118)
where A
0
is the Helmholtz free energy of a reference system.
In the perturbation theories knowledge of the pair distribution function and the Helmholtz
free energy of the reference system is supposed. On one hand the reference system should be
simple (the ideal gas is too simple and brings nothing new; a typical reference system is a fluid
of hard spheres), and the perturbation potential should be small on the other hand. As a result
of a battle between a simplicity of the reference potential (one must know its structural and
thermodynamic properties) and an accuracy of a truncated expansion, a number of methods
have been developed.
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Thermodynamics – Interaction Studies – Solids, Liquids and Gases
Statistical Thermodynamics 17
8.3 Integral equation theories
Among the integral equation theories the most popular are those based on the

approximation for the bridge function. The mostly used closures are in listed in Malijevský &
Kolafa (2008). The simplest of them are the hypernetted chain approximation
B
(r)=0 (121)
and the Percus-Yevick approximation
B
(r)=γ(r) −ln[γ(r)+1] . (122)
Let us compare the perturbation and the integral equation theories. The first ones are simpler
but they need an extra input - the structural and thermodynamic properties of a reference
system. The accuracy of the second ones depends on a chosen closure. Their examples shown
here, the hypernetted chain and the Percus-Yevick, are too simple to be accurate.
8.4 Computer simulations
Besides the above theoretical approaches there is another route to the thermodynamic
quantities called the computer experiments or pseudoexperiments or simply simulations. For
a given pair intermolecular potential they provide values of thermodynamic functions in the
dependence on the state variables. In this sense they have characteristics of real experiments.
Similarly to them they do not give an explanation of the bulk behavior of matter but they serve
as tests of approximative theories. The thermodynamic values are free of approximations, or
more precisely, their approximations such as a finite number of molecules in the basic box or a
finite number of generated configurations can be systematically improved Kolafa et al. (2002).
The computer simulations are divided into two groups: the Monte Carlo simulations
and the molecular dynamic simulations. The Monte Carlo simulations generate the
ensemble averages of structural and thermodynamic functions while the molecular dynamics
simulations generate their time averages. The methods are described in detail in the
monograph of Allen and Tildesley Allen & Tildesley (1987).
9. Interpretation of thermodynamic laws
In Section 2 the axioms of the classical or phenomenological thermodynamics have been
listed. The statistical thermodynamics not only determines the thermodynamic quantities
from knowledge of the intermolecular forces but also allows an interpretation of the
phenomenological axioms.