# Introduction and Theory

In 1873, J.D. Van der Waals published his "Essay on Continuity of the Liquid and Solid States" which profoundly affected the study of Kinetic Molecular Theory.  In this literature, he proposed a radical change to the hypothetical Ideal Gas Law that accounted for discrepencies between the theoretical and experimental behavior of gases.  His work quickly became accepted by his peers, and thus the Van der waals equation of state was born.

Until his publications, the accepted description of gas behavior stemmed from the Ideal Gas Law that proposed a relationship between the pressure, volume, and temperature of a closed system.  This equation, however, conformed to experimental data only at low pressure and high temperatures.  It does so for two reasons: the ideal gas law does not take into consideration a particle's finite volume or  interparticle interactions. Therefore, the need to account for the gas particle's volume became clear to Mr. Van der Waals.

Because a particle occupies a physical volume, the space available to the gas is less than the actual volume of the container.  The volume expression (V) for the Ideal Gas Law PV = nRT thus becomes V = V - (correction factor).  The correction factor clearly depends on the number of particles present since the more particles packed into a smaller area means less actual overall volume.   Therefore, the correction factor is nb where n is the number of moles of gas and b is an emperical constant (meaning it is determined by experiment).  The ideal gas law, taking into account the volume correction, now reads as P(V-nb) = nRT.  Next, and a little less obvious, there existed a need to correct the pressure values of the ideal gas law.

When particles are in close proximity at high pressure, they are very likely to interact with each other in some shape or form.   Collisions would be one example of interaction, however, Van der Waals also hypothesized that the particles experienced a cohesion force that would attract them to each other.   This would make the observed pressure smaller since if the particles attracted each other they would hit the walls with less momentum and thus less force.  The more particles around the walls would mean more interactions and so this correction factor for pressure must be dependant upon the concentration of gas in the container.  Thus, the ideal gas law becomes (P + correction factor)*(V-nb) = nRT.  How the correction factor is derived comes from the fact the assumption that particle interactions occur in pairs.  In a closed container filled with N particles, there exist (N-1) potential partners available for a particle to interact with.  Next, since the each pair is counted twice (i.e. the 1--2 pair is identicle to the 2--1 pair), for N particles there exist  (N(N-1))/2 possible pairs.   When N is large, or on the order of a mole or bigger, N-1 becomes N since 1 is tiny in magnitude compared to a mole.  Clearly, the correction factor must depend on the square of the number of moles of particles and thus the square of the concentration.   Hence the term (a/V)^2 is derived where a is a constant that takes into account the factor of 1/2 from the (N^2)/2 expression.  Finally, the adjusments to the ideal gas law are complete, and we arrive at Van der Waals equation of state:

P = (nRT/(v-b)) -(a/V)^2

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### References:

1. Boltzmann, Ludwig.  Lectures on Gas Theory.  U. of California Press: Berkeley, 1964.

2. Jeans, J.H. The Dynamical Theory of Gases. Cambridge University Press: Cambridge. 1921.

3. Zumdahl, Stephen S. Chemical Principles: Second Edition.  D.C. Health and Co.: Toronto, 1995.