Real gases and Van der waals equation of state

Real gases and Van der waals equation of state


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

At high 
densities, that is at high 
pressures and low temperatures, 
the behavior  of actual or  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 three  values of v for  given 
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 be  only one real value of  v and for T<  Tc there  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
real  values of  volume for a 
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 (isotherm  passing 
through  the  critical point) shows a point of inflexion.
Stated mathematically

(∂P/∂v)T=Tc= 0 and (∂2P/∂v2)T=Tc
= 0

(∂P/∂v)T=Tc = -RTc/(vc –b)2
+ 2a/vc3 = 0

or

RTc/(vc –b)2 = 2a/vc3
(∂
2P/∂v2)T=Tc = 2RTc/(vc-b)3
-6a/v
c4 = 0

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

2/(vc –b) = 3/vc or vc = 3b

At the critical
point, the van der Waal’s equation is given by

Pc = RTc/(vc – b) – a/vc2

 

From these
equations,

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

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