Re: Van der vals eqn of state help!

From: Oscar Lanzi III (ol3_at_webtv.net)
Date: 01/29/05 

Date: Sat, 29 Jan 2005 06:13:07 -0600

Sorry, lost control of my keyboard (remote!) and posted crap, let me try
again.

A method without calculus

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Start by forming a standard polynomial equation:

PV^3 - (b+RT)V^2 + aV - ab = 0

Divide by P:

V^3 - [(b+RT)/P] V^2 + (a/P) V - (ab/P) = 0

Now if this has one root V_crit, then the left side of the equation has
to factor as (V -V_crit)^3. From the binomial theorem:

(V - V_crit)^3 =

V^3 - (3 V_crit) V^2 + 3 (V_crit)^2 - V_crit^3

Then
3 V_crit = (b+RT_crit)/P_crit
3 V_crit^2 = a/P_crit
V_crit^3 = ab/P_crit

Dividing the third equation by the second leads to V_crit = 3b
(therefore 3 times the low-temperature liquid volume b); then putting
V_crit = 3b into either the second or third equaiton leads to an
expression for P_crit. With these known the first equaiton may be
solved for T_crit.

A defect of this equation is that given values of P_crit and T_crit, we
are foced to accept one value of V_crit which may not quite match the
real substance (although it's usually a reasonable first approximation).
More addvanced cubic equations of state have an additional factor, like
the "acentric factor" in the Redlich-Kwong equaion, to fiddle with.

--OL