Non-tabular approaches to calculating properties of real gases

The critical state At the critical state (Tc, Pc), properties of saturated liquid and saturated vapor are identical if a gas can be liquefied at constant T by application of pressure, T·Tc. if a gas can be liquefied at constant P by reduction of T, then P·Pc. the vapor phase is indistinguishable from liquid phase

Properties of the critical isotherm The SLL and SVL intersect on a P-v diagram to form a maxima at the critical point. On a P-v diagram, the critical isotherm has a horizontal point of inflexion.

Departures from ideal gas and the compressibility factor For an ideal gas One way of quantifying departure from ideal gas behavior to evaluate the “compressibility factor” (Z) for a true gas: Both Z<1 and Z>1 is possible for true gases

The critical state and ideal gas behavior At the critical state, the gas is about to liquefy, and has a small specific volume. is very large  Z factor can depart significantly from 1. Whether a gas follows ideal gas is closely related to how far its state (P,T) departs from the critical state (Pc, ,Tc).

Critical properties of a few engineering fluids Water/steam (power plants): CP: 374o C, 22 MPa BP: 100o C, 100 kPa (1 atm) R134a or 1,1,1,2-Tetrafluoroethane (refrigerant): CP: 101o C, 4 MPa BP: -26o C, 100 kPa (1 atm) Nitrogen/air (everyday, cryogenics): CP: -147o C, 3.4 MPa BP: -196o C, 100 kPa (1 atm)

Principle of corresponding states (van der Waal, 1880) Reduced temperature: Tr=T/Tcr Reduced pressure: Pr=P/Pcr Compressibility factor: Principle of corresponding states: All fluids when compared at the same Tr and Pr have the same Z and all deviate from the ideal gas behavior to about the same degree.

Generalized compressibility chart 1949 Fits experimental data for various gases

Use of pseudo-reduced specific volume to calculate p(v,T), T(v,p) using GCC Z Source: http://opencourseware.kfupm.edu.sa/colleges/ces/me/me203-061/files%5C2-Lectures_Ch02b_Ideal_gas.pdf

Nelson-Obert generalized compressibility chart 1954 Based on curve- fitting experimental data

Equations of state

Some desirable characteristics of equations of state Adjustments to ideal gas behavior shoujd have a molecular basis (consistency with kinetic theory and statistical mechanics). Pressure increase leads to compression at constant temperature Critical isotherm has a horizontal point of inflection: Compressibility factor (esp. at critical state consistent with experiments on real gases.)

Some equation of states Two-parameter equations of state Virial equation of states Z=1+A(T)/v+B(T)/v2+…. (coefficients can be determined from statistical mechanics) Multi-parameter equations of state with empirically determined coefficients: Beattie-Bridgeman Benedict-Webb-Rubin Equation of State Often based on theory

Two-parameter equations of states Examples: Van der waals Dieterici Redlich Kwong Parameters (a, b) can be evaluated from critical point data using Van der Waals:

Critical compressibility of real gases Source: http://books.google.com/books?id=SdIQtl7boA8C&pg=PA141&dq=%22critical+compressibility%22+hydrocarbons&hl=en&ei=jUg5TcfvBsPXrQfVjqjyCA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCwQ6AEwAQ#v=onepage&q=Table%203.1&f=false

First law in differential form, thermodynamic definition of specific heats