Author: admin
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Internal Energy Departure Function
schematically compares a real gas isotherm and an ideal gas isotherm at identical temperatures. At a given {T,P} the volume of the real fluid is V, and the ideal gas volume is Vig = RT/P. Similarly, the ideal gas pressure is not equal to the true pressure when we specify {T,V}. Note that we may characterize the departure from ideal gas…
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The Departure Function Pathway
Suppose we desire to calculate the change in U in a process which changes state from (VL, TL) to (VH, TH). Now, it may seem unusual to pose the problem in terms of T and V, since we stated above that our objective was to use T and P. The choice of T and V as variables is because we must work often with equations of state that are functions…
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Practice Problems
P7.1. For Tr < 1 and Pr ≈ Prsat, the Peng-Robinson equation of state has three roots corresponding to compressibility factors between zero and 10. The smallest root is the compressibility factor of the liquid. The largest root is the compressibility factor of the vapor and the middle root has no physical significance. This gives us a general method for finding the…
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Summary
The simple physical observations and succinct mathematical models set forth in this chapter provide powerful tools for current chemical applications and excellent examples of model development that we would all do well to emulate. This chapter has illustrated applications of physical reasoning, dimensional analysis, asymptotic analysis, and parameter estimation that have set the standard for…
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The Molecular Basis of Equations of State: Molecular Simulation
Molecular simulation provides a numerical connection between the intermolecular potential model and the macroscopic properties, but it does so one state point at a time. For an equation of state, we need an equation that makes this connection over all state points. The key to making this kind of connection is to consider the average…
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The Molecular Basis of Equations of State: Concepts and Notation
In the previous sections we alluded to equations of state as empirical equations that may have appeared by magic. In this section and the next two, we attempt to de-mystify the origins behind equations of state by systematically describing the current outlook on equation of state development. It may seem like overkill to develop so…
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Matching the Critical Point
The capability of a relatively simple equation to represent the complex physical phenomena illustrated in Figs. 7.5–7.6, and as shown later in Figs. 7.7 and 7.9, is a tribute to the genius of van der Waals. His method for characterizing the difference between subcritical and supercritical fluids was equally clever. He recognized that, at the critical point, Figure 7.7.…
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Implications of Real Fluid Behavior
There is one implication of non-ideal fluid behavior that should be clear from the equations presented above: Real fluids behave differently from ideal gases. How differently? An example provides the most straightforward answer to that question. Here we adapt some of the derivatives from Chapter 6. Example 7.6. Derivatives of the Peng-Robinson equation Determine , , and for the…
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Cubic Equations of State
To apply the relationships that we can develop for relating changes in properties to CP, CV, P, T, V, and their derivatives, we need really general relationships between P, V, and T. These relationships are dictated by the equation of state (EOS). Constructing an equation of state with a firm physical and mathematical foundation requires considering how the intermolecular forces…
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The Virial Equation of State
At low reduced pressure, deviations from ideal gas behavior are sufficiently small that we can write our equation of state as explicit in a power series with respect to density. That is, where B, C, and D are the second, third, and fourth virial coefficients. This can be considered an expansion in powers of ρ. Coefficients C and D are rarely applied because…