# Real gases and Van der waals equation of state

The ideal gas law is only an approximation to the actual behavior of
gases.

At highย
densities, that is at highย
pressures and low temperatures,ย
the behaviorย  of actual orย  real gases deviate from that predicted by the
ideal gas law. In general, at sufficiently low pressures or at low densities
all gases behave like ideal gases.

An equation of state taking account the volume occupied by the
molecules and the attractive forces between them.

(P+a/v2 )(v-b) = RT

where a and b are van der
Waals constants.

The equation is cubic in volume and in general there
will be threeย  values of v forย  givenย
values of T and P.

However in some
range of values of P and T there is only one real value v.

For T >Tc (critical temperature) there
will beย  only one real value ofย  v and for T<ย  Tc thereย  will be three real values.

In Figure, the solid curve represents the value predicted by the van
der Waals equation of state and the points represent the experimentally
determined values.

It can be observed that at temperatures greater than
critical, there is only one real value of volume for a given P and T.

However at temperatures less than the critical, there are three
realย  values ofย  volume for aย
given value of P and T.

The experimental values differ from those predicted by van der Waals
equation of state in region 2345 if T<Tc.

One can use the criterion that the critical isotherm (isothermย  passingย
throughย  theย  critical point) shows a point of inflexion.
Stated mathematically

(โP/โv)T=Tc= 0 and (โ2P/โv2)T=Tc
= 0

(โP/โv)T=Tc = -RTc/(vc โb)2
+ 2a/vc3 = 0

or

RTc/(vc โb)2 = 2a/vc3
(โ
2P/โv2)T=Tc = 2RTc/(vc-b)3
-6a/v
c4 = 0

2RTc/(vc-b)3 = 6a/vc4

2/(vc โb) = 3/vc or vc = 3b

At the critical
point, the van der Waalโs equation is given by

Pc = RTc/(vc โ b) โ a/vc2

ย

From these
equations,

ย  ย  ย  ย  ย  ย  ย  ย  ย ย  a = 27R2Tc2/64
Pc and b = RTc/8Pc