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•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.

2

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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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

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.

E m r/ 3 (10.1)

If we place another atom at r, then E will distort its orbit and produce a

dipole moment

Em (10.2)

So these two atoms produce interaction energy

6/')( rmmmrU E' (10.3)

(10.4) m r2 6/

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Although m and m each has a zero average, the average of m2 is nonzero, and

this is the reason for the r-6 attractive potential.

The short-distance repulsion and attraction result in the interaction potential

as shown in Fig. (10.1), with a minimum - at r=r0.

The “6-12 potential or Lenard -Jones potential” is commonly used as an

approximation.

612

r

b2

r

brU )( (10.5)

For instance, the function U(r) given by this formula exhibits a “minimum”, of

value -, at a distance (r0 =b), and rises to an infinitely large (positive) value for

r<b and to a vanishingly small (negative) value for r>>b .The portion to the

left of the “minimum” is dominated by the repulsive interaction.

0 r

-

r0

U(r )

0 r

-

r0

U(r )

5

0 r

-

r0

U(r )

0 r

-

r0

U(r )

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.

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Table 10.1. Minimum interaction energy and its distance

  r0(A) (J)

He 2.2 110 -22

H2 2.7 4

Ar 3.2 15

N2 3.7 13

CO2 4.5 40

Some Lennard-Jones potential examples of application are listed in Table 10.1. Even this crude interaction model has extensive applications. This model can explain many properties of gases, solids and liquids quite well.

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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 of obtaining information about intermolecular

potential is from the transport properties of a gas, for example

from measurements of viscosity.

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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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”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.

A second way of learning about intermolecular potential is from the equation of state; by studying deviations from the ideal gas ideal gas equationequation PV=NkT.PV=NkT.

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.

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We shall not discuss these “real” problems“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 density of the gas

is increased.is increased. Then, the equation of state may be written in the

form

1

13

)(l

l

l vTa

kT

P v (10.6)

where v(=V/N) denotes the volume per particle in the system. The expansion (10.6) is known as the virial expansionvirial expansion of the system while the numbers aall(T)(T) are referred to as the virial virial

coefficientscoefficients.. (For detailed description of virial equationvirial equation and derivations of virialvirial coefficients see R.K Pathria, Statistical Mechanics).

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11 a (10.7)

a e r drU r kT2 3

2

0

21

( )( )/

(10.8)

a f f f d r d r3 6 12 13 233

123

1300

1

3

(10.9)

and so on. Here ffijij is the two-particle functiontwo-particle function defined by the relationship

The virial coefficients have the following form:

f eijU kTij /

(10.10)

where the potential Uij is a function of the relative position vector rrijij(=(=rrjj--rrii););

however, if the two-body force is a central oneif the two-body force is a central one,, then the function Uij depends

only on the distancedistance rrijij between the particles.

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In the absence of interactions, the function ffijij is identically equal to zero; 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 aa1111 the lowest-order virial coefficient

that we need to consider here is aa22 , which is given by formula (10.8).

a e r drU r kT2 3

2

0

21

( )( )/

U(r)U(r) being the potential energy of interparticle interaction. One of the typical

semi-empirical potential function is the Lennard-Jones potential (10.5).

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U rb

r

b

r( )

12 6

2

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

)rfor(r U(r) 0(10.11)

which amounts to attributing an impenetrable core, of diameter r0, 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

2U rrUrU 06

00 ,)/()( (10.12)(for r r0)

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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 r drU

kT

r

rr dr

r

r

2 32 0 0

62

0

21

0

0

exp (10.13)

The first integral is quite straightforward; the second one is considerably

simplified if we assume that

1/kT)(U0 (10.14)

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which makes the integrand very nearly equal to -(U-(U00/kT)(r/kT)(r

00/r)/r)66. Equation

(10.13) then gives

Substituting this expression for aa22 into the expansion (10.6), we obtain as a

first-order improvement on the ideal-gas law

ar U

kT203

302

31

(10.15)

v

)T(B

v

kT

kT

U

v

r

v

kTP 20

30 11

3

21

(10.16)

The coefficient BB22,, which is also some times referred to as the second virial

coefficient of the system, is given by

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(b) (UU00/kT)<<1/kT)<<1. We cannot expect, therefore, formula (10.17) to be a

faithful representation of the second virial coefficientsecond virial coefficient of a real gas.

Nevertheless, it does correspond, almost exactly, to the van der Waals van der Waals

approximationapproximation to be equation of state of a real gas. This can be seen by

rewriting eqn. (10.16) in the form

kT

UraB 0

303

22 13

2 (10.17)

130

30

20

30

3

21

3

21

3

2

v

r

v

kT

v

r

v

kT

v

UrP

(10.18)

In our derivation it was explicitly assumed

(a) the potential function U(r)U(r) is given by the simplified expressions

(10.11) and (10.12), and

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Pa

vv b kT

2 ( )P

a

vv b kT

2 ( ) (10.19)

which readily leads to the van der Waals equation of state

where

ar u

2

303

0ar u

2

303

00

30 v4

3

r2b

0

30 v4

3

r2b

and (10.20)

We note that the parameter bb in the van der Waals equationvan der Waals equation of state is

four times the actual molecular volumeactual molecular volume vv00, the letter being the “volume of “volume of

sphere of diameter sphere of diameter rr00”.

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Note that in this derivation we have assumed that b<<vb<<v00, 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 van

der Waals constantsder Waals constants aa and bb 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. 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 eeii be

the electric charge of the i-th particlei-th particle, then the electrostatic potential of a

collection of particles is

Ue e

ri j

i ji j

12

,,

(10.21)

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Since there is no length scale, we can enlarge or contract the body. Let

rrijij==rrijij,, then UU will acquire a factor 1/1/ in the new scale. If UU is negative

then all particles will coalesce into one point and U U - - .

Obviously, other effects must exist; otherwise, the electrostatic interactionelectrostatic interaction alone cannot describe stable matter.

The thermodynamic limit to be established requires the following conditions:

If UU is positive then the charges will fly-off to infinity to lower the energy.

a) the total charge is zero, i.e.

b) kinetic energy is quantum mechanical,

c) at least one type of charge (positive or negative ) is fermionic.

eii 0

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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.

2 22/ Mrkinetic energy ~ (10.22)

where r=extentr=extent of the position. So, if rr 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 NN particles with positive charge ee and N N with negative

charge -e-e. Also suppose that there is a repulsion at short distancerepulsion at short distance preventing

the charges from approaching each other indefinitely.

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Let be the densities of positive and negative chargesdensities of positive and negative charges. Let ((rr)) be the

electric potentialelectric potential.. Then

n( )r

n ne ne )/kT ne )/kT( ) ((r rr (10.23)

where nn is the average density of the positive and negative charges.

The relation of ((rr)) with the density of the total charge is obey to the Poisson Poisson

equationequation and can be written in the following way

))()(()( rrr nne4(10.24)

Substitution of (10.23) into (10.24) will lead to

/ rD2 (10.25)

re n

kTD2

28

r

e n

kTD2

28

(10.26)

where

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The solution can be presented in the following way

If there are no other particles, a charge at the origin produces a potential e/re/r.. Hence the effect of the particles is to reduce the electric potential far away.

Now we have a new length scale rrDD,, the so-called screening length (Debye (Debye

Huckel radius).Huckel radius). Around the origin there is a cloud of charges of density

( ) /r e

re r rD (10.27)

Dr/r

D

err

e/)nn(e

24

4

(10.28)

This is the screening layer. The integral of (10.28) over 44rr22 is -e-e.. Hence the

charge at the origin is screened and its effect outside rD tends to zero. An

increase in temperature increases rrDD and (10.26) is more accurate. Notice that

from (10.26) we have

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3/23

3/12

)(

1)8(

nrkT

ne

D

(10.29)

The dominator on the right is the number of particles in the screening layer number of particles in the screening layer

raised to the powerraised to the power.. Therefore, the above approximation is valid when r3D n is

large, but remains as a low density approximation.

The functions n(n(rr)) 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

The left hand side is approximately the ration of the interaction energy ee22/r/r to

the kinetic energy kTkinetic energy kT ( here r r distance between the particles ~ nn-1/3-1/3).).

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Therefore the(10.30) can be regarded as a conditional probability.

Now if the nn((rr)) are conditional distributions. The density correlation

functions can be calculated in the following way

Drrc e

rT

ne /22

)0()(< r (10.31)

=(Probability of a molecule at =(Probability of a molecule at rr=0)=0)(the (the

conditional probability that there is a particle at conditional probability that there is a particle at rr given that given that

there is a molecule at 0).there is a molecule at 0). n nn(n(rr0). 0).

( ) ( )r 0

n n( ) ( ) ( ) /r r0 0 (10.30)

This is the density distribution given that a particle is at rr=0.=0. Notice that

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In this case, the thermodynamic quantities will be written in the following

way

U N V T U 3 2 1 2 1 2/ / / , E = E 0(10.33)

From this result, we can calculate the specific heat and thermodynamic

potential, etc. Notice that

2/12/332/12

3

23

)()8(8

=

))()())(()((2

kTneVNr

e

πr

kT-V

r

erd

VU

DD

00rr(10.32)

It can be seen that the correlation length of this gas is rrDD. Originally, the

electrostatic interaction had no length scale; and the new scale rrDD is a

function of density, temperature, and electric charge.

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If these charge particles are the solute in solution, e.g NaNa++ and Cl 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 ee22 must be

divided by the dielectric constant . In water =80=80 and quantities in (10.34)

now refer to ions. The pressure pp becomes the osmotic pressure.

The subscript 00 means quantities evaluated when e=0e=0.

NU

VUpp

/

/

0

31

0

T/UccT

EVV 2

10

S S U V 013 /

UFF 32

0

(10.34)