7: PT Behavior and Equations of State (EOS) II
- Page ID
- 447
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Module Goal: To introduce you to quantification in fluid phase behavior.
Module Objective: To introduce you to the concept of Z-factor and the van der Waals equation of state.
- 7.1: Z-Factor
- This page explains how the ideal gas model is adjusted for real gases using the compressibility factor (Z) to account for deviations from theoretical predictions. It introduces the equation of state for real gases: PV = ZnRT, where Z = 1 signifies ideal behavior. Achieving this ideal behavior is complex, and while the Katz-Standing Method was a traditional estimation technique for Z, contemporary approaches increasingly rely on computational methods instead of graphical ones.
- 7.2: Definition of Equation of State (EOS)
- This page discusses the concept of an Equation of State (EOS), which relates pressure, volume, and temperature in a system, often in an empirical form. Most modern EOS are semi-empirical, requiring data fitting for pure substances and adjustments for mixtures. The complexity and accuracy of EOS increase with the number of parameters used.
- 7.3: Purpose and Historical Development
- This page outlines the evolution of Equations of State (EOS) in thermodynamics, essential for volumetric and thermophysical data and vapor-liquid equilibrium calculations. It identifies three historical phases: foundational laws by Boyle and Charles, monumental contributions by van der Waals and Gibbs, and incremental improvements including Pitzer's acentric factor and the Peng-Robinson EOS.
- 7.4: The van der Waals Equation
- This page discusses the van der Waals equation of state (vdW EOS), which enhanced the understanding of real gas behavior beyond ideal gas laws. It emphasizes concepts like matter continuity, the Principle of Corresponding States, and modern cubic equations of state. The vdW EOS addresses molecular volume and cohesive forces with parameters "a" and "b" that vary by substance, relating to critical properties.
- 7.5: Action Item
- This page discusses the compressibility factor (Z) of gases, its relation to ideal gas behavior, and conditions under which Z equals one. It encourages exploration of the Standing-Katz Compressibility Factor Plot for natural gases and highlights Z's behavior at varying pressures. Additionally, it introduces the van der Waals equation of state and methods to calculate Z, particularly as pressure approaches zero, emphasizing the differences in Z's values under low and high-pressure scenarios.