1. Non-tabular approaches to calculating properties of real gases

2. 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

3. 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.

4. 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

5. 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).

6. 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)

7. 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.

8. Generalized compressibility chart
1949
Fits
experimental
data for
various gases

9. Use of pseudo-reduced specific volume to calculate p(v,T), T(v,p) using GCCZ
Source:

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

11. Equations of state

12. 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.)

13. 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

14. 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:

15. Critical compressibility of real gases
Source:

16. First law in differential form, thermodynamic definition of specific heats