University of Warwick, Department of Physics
PX 119 Matter: some associated notes
2. Interactions, non-ideal gases and the Liquid State
Force and Potential
Interactions between molecules are the crucial feature driving what would otherwise be
gases to form liquids and solids. A very general way to represent the interaction is in
terms of its potential energy. For simplicity we will focus on the pairwise additive
approximation in which the potential energy is additive over pairs of interacting atoms:
this means we consider every pair of atoms separately. (One atom may be party to many
pairs!)
In the pair approximation two atoms will share a [contribution to the total] potential
energy U r  varying with their [nucleus to nucleus] separation r. The corresponding law
for the force is simply related by calculus.
Suppose we hold atom one fixed at the origin and we hold atom two at distance r ; in
doing so we will have to apply to atom two a force Fexternal   F r  where where F (r ) is
the force on atom two due to the interaction with atom one. (This is so that total force on
atom two is balanced; a similar situation obtains for atom one of course.) Now let us
slowly move atom two by [infinitessimally] small distance dr : the work done must go
into potential energy so
dU  Fexternaldr   F r dr .
It follows that the force is minus the gradient of potential energy,
dU
F r   
.
dr
U(r )
F(r)
equilibrium separation
r
binding energy
Above is our previous sketch of the potential energy with the corresponding force
F (r ) shown. The force is zero when the potential energy is at minimum: this is
mechanical equilibrium, and the potential energy rise from this point to infinite
separation is the binding energy of the pair. The maximally negative force corresponds
to the potential being most steeply uphill: this the limiting limiting stretching force
which the interaction will bear.
University of Warwick, Department of Physics
PX 119 Matter: some associated notes
Sources of Interaction
Attractive
Repusive
Origin
Pauli repulsion
quantum mechanics
Order of magnitude
(per atom)
Many x 103 Kelvin
QM (+ electrostatics)
Many x 103 K
Ionic
electrostatics
Many x 103 K
dipolar
electrostatics
Few x 103 K
fluctuations
Few x 10 K
Covalent, metallic
van der Waals
The electrostatics and quantum mechanics can be left to later modules. We will briefly
discuss the van der Waals interaction because it is the crucial source of attractive
interaction between neutral atoms, and this in turn is what stabilises simple liquids.
I need you to take on trust two simple properties of electric dipoles:
 Dipole p has energy of interaction -p.E with the electric field E due to other
sources;
 Dipole p causes electric field E(r)  p/r 3 at distance r away
and likewise two atomic properties:
 Atoms have fluctuating dipole moments related to the motion of their electrons;
 An atom experiencing electric field E will have (in addition) an induced dipole
moment p   E
Now the story is as follows.
Let atom one have a (fluctuation) dipole p1 ; then atom two will experience an electric
field E 2  p1/r 3 and so will have an induced dipole moment
p induced 2   E2   p1 / r 3 .
This in turn leads to an electric field felt back by atom one which is
E 1  pinduced 2 / r 3   p1 / r 6 .
Finally Atom one then has an energy
2
 p1  E 1   p1 / r 6
Notice two features: the energy is definitely negative, and attractive interaction, and it
falls off as 1 / r 6 .
[ We can take this story a little further if I give you the full expression
  h
U  1 2 2 ,
40 r 3
where  is the range of photon frequencies which can contribute to the fluctuation
interaction between the two atoms.


University of Warwick, Department of Physics
PX 119 Matter: some associated notes
Then there is a nice tie-in with the relativity module. When r / c  1/ the dipole on
atom one will have changed before it feels the response of atom two, and so at large
enough distances this limits the range of frequencies which can contribute in our
interaction calculation above. Substituting this frequency limit   c / r then gives
the retarded van der Waals interaction,
with U  1 / r 7 .
When atoms (or simple molecules) are in solution the van der Waals interaction
systematically favours precipitation. For dissolved species X to bond with itself, the
net energy change is:
2
U net  U XX  2U XS  U SS   X   S   0
as bonds from X to solvent S are lost and from S to S gained. This is particularly
important in colloids, where the particles are large (up to 1m ) and the vdW
interaction becomes larger than kT at room temperature. ]
Attractions -> liquid phase
Classical understanding of this is via the van der Waals Equation [of state], which should
not be confused with his interaction above!
Our starting point will be the ideal gas equation written as
pint ernalVeffective  NkT
with key modifications as below.
(1) We assume a short range attractive interaction between nearby atoms. The energy
associated with this for each molecule will be proportional to density of others, giving for
the gas as a whole
U attraction  aN 2 / V .
The effect of this on the pressure can be found from a simple work argument:
pint ernal dV  pdV  dU attraction   p  a N 2 V 2 dV :
when the gas expands the ideal gas pressure inside does work both against the walls and
contributing to U attraction .
(2) Molecules exclude each other from
their immediate vicinity, known as the
Excluded Volume Effect. Around each
atom no other atom centre can approach
closer than one atomic diameter, so
around it there is an exclusion zone of
8vatomic .
x
University of Warwick, Department of Physics
PX 119 Matter: some associated notes
There is a subtle overcount here. For any particular pair of atoms, it suffices to count A
excluding B and not both ways round. The result is that each atom sees an effectively
reduced volume Veff  V  Nb where the excluded volume per atom is b  4vatomic .
Assembling this, and working on a per molecule basis for the volume, leads finally to the
van der Waals equation of state,
p  a V 2 V  b   kT .
kT
a
 2 helps to understand the shape of the isotherms.
Rewriting this as p 
V b V


p
High T
Tc
Low T
b
V
The first term blows up to infinity as V  b , and at high enough T it dominates the
whole curve. As T is lowered, the second term becomes more important and for T  Tc
this changes the slope of p vs V for intermediate values of V.
What the equation appears to predict is that at low enough temperatures pressure
increases as volume increases, which we will now see is unphysical and leads to
separation into liquid and gas.
Imagine a box containing substance with this strange behaviour, and supplied
with a central partition.
If the partition moves slightly to the right, the volume goes up on the LHS and so
does the pressure, whilst on the RHS volume and so pressure go down. The
result is a pressure imbalance acting to further displace the partition.
Clearly this situation is unstable, and what happens is that the partition moves
rightwards until the two sides are at different densities which have the same
pressure and the conventional sign of pressure vs volume.
University of Warwick, Department of Physics
PX 119 Matter: some associated notes
Returning to our isotherms, the result is that for any volume per molecule in between
points A and B on the daigram below, the system separates into some liquid at the higher
density corresponding to A and some gas at the lower density corresponding to B.
p
B
A
V
There is one remaining puzzle because there appears to be some choice about the line AB
as the dotted lines in the figure suggest. Maxwell pointed out that by taking the sample
round a (figure of eight) cycle, from A to B along the curve and then back along the
straight line, there had better be no net work done or else this or the reverse would be a
free source of energy, that is
 pdV  0
meaning that the two shaded areas in the figure have to be equal.
[A century later Pippard objected that as the curve from A to B was unphysical,
"Maxwell's Construction" was unrealisable for any experimental material!
Nevertheless the work of Gibbs (long before Pippard) clearly establishes that
there are other routes between points A and B that are realisable, and lead to the
same answer as Maxwell.]
To complete the picture let us go back to the phase diagram in the pressure-temperature
plane.
p
B
A
Tc
T
Each isotherm is a vertical line in the phase diagram, as shown dotted, with the points A
and B coinciding across the liquid-gas phase boundary.
The vdW equation is a qualitative success at explaining the liquid gas transition. It is
only approximate, however. It more drastically fails at high densities, predicting the
University of Warwick, Department of Physics
PX 119 Matter: some associated notes
pressure to diverge when the volume per molecule has reduced to V  b  4vatomic , which
is much too soon. The error is because when particles are close together, their volumes
of exclusion to others overlap, and the total excluded volume is thus overestimated in the
vdW calculation.
x
This reduction in exclusion to others by two coming close is a real effect. In colloids it
drives depletion flocculation, and in simple liquids it is the key driving force for the
structure seen in the radial distribution function.
[At the critical point, where the distinction between liquid and gas collapses, much
subtle behaviour is also missed by the vdW equation.]
Repulsions and the Solid Phase.
We have seen above how it is fundamentally intermolecular attraction which drives the
liquid phse to exist. We will now see that the repulsive interactions drive solidification.
Repulsions -> solid phase;
Lindemann criterion;
Hard spheres and Alder & Wainwright
Non-liquids - C60 and colloids
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