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The van der Waals equation of state is no longer used by engineers for calculating real gas behavior. As we have seen, it has a great deal of pedagogical use, and so we still introduce it as the first correction to the ideal gas equation. However, other equations of state have been developed for engineering purposes. These equations of state do a progressively better job of modeling the liquid phase (liquid desnity) with increasing complexity. The two most commonly used cubic equations of state are the Redlich-Kwong (RK) and Soave modification of the Redlich-Kwong equation of state (RKS). These equations are shown below. You will note the very strong similarity to the van der Waals equation of state - they were developed by modifying the van der Waals equation of state so that it treats the attractions, hence the liquid phase, better. Because molecules are not spherical, the RKS eos also utilizes the acentric factor w (Greek letter omega). The acentric factor is commonly used to correlate behavior of non-spherical molecules, and it can be viewed as a constant that you look up in the DIPPR database for the specific fluid in question, just like you would look up Tc and Pc. A non-cubic eos, the Bennedict-Webb-Rubin (BWR) eos, is also shown here. The BWR eos will give better accuracy in the dense gas and liquid regimes, but it also requires 8 constants that must be obtained from experimental data for the specific fluid that you are modeling. While there are tables of a BWR constants available for a few common compounds, the cubic eos are much easier to use because all that is required are the critical constants and the acentric factor.
The sections below show the various equations of state in their pressure form and their compressibility form. They also list the constants involved in the equation in terms of fundamental constants that one would look up in the DIPPR database; i.e., Tc, Pc, etc.
Pv = RT z = 1
Van der Waals
(P + a/v2)(v - b) = RT z = v/(v - b) - a/(RTv). P = RT/(v - b) - a/v2 .
b = (RTc)/(8Pc)
. .E = aa/RT
.a = 0.42747R2Tc2.5/Pc b = 0.086640RTc/Pc .
a = 0.42747R2Tc2/Pc b = 0.086640RTc/Pc .
m = 0.48505 + 1.5517w - 0.1563w2 a = [1 + m(1 - Tr1/2)]2
Why are there so many equations of state?
The equations of state given above are some of the more common, but there are others that engineers use. Why are there so many? Modeling of the PVT behavior of a fluid over the whole range of temperatures, densities and pressures is not an easy task. Recall the complex behavior that the vdw eos had in order to try and model the coexistence curve where saturated liquid and saturated vapor are in equilibrium. At low pressures, the isotherms are fairly easily modeled with something like the ideal gas eos. However, at higher densities the behavior becomes more difficult to model with a simple equation. If one wants to get the liquid and vapor densities correct, then even more stress is put on the model. None of the eos exactly give the correct PVT behavior of all fluids over the whole domain of desired conditions. Therefore different equations of state have been developed to improve the accuracy in certain regions. For example, the BWR equation will give better liquid densities than the Soave. The Soave equation will give better densities for fluids comprised of non-spherical and polar molecules than the RK eos.
The RK, Soave and BWR equations of state will all give about the same densities for gases. The most significant difference between these equations of state is when they are used to calculate liquid densities, saturated vapor densities, and vapor pressures. This can be seen by looking at the graph below, which is the 320 K isotherm for ammonia as calculated using the vdw, RK, RKS, and BWR. Recall that in order to predict the liquid and vapor molar volumes, the equation of state predicts an isotherm that goes down, up, and then down again on a P vs. v plot. The vapor pressure and equilibrium molar volumes of the liquid and vapor would be found from the horizontal line that bisects the area of this loop. Look at the very different behavior of the 320 K isotherm as predicted by these four equations of state and you can see why they give significantly different values for the vapor pressure and molar volumes of the saturated vapor and liquid.
So, which should you use? In this class, we will mainly use either the
RK or Soave equations of state. These equations give accurate values for
gases, and reasonable approximations for saturated liquids and vapors.
These equations are also used heavily by design engineers. However,
if you were involved in metering or selling a compound based on its temperature
and pressure, you would certainly want to use a more accurate equation
of state such as the BWR or an even more complex equation.
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