Shay Kuuttila, Merrick Miranda, Evan Leonard, Jennifersue Bowker
Pressure, Volume, and Temperature Diagrams for Methane
A Comparison of Peng-Robinson vs. Van der Waals Equation of State
Process Description
Systematically analyzing the Van der Waals equation and other equations of state can
allow a better understanding of the interconnection between various macroscopically
measurable properties of a system, including pressure, temperature, and volume. The
Van der Waals equation and the Peng Robinson equation, both derived from the Ideal
Gas equation, contain similarities and differences, although the parameters of each
equation are significantly different. The first parameter, a, is dependent upon the
attractive forces between molecules while the second parameter, b, is dependent upon
repulsive forces. In both equations a and b are highly dependent on the structure of the
compound; Peng-Robinson works best for hydrocarbons and inorganic gases, while Van
der Waals is inaccurate near the gas-liquid interface. 
Equation 1 Van der Waals
Equation 2 Peng-Robinson
Also, (t) in the Peng-Robinson equation is a function of temperature and since boiling
pressure is a function of temperature, pressure should naturally be more dependent on
temperature in the Peng-Robinson equation. The parameters in the Peng-Robinson
equation are expressible in terms of the critical properties and the acentric factor, ,
which is constant for each gas and can be calculated using the following formula:
  1.0  log 10[ P (Tr ) / Pc]
vap
We used excel to generate an algorithm for the Peng-Robinson equation and made PT,
PV, and VT diagrams. For each of the diagrams we used the following equations to
calculate the parameters and to plot Pr (P/Pc), Tr (T/Tc), and Vr (V/Vc) against one
another appropriately:
For example, on the PV diagram, the user inputs a reasonable temperature, and the
algorithm calculates the corresponding parameters, pressure and volume in the PengRobinson equation. Similarly the user inputs pressure and volume respectively for the
other two diagrams, VT and PT. The program will also calculate and graph the
corresponding Van der Waals diagram with the Peng-Robinson diagram for comparison.
Analysis of PV Diagram
For a constant temperature, a plot of the pressure vs. molar volume for methane will yield
a decreasing function with two points at which the derivative of the functions are zero
(see attached PV Graph).

A positive or zero valued slope signifies the mechanically unstable state of the
compound. The plot varies slightly for different temperatures. As the temperature
approaches the critical temperature the functions of the two EOS’ converge. This occurs
because though the Van der Waals equation does specify P as a function of T, the PengRobinson equation shows dependency of P on both T and a(T). This means that P (for PR) changes at a greater rate as T increases. This can be seen in the P vs. T graph. The
shift is more dramatic for the Peng-Robinson case because the interactions between the
molecules are represented more accurately in this equation, so when the temperature
increases, the molecules internal energy increases (i.e. more rotational and vibrational
movement) and the intermolecular interactions become less apparent. The graphs are
different in value at lower volumes and approach the same curve as the volume increases.
The function only changes drastically around the critical Volume.
At low temperatures and volumes, Methane is a liquid, then the graph shows the
gas/liquid equilibrium. As T approaches Tc, the value of a(T) approaches a, and the two
equations become similar.
Analysis of PT Diagram
For a constant molar volume, one can see that a plot of the temperature vs. the pressure
will give a linear function. As the temperature is increased, the pressure also increases for
a given volume, as both the Van der Waals and Peng-Robinson equations describe. There
are some differences between the two equations of state. The Peng-Robinson equation,
describing the attraction between the molecules of the gas as a function of T and the
molecular size, will have a steeper slope. At low temperatures, the molecular attractions
are a more significant force than the temperature, but at higher temperatures, the
intermolecular interactions are less important as the molecular movements are greatly
increased.
The Van der Waals equation simply relates an attractive force to the molar volume and is
much more generalized and so does not recognize the diminished intermolecular
attractions a small molecule such as methane has. Clearly, the Peng-Robinson is a more
suitable equation for methane.
Analysis of VT Diagram
For a constant pressure, a plot of the molar volume vs. temperature yields an increasing
function. The slope before the critical temperature is very small where methane exists in
liquid/vapor equilibrium, but above the critical temperature, where all of the liquid is
transformed into vapor, the slope is much greater. Near the critical temperature, both
graphs show a very steep increase in the molar volume for small changes in temperature.
This represents the decreasing interactions between molecules as the substance turns
from liquid to gas.
The Van der Waals equation, poorly relating the attractive forces, gives a higher volume
for a given temperature and pressure. The disparity between the two EOS’ decreases for
higher temperatures as once again a(T) approaches a.
Conclusion
Also, at higher T and V’s, above the critical point, the substance is in a gas phase and
intermolecular attractions are not as strong as they are when the substance is in a
liquid/vapor equilibrium, and so the Peng-Robinson EOS is very similar to Van der
Waals. The Peng-Robinson model more accurately shows the relationship of state
parameters as it expresses the attractive forces as both a function of T and molecular size.