Note on van der Waals Reduced Equation of State CHARLES A. STEVENSON Stokely Foods, Ine., Indianapolis, Indiana physical significance. By inspection one root of f(k) = 0 is seen to be unity. The other two positive roots are found fo be 0.382 and 2.618. Thus the roots in question are
T H E van der Waals equation
leads to the reduced equation of state (a 31BP! ( B - '/1) = '/s
+
Y
31 = 0.382 k* = 1 k1 2.618
-
We note immediately that kz = 1 defines the critical state. Note also that The subscript c refers to the critical state. The reduced variables of state or, P, y are pure nurnbers and may assume any positive value. It is a matter of interest to inquire as to the conditions in which these variables assume coincident values. The condition that a = i3 = 7 = k as given by the reduced equation of state is (k 3 / k 9 (k - '/d = 'IS k
+
This equation has four real roots, one being negative and three being positive. Only the positive roots have
-ks= - kr
k~
.
k2
that is, these roots form a geometric progression. Each of these three roots satisfies the original van der Waals equation, and the three states of a van der Wads substance defined by kl, kz, ka lie on the appropriate isotherm. Therefore, the reduced equation of state connects the critical state defined by kz = 1 with two other dnique states-namely, a certain liquid state defined by k, = 0.382 and a certain gaseous state defined by ka = 2.618. All three of these states are characterized by the condition that or = P = 7 .
