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**Non-tabular approaches to calculating properties of real gases**

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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

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**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.

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**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

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**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).

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**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)

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**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.

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**Generalized compressibility chart**

1949 Fits experimental data for various gases

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**Use of pseudo-reduced specific volume to calculate p(v,T), T(v,p) using GCC**

Z Source:

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**Nelson-Obert generalized compressibility chart**

1954 Based on curve- fitting experimental data

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Equations of state

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**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.)

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**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

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**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:

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**Critical compressibility of real gases**

Source:

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**First law in differential form, thermodynamic definition of specific heats**

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Advance Chemical Engineering Thermodynamics

Advance Chemical Engineering Thermodynamics

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