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 >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 (∂²P/∂v²)_T=Tc = 0

(∂P/∂v)_T=Tc = -RT_c/(v_c –b)² + 2a/v_c³ = 0

or

RT_c/(v_c –b)² = 2a/v_c³ (∂²P/∂v²)_T=Tc = 2RT_c/(v_c-b)³ -6a/v_c⁴ = 0

2RT_c/(v_c-b)³ = 6a/v_c⁴

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_c²

 

From these equations,

                   a = 27R²Tc²/64 P_c and b = RT_c/8P_c

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