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A DISSERTATION ON PAIR CORRELATION FUNCTION IN STATISTICAL PHYSICS
PRESENTED BY
AKPOVOKA BARRY EGUOLOR
IDOGHOR A. EMMANUEL
DEPARTMENT OF PHYSICS
UNIVERSITY OF BENIN
BENIN CITY
EDO STATE.
AUGUST,2012.
i
TABLE OF CONTENTS
TITLE PAGE................................................................................................i
TABLE OF CONTENTS..............................................................................ii
ABSTRACT................................................................................................iv
1.1 INTRODUCTION.................................................................................1
1.2 DIFINITION.........................................................................................2
1.2.1 CORRELATION FUNCTION.............................................................2
1.2.2 PAIR CORRELATION FUNCTION....................................................5
1.3 DISTRIBUTION OF PARTICLES WITH CORRELATIONS................9
1.4 CALCULATION OF PAIR CORRELATION FUNCTION BY
COMPUTER SIMULATION METHOD.....................................................19
1.4.1 THE CLASSICAL-MAP HYPERNATTED CHAIN METHOD...........20
(CHNC METHOD)...................................................................................20
1.4.2 MONTE CARLO SIMULATION.......................................................21
1.4.3 RADIATION TECHNIQUES............................................................22
1.5 APPLICATIONS OF THE PAIR CORRELATION FUNCTION IN
STATISTICAL PHYSICS...........................................................................23
1.6 RELATIONSHIP BETWEEN THE RADIAL DISTRIBUTION
FUNCTION, ENERGY, COMPRESSIBILITY AND PRESSURE................24
REFERENCES.........................................................................................41
LIST OF FIGURES
FIG 1:CORRELATION CURVE........................................................................3
ii
FIG 2:CORRELATION CURVE FOR TWO MEASUREMENTS........................4
FIG. 3A.RADIAL DISTRIBUTION FUNCTION(RELATIVE POSITION)...........7
FIG 3B:RADIAL DISTRIBUTION FUNCTION IN A LIQUID MOLECULE
(SCHEMATIC CURVE)……………………………………….
………………………………...8
ABSTRACT
This work is aimed at citing out the distinctive properties,
importance and applications of the pair correlation function
iii
together with its relation to thermodynamic properties in
statistical physics.
iv
1.1 INTRODUCTION
The subject of atomic and molecules distribution functions
has enjoyed a long rich history.
This stems both from the experimental use of radiation
scattering to determine such functions at least at pair level[1,2]
as well as a wide range of theoretical developments motivated by
the presence of exact relations for thermodynamic properties in
term of those distribution functions[3,4]. This combination of
experimental measurements and theoretical insights has been an
indispensable component of condensed-matter physical
chemistry and physics. However, not surprisingly for a scientific
area so characterized by intrinsic complexity; some deep
problems of incomplete understanding still persist.
One of the basic problems concerns pair correlation
realizability. In its simplest version this concerns g(R), the pair
correlation function for a statistically homogenous single-
component many-body system comprising structureless
(Spherically symmetric) particles. In the large system limit, this
1
function is conventionally defined to approach unity as r ∞. By
definition, it cannot be negative.
g(R) 0
(1.0)
By virtue of exhaustive enumeration for system of 15 or
fewer sites subject to periodic boundary conditions, several
conclusion have formulated for the case of a constant pair
correlation beyond the exclusion ranges. These conclude (a) pair
correlation realizability over a nonzero density range, (b)
violation of the kirkwood superposition approximation for many
such realizations, and (c) inappropriateness of the so-called
“reverse monte Carlo” method that uses a candidate pair
correlation function as a means to suggest typical many-body
configurations.
1.2 DIFINITION
1.2.1 Correlation Function
2
A(t)
Time (t)
Let us consider a series of measurement of a quantity of a
random nature at different times.
Fig 1:Correlation curve
Although the value of A(t) is changing randomly, for two
measurements taken at time tI and tII that are close to each other
are good chance that A(tI) and A(tII) have similar values, their
values are correlated[5].
If two measurements taken at time tI and tII that are far
apart, there is no relationship between values of A(tI) and A(tII),
their values are uncorrelated.
The “level of correlation” plotted against time would start at
some value and then decay to a low value[5].
3
A(t)
Time, t
tcorr
Let us shift the data by time tcorr.
Fig 2:Correlation curve for two measurements
and multiply the values in new data set to the values of the
original data set.
Now let us average over the whole time range and get a
single number G(tcorr). If the two data sets are lined up, the peaks
and troughs are aligned and we will obtain a big value from this
multiple-and-integrate operation. As we increase tcorr the G(tcorr)
declines to a constant value.
The operation of multiplying two curves together and
integrating them over the x-axis is called an overlap integral,
4
since it gives a big value if the curves both have high and low
values in the same place.
The overlap integral is also called the Correlation function
G(tcorr) = <A(to)A(to + tcorr)>. This is not a function of time (since
we already integrated over time), this is function of the shift in
time or correlation time tcorr.
Decay of the correlation function is occurring on the
timescale of the fluctuation of the measured quantity undergoing
random fluctuations.
All of the above considerations can be applied to
correlations in space instead of time[5].
G(r) = <A(ro)A(ro + r)>
1.2.2 Pair Correlation Function
The pair correlation function gives the probability to find a
particle in the volume element dr located at r if at r = 0 there is
another particle. It gives the probability of finding the centre of a
particle a given distance from the center of another particle. For
5
short distances, this is related to how the particles are packed
together. For example, consider hard sphere (like marbles). The
spheres cannot overlap, so the closest distances two centers can
be is equal to the diameter of the spheres. However, several
sphere can be touching one sphere, then a few more can form a
layer around them, and so on.
Further away, these layers get more diffuse, and so for a
large distance, the probability of finding two spheres with a
given separation is essentially constant. In that case, its related
to the density, a more dense system has move spheres, thus its
more likely to find two of them with a given distance[9].
The pair correlation function is also called the radial
distribution function which is a special case of the pair
distribution function. The interaction between the molecules in a
liquid or gas cause correlation in their positions. The aim of
almost all modern theories of liquids is to calculate the radial
distribution function by means of statistical thermodynamical
reasoning. Alternatively, the radial distribution function can be
measure directly in computer simulations.
6
r
dr
The radial distribution function can be use in calculating the
energy, compressibility and pressure of a fluid, with a particular
application to a hard sphere fluid[6].
For instance, imaging we have placed ourselves on a certain
molecule in a liquid or gas. Now let us count the number of
molecules in a spherical shell of thickness dr at a distance r, i.e.
we count the number of molecules with a distance between r and
r + dr.
Fig. 3a.Radial distribution function(Relative position)
7
0.5
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 g(r) rrrr
1.0
1.5
2.0
2.5
3.0
3.5
r
g(R)
Simple liquid
Fig 3b:Radial distribution function in a liquid molecule(schematic curve)
Fig 3b implies that the radial distribution function as one
moves away from the origin (reference particle) become constant
with value = 1 i.e. the probability of finding a particle in the
volume element dr located at r if at r = 0 there is another
particle is 1 and as one move closer to the origin, a distance less
than the diameter the radial distribution function becomes zero.
1.3 DISTRIBUTION OF PARTICLES WITH
CORRELATIONS
8
Consider the distribution of particles with correlations.
Correlation may be due to molecular force. In quantum statistics,
the configuration in coordination space depends on the
configuration in momentum space, and thus correlation exists
between particles[7].
Consider a system composed of identical N particle. if r j (j =
1,2, . . .N) denotes the position of these particles, then the
number density at r is given by:
n(1) ( r⃗ )= ⟨n(r⃗ )⟩= ⟨n ⟩
But n ( r⃗ )=(r⃗ j−r⃗ )
n(1) ( r⃗ )=⟨( r⃗ j− r⃗) ⟩ (1.1)
Equation (1.1) above is for one-body density. For two-body
density we have
n(2) ( r⃗ ,r⃗¿ )=⟨ ( r⃗ j−r⃗ )δ(r⃗ k−r⃗1)⟩ (1.2)
Where ⟨ ⟩ means average. It may be a quantum mechanical
expectation values or a time average, or ensemble average, but
at the present moment it is sufficient to suppose that it is an
9
average in a certain sense. Equation (1.2) expresses the
correlation in density at positions r⃗∧r⃗1
The two-body density can be written in terms of one-body
density as
n(2)( r⃗ ,r⃗ 1)=n(1) ( r⃗ )n (1) ( r⃗1 )g ( r⃗ ,r⃗1−r⃗ ) (1.3)
Where g ( r⃗ , r⃗1−r⃗ ) is the pair distribution function for isotropic
substance like a gas or liquid, ⟨n ⟩ = n(1)(r) may be constant except
in the vicinity of container walls. Thus, do not depend on r and r1.
Equation (1.3) becomes
n(2)( r⃗ ,r⃗ 1)=⟨n ⟩2g (R ) (1.4)
Since n(1) ( r⃗ )n(1) (r⃗1 )= ⟨n ⟩ ⟨n ⟩=⟨n ⟩2 andR=|⃗r1−r⃗|
R is not a vector but the magnitude of the vector r⃗∧r⃗1
g(R) is called the radial distribution function which can be
obtained by X-ray scattering.
When the system is large, the correlation usually vanishes
for R ∞ since, correlation is a function of distance.
10
Now we integrate the two-body density with respect to r⃗∧r⃗1.
The sum with the restriction j K is given as the double sum
without the restriction subtracted by a simple sum. Integrating
over a volume V, we denote by Nv the number of particles in the
volume. We have from equation (1.2).
n(2) ( r⃗ , r⃗1¿)=⟨ ( r⃗j−r⃗ )δ( r⃗k−r⃗1)⟩
¿∑j ≠
N
∑k
δ ( r⃗j− r⃗ )δ(r⃗ k−r⃗1)
¿∑j=k
N
∑ δ ( r⃗j− r⃗ )δ ( r⃗k−r⃗ )−∑j=k
δ( r⃗j−r⃗ )
Integrating both sides with respect to r and r1 we have
∬n(2 )(r , r¿¿1)dr dr1=∬∑k∑j
δ ( r⃗j−r⃗ )δ (r⃗k−r⃗1)d r⃗ d r⃗ 1−∬∑j=k
δ (r⃗ j−r⃗❑)d r⃗ d r⃗ 1¿
¿ ⟨∬∑k∑j
δ ( r⃗j−r⃗ ) δ(r⃗ k−r⃗ )d r⃗ d r⃗1⟩−⟨∬∑j=k
δ ( r⃗ j−r⃗ )d r⃗ d r⃗ 1⟩
¿∬∑ ⟨n (r⃗ )⟩ ⟨n(r⃗ 1)⟩ d r⃗ d r⃗1−∫∑ ⟨n(r) ⟩d r⃗ d r⃗1
¿ ⟨ N v ⟩ ⟨N v ⟩− ⟨N v ⟩
¿ ⟨N2v ⟩−⟨N v ⟩
∴∬v
❑
n(2 )(r⃗ ,r⃗¿¿1)dr dr1= ⟨N2v ⟩−⟨N v ⟩¿ (1.5)
11
Where N v=∫v
❑
❑( r⃗j−r⃗ )d r⃗
When the volume v is far from the container walls i.e. when
the volume is small,
⟨ N v ⟩ =⟨n ⟩v (1.6)
{∫v
❑
∑ δ (r j−r )dr1=⟨∑j δ (r j−r )⟩}∫ d r⃗
dr=dxdydz=d3r
∴¿
Since n(1)r=⟨ ⟩
⟨n ⟩ been a constant
From equation (1.4)
n(2) (r,r1) = ⟨n ⟩2g(R)
Subtracting⟨n ⟩2 from both sides and integrating both sides
with respect to r, and r1
∬v
❑
¿¿
12
¿ ⟨n ⟩(2)∬v
❑
[ g(R)−1 ] dr dr1
¿ ⟨n ⟩2∫ dr∫ [g (R )−1 ] dR
∬v
❑
[n(2) ( r ,r 1)−⟨n ⟩2 ]dr dr1= ⟨n ⟩2V∫v
❑
[ g (R )−1 ]dR (1.7)
Now, we could also write
∬v
❑
[n(2) ( r ,r 1)−⟨n ⟩2 ]dr dr1=∬v
❑
n(2) ( r ,r1 )dr dr1−∬ ⟨n ⟩2dr dr1
From equation (1.5) we have
∬v
❑
n(2 )(r⃗ , r⃗¿¿1)d r⃗ d r⃗1= ⟨N 2v ⟩− ⟨N v ⟩ ¿
∴∬v
❑
¿¿
¿ ⟨N v2 ⟩−⟨N v ⟩−⟨n ⟩2V 2
¿ ⟨N v2 ⟩−⟨N v ⟩−⟨ N v ⟩2
∴∬v
❑
[n(2) ( r ,r1 )−⟨n ⟩2 ]dr dr1= ⟨N v2 ⟩− ⟨N v ⟩− ⟨N v ⟩2 (1.8)
Comparing (1.7) and (1.8)
⟨N v2 ⟩−⟨N v ⟩−⟨ N v ⟩2=⟨n ⟩2V∫
v
❑
[g (R )−1 ] dR
13
⟨N v2 ⟩−⟨N v ⟩2=⟨n ⟩2V∫
v
❑
[ g (R )−1 ]dR+⟨N v ⟩
Lets write the expression for
⟨ (N v− ⟨N v ⟩)² ⟩=⟨ (N v− ⟨N v ⟩)(N v− ⟨N v ⟩)⟩
=⟨N v2−N v ⟨N v ⟩−N v ⟨N v ⟩+ ⟨N v ⟩ ² ⟩
=⟨N v2−2N v ⟨N v ⟩+ ⟨N v ⟩ ² ⟩
=⟨N v2 ⟩−2 ⟨N v ⟩ ⟨ N v ⟩+ ⟨N v ⟩ ²
=⟨N v2 ⟩−2 ⟨N v ⟩ ²+ ⟨N v ⟩ ²
¿ ⟨N v2 ⟩−⟨N v ⟩ ²
From (1.9) we can write
⟨ (N v− ⟨NV ⟩ )2 ⟩= ⟨n ⟩2 v∫v
❑
[g (R )−1 ] dR+⟨N v ⟩
Divide both sides by ⟨ N v ⟩2
⟨ (N v− ⟨N v ⟩ )2 ⟩⟨N v ⟩2
= 1
⟨N v ⟩2¿
=1
⟨N v ⟩2¿
14
Since ⟨n ⟩2 v=⟨n ⟩ ⟨n ⟩ v= ⟨n ⟩ ⟨N v ⟩
¿⟨ N v ⟩⟨N v ⟩2
+⟨n ⟩ ⟨N v ⟩
⟨N v ⟩2∫v
❑
[ g (R )−1 ]dR
=1
⟨N v ⟩+
⟨n ⟩⟨N v ⟩∫v
❑
[g (R )−1 ] dR
∴⟨ (N v− ⟨N v ⟩)2 ⟩
⟨N v ⟩2= 1
⟨N v ⟩¿ (1.10)
Equation (1.10) is the mean square fluctuation. It tells us
the relationship between macroscopic quantity (on the left hand
side) and microscopic quantity (on the right hand side).
g(R )usually approaches 1 quite rapidly with increasing R. Thus,
when the volume V of the system is sufficiently large and v is a
small part if it, then the integral on the right-hand side of (1.10)
is independent of v (we assume that v is much larger than the
size of a molecule).
However, when v is comparable with the total volume V, the
above mentioned integral depends on v. If v is equal to V , then Nv
equals N and we have no fluctuation, so that the right hand side
of (1.10) much vanish.
15
In n(2)(r,r1), ⟨ δ (r j−r )∑ K (≠1 )δ (rk−r) ⟩ is a term which expresses
the probability density at r1,when the particle j = 1 is at r.
Therefore n(1)(r1)g(R) is the density at r¹. under the condition that
a particle is at r. For example, for classical independent
particles, when a particle is at r, the remaining N-1 particles are
in the total volume V and thus
⟨n ⟩ g (R )= N−1V
g (R )= N−1⟨n ⟩ v
= N−1N
=1− 1N
Therefore
⟨n ⟩∫v
❑
[ g (R )−1 ] dR=⟨n ⟩∫v
❑
[1− 1N
−1]dR
Since g (R )=1− 1N
⟨n ⟩∫v
❑
[ g (R )−1 ] dR=⟨n ⟩∫v
❑ −1N
dR
=−⟨n ⟩N
∫dR
= −⟨n ⟩V
N
16
Recall N ≈ N v= ⟨n ⟩V
∴ ⟨n ⟩∫v
❑
[ g (R )−1 ]dR=−⟨n ⟩ v⟨n ⟩V
⟨n ⟩∫v
❑
[ g (R )−1 ] dR=−vV
(1.11)
From equation (1.10)
¿¿
¿ 1
⟨N v ⟩ [1+[−vV ]]
¿ 1
⟨N v ⟩ [1+[−vV ]]
¿ 1
⟨N v ⟩[1−P ]
¿¿
Where v/V = P is the probability for a particle to be in the
volume v. By (1.10), the square of the relative fluctuation is thus
(1 – P) ∕ (Nv).
Another extreme case is a system of closely packed hard
spheres where fluctuation is nearly impossible and we must
have, ⟨n ⟩∫ [ g (R )−1 ] dR≈−1
17
Taking the Fourier transform of the deviation of the local
density n(r) from its mean value⟨n(r⃗ )⟩,we can calculate the mean
square of the Fourier component of the density fluctuation. Thus
obtain
{∫v
❑
[n (r )− ⟨n ⟩ ]exp (– if . r )dr ¿2} ¿ ⟨n ⟩ v {1+ ⟨n ⟩V∫ [ g (R )−1 ]exp (−if .R )dR } (1.12)
Equating (1.10, 12) are relations which connect
macroscopic quantities (the left hand side) to the microscopic
radial distribution function g(R) on the right-hand side of these
equation.
1.4 CALCULATION OF PAIR CORRELATION FUNCTION
BY COMPUTER SIMULATION METHOD
The radial distribution can be computed either via computer
simulation methods like the Monte Carlo method or via the
Ornstein-Zernike equation, using the approximative clouser
relations like the Percus-Yevick approximation or the
Hypernetted chian theory. It can also be determined
experimentally, by radiation scattering techniques.
18
1.4.1 THE CLASSICAL-MAP HYPERNETTED CHAIN
METHOD
(CHNC Method)
This is a method used in many-body theoretical physics for
interacting uniform electron liquid in two and three dimensions,
and for interacting hydrogen plasma. The method extends the
famous hypernetted-chain method (HNC) introduced by J.M. Van
Lecuwen et al to quantum fluids as well. The classical HNC,
together with the percus-yevick approximation, are the two
pillars which bear the burnt of most calculations in the theory of
interacting classical fluids. Also, HNC and PY have become
scheme in the theory of fluid, and hence they are of great
importance to the physics of many-particle systems.
The HNC and PY integral equation provides the pair
distribution functions of the particles in a classical fluid, even for
very high coupling strengths. The coupling strength is measure
by the ratio of the potential energy to the kinetic energy. In
classical fluid, the kinetic energy is proportional to the
temperature. In quantum fluid, the situation is very complicated
19
as one needs deal with the quantum operators, and matrix
elements of such operators, which appear in various perturbation
methods based on Feynman diagrams.
In the CHNC method, the pair distribution of the interacting
particles are calculated using a mapping which ensures that the
quantum mechanically correct non-interacting pair distribution
functions is recovered when the coulomb interaction are
switched off. The value of the method lies in its ability to
calculate the interacting pair distribution function g (R )at zero and
infinite temperatures.
Comparison of the calculated g (R )with results from Quantum
Monte Carlo show remarkable agreement even for very strongly
correlated systems.
1.4.2 MONTE CARLO SIMULATION
A trial wave function of the Slater-Jastrow type, with the
long-range part of the two body term modified to account for the
an isotropy of the system. The one-body term is optimized so that
20
the electronic density from vibrational and diffusion Monte Carlo
calculation in the local density approximation and the surface
energies to the results obtained using the Langreth-mehl and
perdew-wang generalized gradient approximation at high
densities. They agree with the results of the Fermi-hypernatted-
chain calculations. The pair correlation function at region near
the surface are tabulated, showing the anisotropy of the
exchange-correlation hole in regions of fast varying densities.
1.4.3 RADIATION TECHNIQUES
In other to obtain a pair distribution function from
crystalline material, 2g of the bulk N1 powder (99.9% Sigma
Aldrich) was measure at room temperature for 15 minutes.
Nickel in a strong neutron scatter which often used as a standard
sample for calibration purpose in neutron diffraction. The
sample was measured in the standard vanadium sample holder
with diameter 6mm. the data was normalized by scattering from
vanadium and background scattering from the empty container
was subtracted. The experimental pair distribution function was
21
obtained by a Fourier transform of the total scattering structure
function up to a high momentum transfer.
1.5 APPLICATIONS OF THE PAIR CORRELATION
FUNCTION IN STATISTICAL PHYSICS
1. The pair correlation function can be used to calculate some thermodynamic quantities like; energy, pressure and compressibility
2. It tells us about the structure of complex isotropic systems.3. It determines the thermodynamic quantities at the level of
the pair potential approximation.4. It can be measured in neutron and x-ray diffraction
experiments.
1.6 RELATIONSHIP BETWEEN THE RADIAL
DISTRIBUTION FUNCTION, ENERGY, COMPRESSIBILITY
AND PRESSURE
Once we know g (r ) ,we can derive all non-entropic
thermodynamic properties.
Energy
22
The simplest is the energy:
U=U∫¿+
32N K BT +
12N
VN ∫
0
∞
dr 4 πr2g (r )φcr ¿ (1.13)
Where
Uint = internal energies of the molecules. The second term
originates from the energy interactions. The average total
potential energy equal 12N times the average interaction of one
particular molecule with all others; the factor ½ serves to avoid
double counting. The distance contribution of all particles in a
spherical shell of thickness dr at a distance r to the average
interaction of one particular particule with all others is
4 π r2(N /V ) g(r )φ(r ). Integrating finally yields equation[6]..
φ (r ) is called the pair interaction between its constituent
molecules.
Compressibility
The isothermal compressibility KT is defined as
KT≡−1V ( dVdp )
T ,N (1.14)
23
From thermodynamics, it is known that KT can be linked to
spontaneous fluctuation in the number of particles in an open
volume V. )
⟨N ⟩ ρ KBT KT=¿ (see equation (1.5)
Where the pointy bracket indicate a long time average or an
average over many independent configurations commensurate
with the thermodynamic conditions. (In this case constant
temperature T and Volume V).
∫v
❑
d3 r1∫v
❑
d3r 2ρ2g ( r12)=⟨ N (N−1 ) ⟩=⟨ N2 ⟩−⟨N ⟩
Where r12 = 1r1 – r21
We can use this to link the compressibility to the radial
distribution function.
⟨N ⟩ ρ KBT KT=ρ∫v
❑
d3 r1ρ∫v
❑
d3 r2g (r 12)+ ⟨N ⟩−ρ∫v
❑
d3r 1ρ∫v
❑
d3 r2
¿ ρ∫v
❑
d3 r1ρ∫v
❑
d3 r2(g (r12 )−1)+ ⟨N ⟩
¿ ρ∫v
❑
d3 r1ρ∫R3
❑
d3 r (g (r❑)−1)+ ⟨N ⟩
Dividing through by ⟨N ⟩
24
⟨N ⟩ ρKBT KT
⟨N ⟩ =1
⟨N ⟩ [ NV .Vρ∫R 3
❑
d3r (g (r )−1)+ ⟨N ⟩ ]¿1
⟨N ⟩ [Nρ∫R 3
❑
d3 r (g (r )−1)+ ⟨N ⟩]¿
⟨N ⟩⟨N ⟩ [ ρ∫R3
❑
d3 r (g (r )−1)+1]¿1+ρ∫
R3
❑
d3 r (g (r )−1)
∴ ρKBT KT=1+ ρ∫R 3
❑
d3r (g (r )−1) (1.15)
Equation (1.15), is the so-called compressibility equation that
shows that the compressibility of a fluid is intimately connected
to the radial distribution function of its constituent molecules.
Pressure
We can now consider the pressure of a fluid. If the density of
fluid is not too high, correlations between three or more particles
may be ignored, in which case the radial distribution function is
given by
g (r )≈ exp(−β φ( r )) (1.16)
25
Where φ (r ) is the pair interaction potential. Also, for not too high
densities, the pressure of a fluid is to a good approximation given
by the first two terms in the viral equation.
PV=NKBT (1+B2 (T ) NV ) (1.17)
Where B2(T) is called the second viral coefficient. Our goal now is
to link B2(T) to the radial distribution function g (r ) or pair
interaction φ (r ). This may be achieved by differentiating the viral
equation with respect to v.
PV=NKBT (1+B2 (T ) NV )
ddV
(PV )=−NK BT B2(T ) ddV ( NV )
vdpdV
+P dVdV
=−NK BT B2(T ) NV 2
(1.18)
( dpdV )N , T
V +P=−NKBT B2(T ) NV 2
Recall, KT≡−1V ( δVdP )
T , N
(1.14)
26
−KTV ≡( δVdP )
−1KTV
≡( δVdP )(1.19)
Substitute equation (1.19) into (1.18)
−1KT
+P=−NKBT B2(T ) NV 2 (1.20)
Recall also,
PV=NKBT (1+B2 (T ) NV ) (1.17)
P=NKBT
V (1+B2 (T ) NV )
(1.21)
Substitute (1.21) into (1.20)
∴−1KT
+NKBT
V (1+B2 (T ) NV )=−NK BT B2(T ) N
V 2
−1KT
+NKBT
V+NKBT B2 (T ) N
V 2=−NKBT B2(T ) NV 2
−1KT
+NKBT
V=−NK BT B2(T ) N
V 2−NKBT B2(T ) NV 2
27
−1KT
+ρK BT=−2NKBT B2 (T ) NV 2 (1.22)
Since ρ=NV
Multiply equation (1.22) by KT
−1+ρKBT K T=−2K BT B2(T ) NV 2
KT
−1+ρKBT K T=−2 B2(T ) NV
ρKBT K T=1−2B2(T )NV
(1.23)
Since NK BK T
V≈1
Comparing the two equation (1.15) and (1.23), we can write the
second viral coefficient as a three dimensional integral over the
pair interaction φ(r )
We have
1+ρ∫R3
❑
d3 r (g (r )−1 )=1−2B2(T ) NV
ρ∫R3
❑
d3 r (g (r )−1 )=−2B2(T ) NV
28
Recall ρ=NV
∫R 3
❑
d3 r (g (r )−1 )=−2B2(T )
B2 (T )=−12∫R3
❑
d3 r (g (r )−1 )
B2 (T )=−12∫R3
❑
d3 r (e−βφ(r )−1 ) (1.24)
Since g (r )≈ exp(−β φ( r )) (1.16)
Equation (1.24) is important because it allows us to calculate the
pressure of a fluid knowing only the pair interactionφ (r ) between
its constituent molecules.
If we consider a small volume v << V of a large system in
thermal equilibrium. The fluctuation of the number of molecules
in v is related, in thermodynamics to the isothermal
compressibility [7].
KT=−1V ( dvdP )
T , N(1.14)
⟨ (N v− ⟨N v ⟩)2
⟨N v ⟩2 ⟩= KT KT
V(1.25)
29
Where K = KB = Boltzman constant
T = temperature
v = volume
Recall that,
⟨ (N v− ⟨N v ⟩)2
⟨N v ⟩2 ⟩= 1
⟨N v ⟩ {1+ ⟨n ⟩∫ [ g (R )−1 ]dR } (1.10)
Equating (1.10) and (1.25)
KT K T
v= 1
⟨N v ⟩ {1+ ⟨n ⟩∫ [ g (R )−1 ]dR }
Multiply through by v
KT KT=v
⟨N v ⟩ {1+⟨n ⟩∫ [ g (R )−1 ] dR }
¿ v
⟨N v ⟩+v ⟨n ⟩⟨N v ⟩∫ [g (R )−1 ] dR
¿ v
⟨N v ⟩+∫ [ g (R )−1 ] dR (1.26)
Since v ⟨n ⟩=⟨N v ⟩ (1.16)
⟨ N v ⟩=v ⟨n ⟩
30
v
⟨N v ⟩= vv ⟨n ⟩
= 1⟨n ⟩
equation (1.26) becomes
¿ 1⟨n ⟩
+∫ [g (R )−1 ] dR
KT KT=1
⟨n ⟩+∫ [ g (R )−1 ]dR (1.27)
Equation (1.27) is the well known Ornstein-Zernike relation or
compressibility equation. This relation applies to a quantum-
mechanical systems as well.
For usual liquids, compressibility is very small
(KT KT ⟨n ⟩ ≪1) which mean ⟨n ⟩∫ [ g (R )−1 ] dR=1
For classical gas subject to Boyles-Charles law
¿
KT KT=1
⟨n ⟩
−KTV ( dvdP )= 1
⟨n ⟩ (1.28)
Recall equation (1.6)thus, (1.28) becomes,
31
−KTv ( dvdP )= v
⟨ N v ⟩
−KT
v2 ( dvdP )= 1
⟨ N v ⟩
−⟨ N v ⟩ KT dv
v2=dp
Integrating both sides, we have
−⟨ N v ⟩ KT∫ dv
v2=∫dp
−⟨ N v ⟩ KT−v=p
⟨N v ⟩ KTv
=p
P= ⟨n ⟩KT (1.29)
Equation (1.29) is the Boyles-Charles law which is obtained if we
assume there is absence of correlation.
If we assume a classical gas of hard spheres, molecules
cannot come closer than their diameter D. for dilute gas we may
approximately put.
g(R){0(R<D)1(R>D)
32
Recall, equation (1.27)
KT KT=1
⟨n ⟩+∫ [ g (R )−1 ]dR
Putting g (R )=1, we have
∫(g (R )−1)dR
¿∫ (1−1 )dR=∫ dR (0 )=0
Putting g(R) = 0 we have,
¿∫ [0−1]dR=∫−dR=−∫ dR=v
Where v=4 πD3
3 (volume of a sphere)
equation (1.27) becomes
KT KT=1
⟨n ⟩ −4 πD3
3
−KTv ( dvdp )= 1
⟨n ⟩ −4πD3
3
Multiplying through by 1/v we have
−KTv ( dvdp )= 1
⟨n ⟩ v−4 πD
3
3 v
¿ 1⟨N v ⟩
−4 πD3
3v
33
¿V−4πD3 ⟨ N v ⟩3V ⟨ N v ⟩
(1.30)
Let q=4/3 πD3 ⟨N v ⟩
Equating (1.30) becomes
−KT
v2 ( dvdp )= v−qv ⟨N v ⟩
−KTv ( dvdp )= v−q
⟨N v ⟩
−KT ⟨N v ⟩dvv (v−q)
=dp
Integrating both sides we have
−KT ⟨ N v ⟩∫ dvv (v−q)
=∫ dp
Using integration by partial fraction on ∫ dvv (v−q), we have
∫ dvv (v−q)
=∫( Av + Bv−q )dv
¿∫ A (v−q )+B(v)v (v−q)
¿∫ Av−Aq+Bvv (v−q)
34
¿∫ Av+Bv−Aqv (v−q)
(A + B)v = 0A + B = 0−Aq=1
A=−1q
∴ A+B=−1q
+B=0
B=−1q
∴∫ [ Av + Bv−q ]dv=∫ [−1vq + 1
q (v−q) ]dv
∫ [ 1q (v−q)
− 1vq ]dv
∫ dvv (v−q)
=1q∫ [ 1
v−q−1v ]dv
∴−KT ⟨ N v ⟩
q[¿ (v−q )−¿v ]=p
−KT ⟨N v ⟩q
⌊∈( v−qv )¿¿
KT ⟨N v ⟩q
⌊∈( vv−q )¿¿ (1.31)
35
We have
¿( vv−q )=¿ [ 1
1−q/ v ]
Putting t=qv
We have
¿( 11−q/ v )=¿ [ 11−t ]
Expanding ¿ [ 11−t ] in series form and neglecting higher terms we
have
¿ [1+z ]=z− z2
2+ z3
3− z 4
4+. . .
∴∈[ 11−t ]=¿ [1+ t1−t ]=t (1−t)−1−
t2(1−t)−2
2+ .. .
But (1 – t)-1 = 1 + t + t2 + . . . ∴∈[1+ t
1−t ]=t (1+t+t¿¿2+ .. .)− t2
2(1+2t ¿¿2)+ .. .¿¿
Considering only lower terms we have
t+ t2− t 2
2
36
¿ [ 11−t ]=t (1+ t2 ) (1.32)
Substituting equation (1.32) into (1.31)
KT ⟨N v ⟩q
⌊ t (1+ t2 )¿¿
Recall t=qv
KT ⟨N v ⟩q
⌊ qv (1+ q
2v )¿¿
Also q=43 π D 3 ⟨ N v ⟩
KT ⟨N v ⟩v
⌊1+4 π D3 ⟨N v ⟩3.2 v
¿¿
KT ⟨N v ⟩v [1+2 π D3 ⟨N v ⟩
3 v ]=p
KT ⟨N v ⟩v [(1+ 2π D3 ⟨N v ⟩
3v )−1]=p
KT ⟨N v ⟩v
1
[1+ 2π D3 ⟨N v ⟩3v ]
−1=p
p=KT ⟨ N v ⟩
v ⌊1+(−2π D3 ⟨ N v ⟩3v )¿
¿
Since the series expansion of
37
[1+ 2 π D 3 ⟨ N v ⟩3 v ]
−1
=[1+(−2π D3 ⟨N v ⟩
3v ]And also neglecting higher terms
∴ p=KT ⟨N v ⟩
[v−2π D3 ⟨N v ⟩3 v ] (1.33)
let b=2 π D 3 ⟨ N v ⟩
3
Equation (1.32) becomes
p≈KT ⟨N v ⟩v−b
(1.34)
Equation (1.34) is the Vander waal’s equation that expresses the
pressure in a classical gas of hard sphere. Vander waal’s
equation is valid for quantum mechanicals because correlation
exist even when there are no molecular forces in which case
Boyles-Charles law is not applicable.
REFERENCES
38
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40