The Advanced Chemical

Engineering Thermodynamics

The thermodynamics properties of

fluids (II)

Q&A_-10- 11/17/2005(10)

Ji-Sheng Chang

Property relations

� The residual Gibbs free energy

� The definition of residual properties

� MR(T,P) = M(T,P) - M

IG(T,P)

� R: Residual property

� IG : Ideal gas behavior property

� MR(T,P) = M(T,P) - M

IG(T,P),

M are the properties as V, G, H, S, �

Property relations

� The residual properties

� VR(T,P) = V(T,P) - V

IG(T,P) = V - RT/P;

Basic thermodynamics relationship for calculations

� GR(T,P) = G(T,P) - G

IG(T,P);

More important in traditional Chemical Engineering Thermodynamics

� HR(T,P) = H(T,P) - H

IG(T,P)

Property relations

� Deriving the residual properties in terms of the PVT properties.

� From the generating function of Gibbs free energy, V

R/RT= [∂(GR

/RT)/∂P]T,

� Then d(GR/RT) = (V

R/RT)dP,

� to integrate from P=0 to P at constant T.

Property relations

� The integrated result in terms of PVT property as the formula (6.45)

T) (constant ,)1(0∫ −=P

R

P

dPZ

RT

G

T) (constant ,)1

(0∫ −=P

R

dPPRT

V

RT

G

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)

T) (constant ,0∫

∂∂

−=P

P

R

P

dP

T

ZT

RT

H

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)

T) (constant ,)1(0 0∫ ∫ −−

∂

∂−=

P P

P

R

P

dPZ

P

dP

T

ZT

RT

S

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 = ZρRT; as ρ = 1/V

� dP=RTd(Zρ)=RT(Zdρ+ρdZ) at constant T

Property relations

� dP/P=dρ/ρ+dZ/Z,

� as the formula (6.58)

� the alternative form.

T) (constant ,ln1)1(0

ZZd

ZRT

GR

−−+−= ∫ρ

ρρ

T) (constant ,ln1)1( ZZV

dVZ

RT

G VR

−−+−= ∫∞

T) (constant ,)1(0∫ −=P

R

P

dPZ

RT

G

Property relations

� Calculating the residual properties � The PVT data,

Using graphical integration method to calculate the residual Gibbs free energy from steam data.

T) (constant ,1

0∫

−=P

R

dPP

Z

RT

G

Phase diagram

0 200 400 600 800 1000T/K

0

100

200

300

400

500

P/b

ar

Solid Liquid

Gas

PC,TC

1. PT phase diagram of water

2. The Solid/Liquid equilibrium curve

for water

for most component

200OC path

Critical properties

PC=220.55 bar

TC=647.1 K

TTri=273.16 K

Vapor/Liquid equilibrium curve

� The Antoine equation

� The constant of Antoine

equation for water

� A=16.3872

� B=3056.96

� C=217.625

CCT/

BA/kPaPln

sat

+°−=

Property

� Critical properties of water

� PC=220.55 bar

� TC=647.1 K

� PTri=0.611 kPa

� TTri=273.16 K

� Tn=373.15 K at P= 1.01325 bar = 1 atm

Property relations

� Problems

� Calculate the residual Gibbs

energy of the 200°C superheat steam at 10 bar, using the PVT

data from steam tables

Properties

P/kPa P/bar V/cm3g-1 V/cm3mol-1 Z (Z-1)/P

1 0.01 218350 3933575 0.999951 -0.00493

50 0.5 4356.0 78473.34 0.997432 -0.00514

100 1 2172.3 39133.98 0.994822 -0.00518

200 2 1080.4 19463.41 0.989555 -0.00522

300 3 716.35 12905.05 0.984174 -0.00528

400 4 534.26 9624.694 0.978674 -0.00533

500 5 424.96 7655.654 0.973069 -0.00539

600 6 352.04 6342.001 0.967317 -0.00545

700 7 299.92 5403.059 0.961455 -0.00551

800 8 260.79 4698.132 0.955446 -0.00557

Properties

P/kPa P/bar V/cm3g-1 V/cm3mol-1 Z (Z-1)/P

900 9 230.32 4149.215 0.949291 -0.00563

1000 10 205.92 3709.649 0.943027 -0.0057

1100 11 185.92 3349.349 0.936579 -0.00577

1200 12 169.23 3048.678 0.930002 -0.00583

1300 13 155.09 2793.946 0.923321 -0.0059

1400 14 142.94 2575.064 0.916447 -0.00597

1500 15 132.38 2384.826 0.909367 -0.00604

1550 15.5 127.61 2298.894 0.90582 -0.00608

1554.9 15.55 127.20 2291.508 0.905764 -0.00606

Residual Gibbs free energy

properties

0 4 8 12 16 20P [ bar ]

-0.008

-0.006

-0.004

-0.002

0

(Z-1

)/P

[

ba

r-1

]

GR/RT of superheat steam

200OC

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.

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

Deriving the

formulas

� Second virial coefficient

� From PVT data

� From generalized

correlation equations

� Based molecular

thermodynamic, it is integrated from

potential function

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.

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.

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

Theorem of corresponding states

� Three-parameter theorem of corresponding states

� Given T and P

� Finding Tc, Pc, ω

� Calculate Tr, Pr

� Using Tr, Pr, ω to finding the reduced properties from Table E.1 to

E.16

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.

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.

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.

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.