Equations of State

    In a microscopic system, the equation of state for an ideal gas of low density can be described by:

                                      (Schroeder 7)

This equation gives us a linear relationship between pressure, volume, temperature, and the number of molecules in the system.  Most gasses are not ideal and involve more complex interactions.  For this reason, the Van der Waals equation is used:

                (Schroeder 180)

The value a is a constant which is introduced to compensate for the short-range attractive forces between molecules.  b is a constant introduced to compensate for the short-range repulsion forces between molecules.

      While these equations hold true for most systems of gasses, they don’t aid in the visualization of these properties.  For this reason, we chose to model such a system in the macroscopic world.  We observed the force a "gas" would exert in containments of varying areas.  We also observed the interaction of two "gasses" when enclosed with a moveable wall.

      Although our system modeled a real system, it fell short on many of the important properties.  Temperature was not a parameter in this experiment.  On the microscopic level, temperature is a source of energy.  A change in temperature can effect the pressure, volume, and even speed distribution.  The lack of this parameter causes many of the fundamental laws to fall apart.  A few of the laws which are violated are:

• Van der Waals equation of state cannot be used to describe the system.  Not only is temperature not a variable, but there are no repulsion or attraction forces between the molecules  (Schroeder 180).  Our system is also in two dimensions instead of three. 

• The Equipartition Theorem, which states that the total energy of a system can be defined by the product of the number of molecules, the temperature, k, and the degrees of freedom of a system, is not valid  (Schroeder 14).  The macroscopic system is much too large to establish the degrees of freedom.  The Equipartition Theorem can be applied only to systems where the energy can be described by degrees of freedom.

• The macroscopic model is not in Thermal Equilibrium.  In order to reach Thermal Equilibrium, the two systems need to be in contact long enough to reach the same temperature.  (Schroeder 3)  Our system has mechanical interaction, but it is not converted into a source of energy or interchanged. 

• Maxwell’s Distribution, which states that all speeds are not equally probable, is not valid (Schroeder 242).  Maxwell’s Distribution relies on temperature being the source of energy.  In the macroscopic system, the speed depends on the area of the containment, not on the temperature or the energy. 

    In our macroscopic system, the equation of state that we could obtain fails because it can not take into account many of the interactions that occur in the microscopic levels.  The parameters we evaluated were force, volume, and the number of molecules.  Force is the change of momentum over time.  Force is related to pressure in that pressure is the force per unit length.

Pressure Fluctuation Machine

    When two systems of gases interact, they exhibit certain properties that are not easily understood.  In this section of the lab, we created, on a macroscopic level, a model of the microscopic molecular interactions.  By creating a system with varying area and adding various numbers of motorized “molecules,” we were able to observe the most probable area of each state.

    When two gasses of equal molecules and total energy interact, they move toward equilibrium.  In our case, the variant parameter was the area.  The force each “gas” exerted on the piston would cause the area to change until the forces exerted by each “gas” were equal.  To calculate the most probable state, we relied on the principles of statistical mechanics. 

    In order to calculate the probability of each system, we first had to calculate their individual multiplicities.  Our first step was in simplifying the system for calculations.  We converted our system into one that mirrored a lattice gas.  Since each squiggle ball has a diameter of 4 cm, we started by dividing our area into 32 squares each with a side of 4 cm.

In order to find the multiplicity of the left system, we used the following equation:

k refers to the position of the piston while NL is the number of molecules in the left system.  The multiplicity of the right system was found by using the equation:

We inserted the 32 in front of the k because the origin of this system is not zero.  The actual k of the right side is the total system minus the area of the left side.  NR refers to the number of molecules in the right system.

    The total multiplicity of the system is the number of microstates available to the system.  This is found by the equation:

We subtracted the total number of balls because they are not states that vary in the system.  The one is a compensation factor to help scale our data.  The probability of the entire system is then easily calculated with the following equation:

These equations were all derived based on the theory provided in Thermal Physics by David Schroeder.  While we based on a previous experiment, we found their equations of multiplicity difficult to understand. 

    Although many concepts behind this experiments is lacking, it is still functional.  The experiment allows students to obtain a visual representation of what is happening on the microscopic level.  We recommend this procedure for use in a beginning lab or in a class demonstration.