VOLUME 33, NO. 9, SEPTEMBER, 1956
a
A SIMPLE MODEL FOR VAN DER WAALS "a" STEPHEN S. W I N T E R Northeastern University, Boston, Maasaohusetts
THEfallacy of treating the internal-pressure correc- reduced. The measured pressure is therefore less than tion in van der Waals equation on the basis of a cumulative force field from the body of the gas, which neglects compensating attractions by the molecules of the wall, has recently been discussed by Swinbourne.' The alternative presented, however, appears beyond the scope of an undergraduate physical chemistry course, though i t is considerably more elegant than the simple model presented below. The treatment outlined here has the advantage of being qualitatively correct and simply described. It still leads to the correct n2a/V2term of the van der Waals equation. The internal pressure of a gas is a measure of the attractive forces between molecules. As a result of this attraction, collisions do not occur instantaneously. Instead, the colliding molecules tend to remain near one another for a short period of time. Molecules therefore require a slightly longer time than that calculated from their root-mean-square velocity to traverse one of the linear dimensions of the container. According t o the kinetic picture of a gas, the pressure results from the collisions of the molecules with the walls of the container. Because of the delay during collision, the number of molecules reaching the wall is ' SWINBO~RNE, E.S., J. CHEM.EDUC., 32,366 (1955).
that postulated by the kinetic theory of the ideal gas. The correction, P,,,, to the ideal pressure depends first upon the number of molecnles that reach the wall, and second upon the number of collisions each of them has made in their flight across the container. Both of these quantities are proportional to the density of molecules, or n/V. Piatis therefore proportional, with the proportionality constant a, to n2/V2. Then,
The pressure, P, in the ideal gas equation is the ideal pressure. It should be replaced by P,.,., nza/V2. Making this substitution in :
+
PV
leads to: (P
=
nRT
+ n h / V Z )V = nRT
which, with the usual volume correction, gives the van der Waals equation: (P
+ n h / V a )(V - nb) = nRT
