Building Bridges in the First Two Years

Project #1: Equations of State

The simplest equation of state is that of an ideal gas:

where P= pressure, V= volume, and T= temperature, are the variables of interest. The remaining parameters are n, the number of moles in the volume, and R, the universal gas constant.

A more realistic equation of state is the van der Walls equation (call it vdW for short)

Here, a and b are constants that depend on the type of gas in the volume.

The purpose of this project is to use Calculus and Maple to explore and illustrate some of the basic properties of this equation of state. Here are some questions to guide you along.

1. Solve for pressure as a function of everything else. Fix n, R, as well as T and plot pressure versus volume for each equation of state. When do the models agree most closely? What are some of the key differences between the two models?

   You will have to choose units (R = 0.082056, in units ). You will also have to choose values for a and b in the vdW equation. (You can find a table of values in your chemistry book.) Experiment with different temperatures and different ranges for P and V in your graphs.
   
   

2. The chemistry text states that ``as V becomes large... the van der Waals equation of state approaches and eventually becomes the same as the equation of state for an ideal gas.'' (Did you see this in your graphs for part (1)?)

   One way to illustrate this is to define the compressibility factor

   

   and plot Z as a function of V for the vdW equation. What happens to Z as V increases? Does your answer depend on the values of a and b in vdW?
   
   

3. The van der Waals equation has some strange properties at low temperatures (and/or low volumes and pressures). Look at P as a function of V for diffenent values of T to illustrate these properties. For temperatures above a critical temperature, pressure is a strictly decreasing function of volume. Use what you know about derivatives (first and second) to find the critical temperature. (Do this for at least two examples--carbon dioxide ( ) and chlorine ( ) are nice.)

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