1. Q&A_-10- 11/17/2005(10) Ji-Sheng Chang
The Advanced Chemical Engineering Thermodynamics The thermodynamics properties of fluids (II)
Q&A_ /17/2005(10)
Ji-Sheng Chang

2. Property relations The residual Gibbs free energy
The definition of residual properties
MR(T,P) = M(T,P) - MIG(T,P)
R: Residual property
IG : Ideal gas behavior property
MR(T,P) = M(T,P) - MIG(T,P), M are the properties as V, G, H, S, …

3. Property relations The residual properties
VR(T,P) = V(T,P) - VIG(T,P) = V - RT/P; Basic thermodynamics relationship for calculations
GR(T,P) = G(T,P) - GIG(T,P); More important in traditional Chemical Engineering Thermodynamics
HR(T,P) = H(T,P) - HIG(T,P)

4. Property relations
Deriving the residual properties in terms of the PVT properties.
From the generating function of Gibbs free energy, VR/RT= [(GR/RT)/P]T,
Then d(GR/RT) = (VR/RT)dP,
to integrate from P=0 to P at constant T.

5. Property relations
The integrated result in terms of PVT property as the formula (6.45)

6. Property relations
The residual enthalpy, HR/RT= - T [(GR/RT)/T]P, The integrated result in terms of PVT property as the formula (6.46)

7. Property relations
In previous section S = H/T - G/T, then the residual entropy SR/R = HR/RT - GR/RT The result in terms of PVT property as the formula (6.48)

8. Property relations Using the independent property of T and V
More commonly as the form P=P(V,T) of the PVT equations of state, especially the cubic equations of state.
P = ZRT/V, P = ZRT; as  = 1/V
dP=RTd(Z)=RT(Zd+dZ) at constant T

9. Property relations dP/P=d/+dZ/Z, as the formula (6.58)
the alternative form.

10. Property relations Calculating the residual properties
The PVT data, Using graphical integration method to calculate the residual Gibbs free energy from steam data.

11. Phase diagram 1. PT phase diagram of water
2. The Solid/Liquid equilibrium curve
for water
for most component
Solid
Liquid
Gas
PC,TC
200OC path
Critical properties
PC= bar
TC=647.1 K
TTri= K

12. Vapor/Liquid equilibrium curve
The Antoine equation
The constant of Antoine equation for water A= B= C=

13. Property Critical properties of water PC=220.55 bar TC=647.1 K
PTri=0.611 kPa
TTri= K
Tn= K at P= bar = 1 atm

14. Property relations Problems
Calculate the residual Gibbs energy of the 200C superheat steam at 10 bar, using the PVT data from steam tables

15. Properties P/kPa P/bar V/cm3g-1 V/cm3mol-1 Z (Z-1)/P 1 0.01 218350 50
0.5
4356.0
100
2172.3
200 2
1080.4
300 3
716.35
400 4
534.26
500 5
424.96
600 6
352.04
700 7
299.92
800 8
260.79

16. Properties P/kPa P/bar V/cm3g-1 V/cm3mol-1 Z (Z-1)/P 900 9 230.32
1000 10
205.92
1100 11
185.92
1200 12
169.23
1300 13
155.09
1400 14
142.94
1500 15
132.38
1550
15.5
127.61
1554.9
15.55
127.20

17. Residual Gibbs free energy properties

18. Example 1 Calculating the residual properties
The PVT equations of state
Using the two term virial equation of states, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the carbon dioxide at the states of temperature of 25C and pressure of 1 bar.

19. Deriving the formulas The equations of residual Gibbs free energy
Three type of the two term virial equation
The residual Gibbs free energy for a gas at each P and T

20. Deriving the formulas Second virial coefficient From PVT data
From generalized correlation equations
Based molecular thermodynamic, it is integrated from potential function

21. Example 2 Calculating the residual properties
The PVT equations of state
Using the two term virial equation of states, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the methane at the states of temperature of 25C and pressure of 1 bar.

22. Example 3 Calculating the residual properties
The PVT equations of state
(？) Using the two term virial equation of states, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the benzene at the states of temperature of 25C and pressure of 1 bar.

23. Theorem of corresponding states
Two-parameter theorem of corresponding states
Given T and P
Finding Tc, Pc
Calculate Tr, Pr
Using Tr, Pr to finding the reduced properties from Table E.1 to E.16

24. Theorem of corresponding states
Three-parameter theorem of corresponding states
Given T and P
Finding Tc, Pc, 
Calculate Tr, Pr
Using Tr, Pr,  to finding the reduced properties from Table E.1 to E.16

25. Theorem of corresponding states
Based on the corresponding states principles
The Lee/Kesler generalized correlation tables, Appendix E at text books of pages from 695 to 711.
The properties of some pure species tables, Appendix B at text books of pages from 679 to 682.

26. Example 4
Application of the theorem of corresponding states to estimate the thermodynamics properties of the pure components.
At temperature of 25C and pressure of 1 bar, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the carbon dioxide.

27. Example 5
Application of the theorem of corresponding states to estimate the thermodynamics properties of the pure components.
At temperature of 25C and pressure of 1 bar, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the methane.

28. Example 6
Application of the theorem of corresponding states to estimate the thermodynamics properties of the pure components.
(？) At temperature of 25C and pressure of 1 bar, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the benzene.