Transcript Document
Lecture 10
•Non Ideal systems. •Intermolecular interactions. •Short Distance and Long Distance Interaction. •Lenard-Jones potential. •Corrections to the Ideal Gas Law. •Van der Waals equation. •The Plasma Gas and Ionic Solutions. •The Debye-Huckel radius.
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When we discuss the ideal gas , we regard the interaction between molecules as causing collisions, but we neglect the duration of collision and other details. As a result, collisions only lead to changes of the moment of the molecules. However, the effect of interaction is more than this. The interaction between molecules naturally depends on the structure of the molecules. For simplicity, we shall only consider the interaction between simple molecules..
If two molecules are so close that their electronic shells touch, a strong repulsion is produced because of the fermionic character of the electrons If the two molecules are very far apart, there is a weak attractive force .
This is mainly due to the electric dipole interaction . An atom alone has no electric dipole (i.e. the center-of-mass of the electrons coincides with the molecules), but this is only a time averaged property - the dipole changes rapidly, averaging to zero.
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If there are two atoms, the electrons will mutually interact. The interaction energy is proportional to r -6 , where r is the distance between the atoms or molecules. This r -6 interaction can be understood as follows: Consider an atom at the origin. It possesses a fast rotating dipole
m
. The electric field at
r
is
E
m r
3 (10.1) If we place another atom at produce a dipole moment
m
E
r
, then
E
will distort its orbit and (10.2) So these two atoms produce interaction energy
U
(
r
)
m
'
E
m
'
m
/
r
6 (10.3)
m
2 /
r
6 (10.4) 3
Although m and m each has a zero average, the average of nonzero, and this is the reason for the r -6 attractive potential.
m 2 The short-distance repulsion and attraction result in the interaction potential as shown in Fig. (10.1), with a minimum The “6-12 potential or Lenard -Jones potential” at r=r 0 .
is commonly used as an approximation.
is U(r )
U
(
r
)
b r 12
2 b r 6
(10.5) For instance, the function U(r) given by this formula exhibits a “minimum”, of value large (positive) value , at a distance ( r 0 =b), for r>b .
The portion to the left of the “ minimum” value is dominated by the repulsive interaction .
-
4
U(r )
r 0
r 0
-
That comes into play when two particles come too close to one anther, while the portion to the right of the minimum is dominated by the attractive interaction that operates between particles when they are separated by the respectable distance. 5
Some Lennard-Jones potential Table 10.1. Even this crude interaction model has extensive applications. This model can explain many properties of gases, solids and liquids quite well.
examples of application are listed in
Table 10.1. Minimum interaction energy and its distance
r 0 (A)
(J)
He 2.2
1
10
-22 H 2 Ar N 2 CO 2 2.7
3.2
3.7
4.5
4 15 13 40
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A weak attractive force between atoms or nonpolar molecules caused by a temporary change in dipole moment arising from a brief shift of orbital electrons to one side of one atom or molecule, creating a similar shift in adjacent atoms or molecules.
One way potential of obtaining information about is from the example from measurements of viscosity.
intermolecular transport properties of a gas , for Transport properties depend directly on molecular collisions , i.e. on the law of force governing these collisions. For example, in elementary treatments of transport properties one employs the mean free path, which depends on the collision cross-section.
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A second way of learning about intermolecular potential from the equation of state ; by studying deviations from the
ideal gas equation
PV=NkT.
is It is this aspect which we shall consider in this lecture, generalizing the perfect gas treatment of lectures 6 and 7. We can no longer ascribe a “
private”
energy to each molecule of a real gas since the molecules of a real gas interact with each other. The fact that the molecules interact makes this a very hard problem. For interacting molecules condensation from the gas to the liquid occurs at appropriate densities and temperatures, and one possesses only a very limited understanding of such phase transitions. 8
We shall not discuss these
“real” problems
which occur at high densities, but shall limit ourselves to considering how deviations from perfect gas behavior first show up as the
density of the gas is increased.
Then, the equation of state may be written in the form
P
v
kT
l
1
a l
(
T
) 3
v
l
1 (10.6) where
v(=V/N)
denotes the volume per particle in the system. The expansion (10.6) is known as the
virial expansion
of the system while the numbers referred to as the
virial coefficients
.
a l (T)
are (For detailed description of
virial equation
and derivations of
virial
coefficients see R.K Pathria, Statistical Mechanics).
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The virial coefficients have the following form:
a
1 1
a
3
a
2 2 3 1 3 6 0 0 (
e
( )/
kT
0 1 ) 2
r dr
12 13 23 3 12 3
f f f d r d r
13 and so on. Here relationship
f ij
is the two-particle function defined by the (10.7) (10.8) (10.9)
f ij
e
U ij
/
kT
(10.10) where the potential U ij is a function of the relative position vector
r
ij (=r j -r i );
function U ij however, if the two-body force is a central one , depends only on the distance r ij then the between the particles.
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In the absence of interactions, the function
f ij
zero; is identically equal to in the presence of interactions, it is non-zero, but at sufficiently high temperatures it is quite small in comparison with unity. If a given physical system does not show great departures from the ideal-gas behavior, the equation of state of the system is given adequately by the first few virial coefficients. Now, since
a 1
1
the lowest-order virial coefficient that we need to consider here is
a 2
which is given by formula (10.8).
,
a
2 2 3 0 (
e
( )/
kT
1 )
U(r)
being the potential energy of interparticle interaction . One of the typical semi-empirical potential function is the (10.5).
Lennard-Jones potential 11
b r
12
2
b r
6
For most practical purposes, the precise form of the repulsive part of the potential is not very important; it may as well be replaced by the crude approximation
U(r)
for(r
r 0 )
(10.11) which amounts to attributing an impenetrable core, of diameter r 0 , to each particle. The precise form of the attractive part of the potential is, however, generally significant; in view of the fact that there exists good theoretical basis for the sixth-power attractive potential, this part may be simply expressed as
U
(
r
)
U 0
(
r 0
/
r
)
6
,
U 0
2
(for
r
r
0
) (10.12) 12
The potential given by the expressions (10.11) and (10.12) may, therefore, be used if one is only interested in a qualitative assessment of the situation and not in a quantitative comparison between the theory and the experiment.
Substituting (10.11) and (10.12) into (10.8) we obtain for the second virial coefficient
a
2 2 3
r
0 0 2
r dr
r
0 1 exp
U kT
0
r
0
r
6 2
r dr
(10.13) The first integral is quite straightforward; the second one is considerably simplified if we assume that
(U 0 /kT)
1
(10.14) 13
which makes the integrand very nearly equal to Equation (10.13) then gives
-(U 0 /kT)(r 0 /r) 6
.
a
2
2 3
r
0 3 3
1
U kT
0 (10.15) Substituting this expression for as a first-order
a 2
into the expansion (10.6), we obtain improvement on the ideal-gas law
P
kT v
1 2
r
0 3 3
v
1
U
0
kT
kT v
1
B
2
( T ) v
(10.16) The coefficient
B 2 ,
which is also some times referred to as the second virial coefficient of the system, is given by 14
B
2
a
2
3 2
r
0 3 3 1
U
0
kT
In our derivation it was explicitly assumed (10.17) (a) the potential function
U(r)
(10.11) and (10.12), and is given by the simplified expressions (b) (
U 0 /kT)<<1
.
We cannot expect, therefore, formula (10.17) to be a faithful representation of the second virial coefficient of a real gas. Nevertheless, it does correspond, almost exactly, to the van der Waals approximation to be equation of state of a real gas. This can be seen by rewriting eqn. (10.16) in the form
P
2
r
0 3
U
0 3
v
2
kT v
1 2
r
0 3 3
v
kT v
1 2
r
0 3 3
v
1 (10.18) 15
which readily leads to the van der Waals equation of state
P
a v
2 (
v b
)
kT
where
a
2 3
r u
0 0 3 and
b
2
r 0 3 3
4 v 0
(10.19) (10.20) We note that the parameter
b
in the van der Waals equation of state is four times the actual molecular volume
v 0
, the letter being the “volume of sphere of diameter
r 0
”.
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Note that in this derivation we have assumed that
b<
, which means that the gas is sufficiently dilute for the mean interparticle distance to be much larger than the effective range of the interparticle interaction.
Finally, we observe that, according to this simpleminded calculation, the van der Waals constants
a
and
b
are temperature-independent, which in reality is not true. A realistic study of the second virial coefficient requires the use of the realistic potential such as the one given by Lennard-Jones, for evaluating the integral appearing in (10.13).
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Electrostatic interactions.
The electrostatic interaction is very strong at short distances and is weaker at large distances. However, it can interact with many particles at the same time and therefore cannot be analyzed by a simple expansion.
The electrostatic interaction energy is inversely proportional to the distance between the charges and does not have a characteristic length scale. Let
e i
be the electric charge of the
i-th particle
electrostatic potential of a collection of particles is , then the
U
1 2
e e i j r
(10.21) 18
Since there is no length scale, we can enlarge or contract the body. Let
r ij =
r
ij ,
then
U
will acquire a factor
1/
in the new scale. If
U
negative then all particles will coalesce into one point and
U
-
is . If
U
is positive then the charges will fly-off to infinity to lower the energy. Obviously, other effects must exist; otherwise, the interaction alone cannot describe stable matter.
electrostatic The thermodynamic limit to be established requires the following conditions: a) the total charge is zero, i.e.
i e i
0 b) kinetic energy is quantum mechanical, c) at least one type of charge (positive or negative ) is fermionic.
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Condition (a) means that uncompensated charges will be dispersed (to the surface of the body if possible). Condition (c) avoids the coalescing of all the particles into one point. The requirement of quantum mechanics is this: If the position of a particle is restricted, then its kinetic energy is increased, i.e.
kinetic energy ~ 2 / 2
Mr
2 (10.22) where
r=extent
of the position. So, if
r
is too small, (10.22) will be larger than the potential energy . Thus condition (b) is not enough. It has to be supplemented by the exclusion principle, i.e. condition (c).
Suppose we have
N
particles with positive charge
e
and
N
with negative charge
-e
.
distance Also suppose that there is a repulsion at short preventing the charges from approaching each other indefinitely.
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Let
n
( ) be the the densities of positive and negative charges . Let electric potential .
Then
(r)
be
n
ne
e
(
r
)/kT
ne
(
r
)/kT
n
(10.23) where
n
is the average density of the positive and negative charges. The relation of
(r)
Puasson equation with the density of the total charge is obey to the and can be written in the following way (
r
)
4
e
(
n
(
r
)
n
(
r
)) (10.24) Substitution of (10.23) into (10.24) will lead to /
r D
2 (10.25) where
r D
2
8 2
e n kT
(10.26) 21
The solution can be presented in the following way ( )
e e
r
/
D
(10.27) If there are no other particles, a charge at the origin produces a potential
e/r .
Hence the effect of the particles is to reduce the electric potential far away. Now we have a new length scale screening length (Debye Huckel radius).
r D ,
the so-called Around the origin there is a cloud of charges of density
e ( n
n
)
/
4
e
4
r D
2
r e
r / r D
(10.28) This is the screening layer. The integral of (10.28) over
4
r 2
is
-e .
Hence the charge at the origin is screened and its effect outside tends to zero. An increase in temperature increases
r D
r D and (10.26) is more accurate. Notice that from (10.26) we have 22
( 8
e
2
n
1 / 3 ) 1 (10.29)
kT
(
r D
3
n
) 2 / 3 The left hand side is approximately the ration of the interaction energy
e 2 /r
to the kinetic energy kT ( here
r
distance between the particles ~
n -1/3 ).
The dominator on the right is the number of particles in the screening layer raised to the power .
Therefore, the above approximation is valid when r 3 D n is large, but remains as a low density approximation. The functions
n(r)
are conditional distributions. The condition is that there is a charge at the origin. Let us consider the evaluation of the density correlation function in this case. More detailed we’ll consider the correlation function in the following lectures.
Let 23
n
(
r
0 ) 0 /
n
(10.30) This is the density distribution given that a particle is at
r=0.
Notice that
=(Probability of a molecule at r=0)
(the conditional probability that there is a particle at r given that there is a molecule at 0).
n
n(r
0).
Therefore the(10.30) can be regarded as a conditional probability.
Now if the
n
(r)
are conditional distributions. The density correlation functions can be calculated in the following way <
(
r
)
( 0 )
c
e
2
n
2
rT e
r
/
r D
(10.31) 24
It can be seen that the correlation length of this gas is
r D
.
Originally, the electrostatic interaction had no length scale; and the new scale
r D
is a function of density, temperature, and electric charge.
U
V
2
d
3
r e
2
r
=
-V kT
8
πr D
3 ( (
r
) (
r
))( (
0
) (
0
))
e r D
2
N
V
( 8 ) 1 / 2
e
3
n
3 / 2 (
kT
) 1 / 2 (10.32) From this result, we can calculate the specific heat and thermodynamic potential, etc. Notice that
U
N V
/
T
/
, E = E
0
U
(10.33) In this case, the thermodynamic quantities will be written in the following way 25
E
T
c V
c
0
V
1 2
U / T S
S
0 1 3
p
F
p
0 0
F
0 2 3
U
1 3
U
U
/ /
N V
The subscript
0
means quantities evaluated when
e=0
.
(10.34) If these charge particles are the solute in solution, e.g
Na +
and
Cl -
in a saline solution, the above results can be directly applied. The only correction is the electric charge. In water, because the water molecules have an electric dipole moment, the charge of the ions has already been screened partly, so In water
=80 e 2
must be divided by the dielectric constant and quantities in .
(10.34) now refer to ions. The pressure
p
becomes the osmotic pressure.
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