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Van der Waals' Equation of State for a Non-Ideal Gas

Most attempts at modeling real gases take into account two things: (1) particles occupy a finite volume; and (2) particles are strongly influenced by surrounding forces that cause them to react with one another. The van der Waals equation for gas was derived from both of these considerations. Constant is a consideration of the internal "pull" pressure exerted on particles near the container surface that is created by the attractive forces from particles in adjacent particle layers. Constant accounts for the volume excluded by colliding particles (assuming a spherical particle, this equals four times the volume of the particle).
Van der Waals:
ideal gas:
where
= pressure (atm),
= molar volume (L),
= correction term for gas particle interaction ),
= correction term for finite volume of particles (L/mol),
= ideal gas constant = 0.082 (L atm/K mol),
= temperature (K)

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Van der Waals' equation states that as pressure decreases and temperature increases, real gas behavior approaches that of an ideal gas. The plot also shows that at some critical (shown for and ), a horizontal inflexion is created. Below this temperature, with additional analysis, van der Waals' equation can be used to predict phase changes. Graphically, this is done by comparing the area within the concave up section of the plot to that of the concave down section for any given pressure. The pressure at which these areas are equal is the equilibrium state between the gas and liquid phase.
Reference: D. R. Gaskell, Introduction to the Thermodynamics of Materials, 4th ed., New York: Taylor & Francis, 2003.

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