# 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/v^{2} )(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 >T_{c} (critical temperature) there
will be only one real value of v and for T< T_{c} 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<T_{c}.

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 (∂^{2}P/∂v^{2})_{T=Tc}
= 0

(∂P/∂v)_{T=Tc} = -RT_{c}/(v_{c} –b)^{2}
+ 2a/v_{c}^{3} = 0

or

RT_{c}/(v_{c} –b)^{2} = 2a/v_{c}^{3}
(∂^{2}P/∂v^{2})_{T=Tc} = 2RT_{c}/(v_{c}-b)^{3}
-6a/v_{c}^{4} = 0

2RT_{c}/(v_{c}-b)^{3} = 6a/v_{c}^{4}

2/(v_{c} –b) = 3/v_{c} or v_{c} = 3b

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

P_{c} = RT_{c}/(v_{c} – b) – a/v_{c}^{2}

From these equations,

a = 27R^{2}Tc^{2}/64
P_{c} and b = RT_{c}/8P_{c}

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