We have v _{ideal } = RT / P

u_{ real} / u_{ ideal} = compressibility factor = z

Pu = ZRT

Fig. 9.3 shows a plot of Z against P and T, which is also called compressibility factor chart.

Separate charts are needed for every gas. However, one single generalized chart can be developed based on law of corresponding states where we express pressure, temperature and specific volume in dimensionless forms

P _{r} = p / p_{c} , T _{r} = T / T _{c }, u _{r} = u u _{c}

where P,, T, and v, are known as the reduced pressure, reduced temperature and reduced specific volume respectively, and P_{c}, T _{c} and v _{c} are the critical properties of substance.

Now the law of corresponding states may be stated as follows :

“If two or more substances have the same reduced pressure P,, and reduced temperature T,, then they will have the same reduced volume v,”

Mathematically Z = f(P _{r}, T _{r})

= Z R T P_{C} / Z_{C} R T_{C} P = Z / Z_{C }, T _{r} / P _{r}

Where Z_{C} = P_{C} u _{c} / R T_{C} = critical compress1 11ty actor.

When T, is ploted as a function of reduced pressure and Z, a single plot called the

generalized compressibility chart is obtained for a great variety of substances.

We have ( p + a / v^{2}) ( u –b) = RT

or pv^{3 }– ( pb + RT) u^{2} + av – ab = 0

From equations (9.48) and (9.49), we get

We get ( p_{c} + a / v _{c}) ( v c _{–} b) = RT_{C}

or ( p_{c} + a / v _{c}) ( v _{c }– v3 3) = 8a / 9 T _{c} u _{c} . T _{c}

a = 3 pc v _{c}

Therefore the value of R becomes

R = 8 / 3 p_{c }u _{c} / T _{c}

_{ }

Thus we have found out a, band R in terms of critical properties. So van der Waals

equation of state becomes

( p+ 3p_{c }u _{c} / u^{2}) ( u – v c /3) = 8 p_{c} v _{c} / 3 t _{c} . t

or ( p / p_{c} + 3 u^{2} / u^{2} ) ( u / v _{c} – 1 / 3) = 8t / 3 t _{c}

or ( p _{r }+ 3 / u^{2}) ( 3u _{r} -1) = 8t

Equation (9.50) is an expression of the law of corresponding state.

We have from above last equation (9.50)

P _{r} u _{r} = 8 T _{r} u _{r} / 3u_{r }-1 – 3 / u _{r}

For minima isotherms we have

Simplifying

(P _{r} v _{r}) -9P_{r}v,+6P_{r}=O => a parabola

Passing through minima of the isotherms.

When p _{r} = 0

P _{r} u _{r} = 0 and 9.

P _{r }= 9 ( p _{r} u _{r }– (p _{r} u _{r} )2 / 6

Dp _{r} / d (p _{r} u _{r} ) = 9-2 (p _{r }u _{r}) =0

P_{r} u_{r} = 4.5

P_{r }= 9 x 4.5 -4.52 / 6 = 3.375

The parabola has the vertex at P, v, = 4.5 and P, = 3.375 and the parabola intersects

the ordinate at 0 and 9.

Boyle Temperature (T 8) : Every isotherm has a minimum up to the marked T _{B.} The

1 _{B} isotherm has a starting equation P _{r }u _{r} =constants which obeys Boyle’s law at moderate

pressure. The corresponding temperature is called Boyle temperature.

We have ( p _{r} u _{r} + 3 / u _{r}) ( 3u_{r} -1) = 8 T _{r} u _{r}

or ( 9 + 3 / u _{r }( 3u_{r }-1) = 8 T _{r} u _{r}

or 3 ( 3ur 1) ( 3u_{r} +1) (3ur -1) = 8 T _{r} u _{r}

or 3 (9u_{r}^{2} – 1) = 8T_{r} u_{r}2

or T _{r }= 27 / 8 – 3 / 8 u2_{r} for p _{r }u _{r} = 9

Again for p _{r} u _{r} = 0

3 / u _{r} ( 3ur -1) = 8 t _{r} u _{r}

9 / 8 u _{r }– 3 / 8 u _{r }= t