1. CHAPTER 3 Volumetric Properties of Pure Fluids: Part 1
Norhaniza binti Yusof
Faculty of Chemical Engineering
Universiti Teknologi Malaysia, UTM
Johor, Johor Bahru, Malaysia
Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

2. Topic Outcomes Week Topic Topic Outcomes 2-3
Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Topic Outcomes
Week
Topic
Topic Outcomes
2-3
Volumetric Properties of Pure Fluids
Introduction to volumetric properties
PVT behaviour of pure substances
Volumetric properties from equations of state:
Ideal gas equation
Virial equation
Generic cubic equations
Volumetric properties from generalized correlations:
Gases
Liquid
Volumetric properties from thermodynamic Tables and Diagrams
It is expected that students are able to:
Determine the state/phase of a given fluid at given conditions.
Calculate the volumetric properties using the equations of state and generalized correlations for a given system.
Identify the applicability and limitation of every equation of state.
Determine the volumetric properties from thermodynamic tables and diagrams.

3. Scope of Lecture Introduction to volumetric properties
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Scope of Lecture
Introduction to volumetric properties
PVT behaviour of pure substances
Volumetric properties from equations of state

4. PVT Behavior of Pure Substances
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5. PT Diagram for Pure Substances
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PT Diagram for Pure Substances
Ftp = 0 (invariant)
Fliq = 2 (divariant)
Fvap. curve = 1
(univariant)
Pressure
Temperature Pc Tc
Solid region
Fusion curve 2 1 3 A B
Vapor
region
Gas region
Fluid region
Liquid region
Vaporization
curve
Triple
point
Sublimation C
No. of phases
No. of species
DOF for the system
Independent variables (T, P)
F = 2 –  + N
Supercritical
T > Tc
Note: DOF, degree of freedom

6. PV Diagrams for Pure Substances
In a PV diagram, the phase boundaries becomes areas/region
Sol., liq. & gas regions
Liq., liq./vap. regions & vap. with isotherms
Single-phase (sat.) vap. at cond. T
Single-phase (sat.) liqs. at boiling T
Triple point
Subcritical T
→ consist of 3 segments
Note: Sol., solid; liq., liquid; vap., vapor; sat., saturated; cond., condensation; T, temperature

7. PVT Surface for a Real Substance
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PVT Surface for a Real Substance P T V CP
Solid
Liquid
Triple line
Vapor
Gas
Note: CP., critical point

8. PVT Surface (Cont.) Constant temperature line
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PVT Surface (Cont.)
Constant temperature line

9. Pure Substances Phase Any fluid could be in one of below conditions :
Single-phase liquid
Single-phase gas
2-phases, liquid & gas
Generally, if the system is at any T or P;
Condition T and P
Phase
T > Tc
Gas
T < Tc and P > Pc
Liquid
T < Tc and P < Pc
See boiling point
T > Tbpt
T < Tbpt
T = Tbpt
2-phase (saturated)
*For H2O, Tc = 374 oC and Pc = 220 bar
Note: Tc, critical temperature, Pc, critical pressure, Tbpt, biling point temperature

10. Determination of Pure Substances Phase
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Determination of Pure Substances Phase
Boiling point and vapor pressure could be obtained from :
Antoine equation
Cox Chart
E.g.: Antoine equation for H2O;
log10 P (mmHg) = – / (T oC + 228)
Liquid phase: V is almost stable and not depend on P and T.
Gas phase: V of pure substance is depend on T and P.
Mathematic relations between P, V, T of pure substance is called as EQUATION OF STATE

11. Single-Phase Region f (P, V, T ) = 0 PV = RT
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Single-Phase Region
From the regions of the diagram (PV) where a single phase exists,
implies a relation connecting P, V & T →express by function equation
f (P, V, T ) = 0
PVT equation of state (EoS)
Simplest EoS → ideal gas law
Valid for low P
Will be discussed later
PV = RT
To solve the equation,
V = f (P , T ) or P = f (V , T) or T= f (P , V)
Note: P, pressure; EoS; equation of state

12. Example V = f (P, T) V = V (P, T) Hence,
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Example
V = f (P, T)
V = V (P, T)
Hence,
For liquid phase
incompressible fluid
 and  are very small ( ≈ 0).
Partial derivative in the equation
Integration
Volume expansivity, 
Isothermal compressibility, 
Divide by V

13. Notes Gas Ideal gas Non-ideal gas PV = ZRT
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Notes
Gas
Ideal gas
Non-ideal gas
PV = ZRT
Valid at low pressure
Z = 1
Equations of state (from PVT data).
Generalized correlations
Equations of state
Theorem of corresponding states

14. Cubic Equation of State
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Equation of State
Equation of State
Ideal Gas Equation
Virial Equation
Cubic Equation of State
Van Der
Waals
Redlich/
Kwong
Kwong/
Soave
Peng/
Robinson
Generic
Vapor &
Vapor-Like
Roots
Liquid &
Liquid-Like

15. Virial Equations of State
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16. Virial Equations (Gas Phase)
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Virial Equations (Gas Phase)
A useful auxiliary thermodynamic property is defined by the equation
Compressibility factor
There are 2 types of virial equations / virial expansions :
B’, C’, D’, B, C, D etc.
→virial coefficients
Only depend on T.
Obtained from PVT data.
Note: T, temperature

17. Application of the Virial Equations
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Application of the Virial Equations
CH4
Low to moderate P (P < 15 bar)
2 terms
High P up to ≈50 bar (below the Pc)
Truncated to 3 terms
Note: P, pressure; Pc, critical pressure

18. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
The Ideal Gas
Virial expansion arise on account of molecular interaction.
Term B/V arises on account of interactions between pairs of molecules
The C/V2 term, on account of 3-body interactions.
For ideal gas interaction molecular is assume not existed.
As the P of real gas is reduced at constant T
→ V  & the contribution of the terms B/V, C/V2, etc.,  .
As P → 0, V becomes , then Z approaches unity,
= 1
PV = ZRT
(low P only)

19. Internal energy U = U(T)
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Internal energy
Internal energy of a REAL GAS is a function of P & T
P dependency is the result of forces between the molecules.
If such forces did not exist (IDEAL GAS behavior) → the internal energy of gas depends on T only
U = U(T)
(Ideal gas)

20. Property Relations for an Ideal Gas
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Property Relations for an Ideal Gas
Heat capacity at constant V (CV) is a func. of T only
Enthalpy (H) is a func. of T only
Heat capacity at constant P (CP) is a func. of T only
Relation between CP and CV
This equation does not imply that CP and CV are themselves constant for an ideal gas, but only that they vary with T in such a way that their difference is equal to R.

21. Equations for Process Calculations
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Equations for Process Calculations
For a unit mass or a mole of IDEAL GAS in any mechanically reversible closed-system process:
These equations may be applied to the following processes

22. Adiabatic Process Isothermal process: Isobaric process:
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Adiabatic Process
Isothermal process:
Isobaric process:
Isochoric process:

23. Adiabatic Process (Const. Heat Capacities)
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Adiabatic Process (Const. Heat Capacities)
Equations
Other expressions
Work
The process should be mechanically reversible!
Alternative forms

24. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Example 1 (Ideal gas)
One mol of an ideal gas with Cp = (7/2)R and Cv = (5/2)R expands from P1 = 8 bar and T1 = 600 K to P2 = 1 bar by each of the following paths:
Constant volume
Constant temperature
Adiabatically
Assuming mechanical reversibility, calculate W, Q, U, and H for each process.

25. Example 2 (Virial Equation)
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Example 2 (Virial Equation)
Reported values for the virial coefficients of isopropanol vapor at
200 C are:
B = – 388 cm3 mol–1 C = – cm6 mol–2
Calculate V and Z for isopropanol at 200 C and 10 bar by:
The ideal-gas equation
Truncated virial equation to 3 terms

26. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Tutorial 1
For methyl chloride at 100 C the 2nd and 3rd virial coefficients are:
B = – m3 mol –1 C = 25, 200 cm6 mol –2
Calculate the work of mechanically reversible, isothermal compression of 1 mol of methyl chloride from 1 bar to 55 bar at 100 C. Base calculations on the following forms of the virial equations:
Where

27. CHAPTER 3 Volumetric Properties of Pure Fluids: Part 2
N – 1.00 P.M Feb 21, 2013 (Thu)
CHAPTER 3
Volumetric Properties of Pure Fluids: Part 2
Mohd Asmadi Bin Mohammed Yussuf
Faculty of Chemical Engineering
Universiti Teknologi Malaysia, UTM
Johor, Johor Bahru, Malaysia
Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

28. Scope of Lecture Cubic equation of state Generic cubic
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Scope of Lecture
Cubic equation of state
Generic cubic
Example & Tutorial

29. Cubic Equation of State
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30. Generic Cubic Equation of State
The simplest equations capable of representing both liquid and vapor behavior but not for dual-phase condition.
More accurate for a wide range of T and P
Where b , θ , κ , λ and η are parameters which is general depend on T and (for mixture) composition .
Note: T, temperature; P, pressure; cubic equation of state ,15 < p < 50 bar

31. Equation of State Parameters
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Equation of State Parameters
When in polynomial form :
Where
Pc and Tc → pressure and temperature in critical point
a and b → +ve constants.

32. Cubic Equation of State
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Cubic Equation of State
Cubic Equation of State
Van Der
Waals
Redlich/
Kwong
Redlich/
Kwong/
Soave
Peng/
Robinson
Generic
Vapor &
Vapor-Like
Roots
Liquid &
Liquid-Like
Roots

33. van der Waals θ = a η = b κ = λ = 0
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van der Waals
θ = a
η = b
A Generic Cubic EoS (previous slide)
κ = λ = 0
Reduces to van der Waals EoS
Will be discussed later
Note: EoS; equation of state

34. Redlich/Kwong Equation, Solve by cubic equation solver.
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Redlich/Kwong
Equation,
Solve by cubic equation solver.

35. V for Gas & Liquid Phases (RK)
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V for Gas & Liquid Phases (RK)
V for gas phase
V for liquid phase
Multiple &
rearrange
Solve by iteration method, use b as an initial Vo.
Note: RK, Redlich/Kwong

36. Redlich-Kwong-Soave & Peng-Robinson
Acentric factor (App. B)
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37. Parameters for all Cubic Equation of State
EoS
a( Tr) σ ε Ω Ψ Zc
van der Waals 1
1/8
27/64
3/8
Redlich/ Kwong
1/3
Redlich-Kwong-Soave
Peng-Robinson
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Note: Please text book p. 98

38. Generic Cubic Vapor & Vapor-Like Roots of the Generic Cubic EoS
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Generic Cubic
Vapor & Vapor-Like Roots of the Generic Cubic EoS
Solution for V may be by
Trial
Iteration
With the solve routine of a software package
Initial estimate for V is the ideal-gas value →

39. Dimensionless Quantities
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Dimensionless Quantities
An equation for Z equivalent to equation above is obtained through
substitute
2 dimensionless quantities leads to simplification

40. Process continues to convergence
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Determination of Z
Therefore,
Iteration
Start with Z = 1
→ Substituted on the right side.
Calculated Z
→ Returned to the right side
Process continues to convergence
Final Z
→ yields the volume root through
V = ZRT/P

41. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Generic Cubic (Cont.)
Liquid & Liquid-Like Roots of the Generic Cubic EoS
V in the numerator of the final fraction to give :
Starting value V = b.
An equation for Z equivalent to the equation above :

42. Iteration For iteration Z = β. Once Z is known, the volume root is
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Iteration
For iteration Z = β. Once Z is known, the volume root is
EoS which express Z as a function of Tr & Pr
Called generalized
Their general applicability to all gases & liquids.
Note: EoS; equation of state

43. A Generic Cubic Equation of State
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A Generic Cubic Equation of State
An important class of cubic equations results from the preceding equation with the assignments
η = b θ = a ( T ) κ = ( є + σ ) b λ = є σ b2
ε & σ are pure numbers
→same for all substances
a(T) is specific to each EoS
Note: EoS; equation of state

44. Determination of EoS Parameters
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Determination of EoS Parameters
Suitable estimates of EoS parameters are usually found from values for the critical constants Tc and Pc → critical isotherm exhibits a horizontal inflection at the critical point!
At critical point ( Pc, Tc, Vc & Zc ) :
Replace Pc, Tc and Vc
From van der Waals equation :
Then,
Note: EoS; equation of state

45. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
EoS Parameters (Cont.)
An analogous procedure may be applied to the generic cubic equation
Previous slide
Yielding expressions for parameters
Ω and Ψ
Pure numbers,
Independent of substance
Determined for a particular EoS from the values assigned to ε & σ

46. EoS Parameters (Cont.) unity at the Tc
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EoS Parameters (Cont.)
This result may be extended to T other than the critical by Introduction of a dimensionless function
unity at the Tc
Function α(Tr) : an empirical expression
specific to a particular equation of state.
Note: Tc; critical temperature

47. Theorem of Corresponding States
All fluids, when compared at the same Tr & Pr, have approximately the same Z, and all deviate from ideal-gas behavior to about the same degree .
This theorem is very nearly exact for the simple fluids (Ar, Kr, Xe).
However, systematic deviations are observed for more complex fluids.
Note: Z; compressibility factor ; Tr, reduce temperature; Pr, reduced pressure

48. Acentric Factor (Cont.)
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Acentric Factor (Cont.)
Approximate T dependence of the reduced vapor pressure

49. Acentric Factor (Cont.)
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Acentric Factor (Cont.)
Therefore, an acentric factor is introduced as follow:
Acentric factor
At Tr = 0.7
→ = 0 (Ar, Kr & Xe)
This value of  can be determined for any fluid from Tc, Pc and a single vapor-pressure measurement made at Tr = 0.7
App. B lists the values of ω and the critical constants Tc, Pc, and Vc for a number of fluids.

50. Acentric Factor (Cont.)
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Acentric Factor (Cont.)
All fluids having the same value of ω, when compared at the same Tr & Pr, have about the same value of Z, and all deviate from ideal-gas behavior to about the same degree. w

51. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Example 3
Given that the vapor pressure of n-butane at 350 K is bar, find the volumes of (a) saturated-vapor and (b) saturated-liquid n-butane at these conditions as given by the Redlich/Kwong equation.

52. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Example 4
Calculate Z and V for ethylene at 25 C and 12 bar by the following equations:
The Redlick/Kwong equation
The Soave/Redlich/Kwong equation
The Peng/Robinson equation

53. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Tutorial 2
Calculate Z and V for ethylene at 50 C and 15 bar by the following equations:
The Redlick/Kwong equation
The Soave/Redlich/Kwong equation
The Peng/Robinson equation

54. CHAPTER 3 Volumetric Properties of Pure Fluids: Part 3
N02 2-6, –1.00P.M Feb 17, 2013 (Mon)
CHAPTER 3
Volumetric Properties of Pure Fluids: Part 3
Mohd Asmadi Bin Mohammed Yussuf
Faculty of Chemical Engineering
Universiti Teknologi Malaysia, UTM
Johor, Johor Bahru, Malaysia
Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

55. Scope of Lecture Generalized Correlations for Gases
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Scope of Lecture
Generalized Correlations for Gases
Compressibility factor, Z
2nd virial coefficient, B
Generalized Correlations for Liquids
Examples & Tutorials

56. Generalized Correlations for Gases
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57. Pitzer Correlations for the Z
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Pitzer Correlations for the Z
The correlation for Z takes form :
Equally suitable for gases & liquids
Where Z0 and Z1 are functions of both Tr and Pr .
From Table E1–E4, require interpolation, see App. F
Error
Non-polar/slightly polar < 2-3%
Highly polar larger error
When ω = 0 →simple fluids.
Ar, Kr, Xe

58. Lee/ Kesler Correlation for Z0 = F0 (Tr, Pr)
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Lee/ Kesler Correlation for Z0 = F0 (Tr, Pr) Pr
1.2
1.0
0.8
0.6 Z0
0.4
0.05
0.1
0.2
0.5
1.4
2.0
5.0
10.0
Two-phase region
Compressed liquids (Tr < 1.0)
Gases
Tr = 0.7
0.9
4.0
1.5 C

59. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Quantum Gases (H2, He, Ne)
Do not conform to the same corresponding-states behavior as do normal fluids
The correlation is accommodated by use of T-dependent effective critical parameters.
E.g. Hydrogen
T = absolute temperature in K

60. Pitzer Correlations for B
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Pitzer Correlations for B
Correlations for 2nd Virial Coefficientn(comb Eq 3.61 and 3.63)
Validity at low to moderate P
Simple and recommended
Tr > Tr ≈ 3, there appears to be no limitation on the P
Most accurate for non-polar species

61. Virial Correlations (Cont.)
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Virial Correlations (Cont.)
The simplest form of the virial equation :
Compare
2nd correlations, yields
2nd virial coefficients = f(T) only
B0 & B1 = f(Tr) only
Note: f, function

62. Comparison of Correlation for Z0
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Comparison of Correlation for Z0
Region above the dashed line the 2-correlations differ by <2%
Points: Lee/ Kesler correlation
0.0
0.5
1.0
1.5
2.0
2.5 Pr
0.9
0.8
0.7 Z0
Tr = 0.8
1.1
1.3
1.8
2.4
4.0
Straight lines: virial-coefficient correlation

63. App. Validity of the Ideal-Gas Eqn.
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App. Validity of the Ideal-Gas Eqn. 10 1
0.1
0.01 Pr
0.001 2 3 4 Tr
Z0 = 0.98
Z0 = 1.02
Region where Z0 lines between 0.98 &1.02
Ideal-gas eqn. is a reasonable approximation to reality

64. Generalized Correlations for Liquid
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65. Molar Volume of Sat. Liquids
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Molar Volume of Sat. Liquids
The generalized cubic equation of state →low accuray
The Lee/Kesler correlation includes data for subcooled liquids
Suitable for non-polar & slightly polar fluids
Estimation of molar volumes of saturated liquids
Data required → crirical conts. (App. B)
Results accurate → 1 or 2%

66. Generalized Density Correlation
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Generalized Density Correlation
2-parameter corresponding-states correlation for estimation of liquid volumes.
Density at the critical point
Known volume
Required volume
Reduced densities read from next page
Give good results and requires only experimental data (usually available).

67. Generalized Density Correlation (Cont.)
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Generalized Density Correlation (Cont.)
Tr = 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Tr = 0.95
0.97
0.99
Saturated liquid
3.5
3.0
2.5
2.0
r
1.5 1 2 3 4 5 6 7 8 9 10 Pr

68. Examples & Tutorials
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69. Example 4 (Gases) Solution 4:
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Example 4 (Gases)
Determine the molar volume of n-butane at 510K and 25 bar by, (a) the ideal-gas equation; (b) the generalized compressibility-factor correlation; (c) the generalized virial-coefficient correlation.
Solution 4:
The ideal-gas equation
The generalized compressibility-factor correlation

70. Solution 4 (Cont.) The generalized virial-coefficient correlation
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Solution 4 (Cont.)
The generalized virial-coefficient correlation

71. Example 5 (Gases) Solution 5:
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Example 5 (Gases)
What pressure is generated when 1 (lb mol) of CH4 is stored in a volume of 2 ft3 at 122 F using (a) the ideal-gas equation; (b) the Redlich/Kwong equation; (c) a generalized correlation.
Solution 5:
The ideal-gas equation
The RK equation

72. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Solution 5 (Cont.)
The generalized compressibility-factor correlation (high P)
Z starts at Z0 = 1 Pr
Tr = 1.695
Table E.3 & E.4
Z0 & Z1
Z = Z0 + Z1
Final Z
Converges
(Z = 0.890)
Pr1 = 4.68

73. Example 6 (Gases) Solution 6:
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Example 6 (Gases)
A mass of 500 g of gases ammonia is contained in a cm3 vessel immersed in a constant-temperature bath at 65 C. Calculate the pressure of the gas by (a) the ideal-gas equation; (b) a generalized correlation .
Solution 6:
The ideal-gas equation

74. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Solution 6 (Cont.)
The generalized virial-coefficient correlation (low P, Pr ~ 3 )

75. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia
Tutorial 3
Calculate the molar volume of saturated liquid and the molar volume of saturated vapor by Redlich /Kwong equation fo one of the folowing and compare results with values found by suitable generalized correlations.
Propane at 40 C where Psat = bar
n-Butane at 100 C where Psat = bar
Tutorial 4
The vapor-phase molar volume of a particular compound is reported as 23,000 cm3mol–1 at 300 K and 1 bar. No other data are available. Without ideal-gas behavior, determine a reasonable estimate of the molar volume of the vapor at 300 K and 5 bar.

76. Example 7 (Liquid) Solution 7:
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Example 7 (Liquid)
For ammonia at 310 K, estimate the density of (a) the saturated liquid; (b) the liquid at 100 bar
Solution 7:
Apply the Rackett equation at the Tr
Exp. = cm3 mol–1
→ differs = 2.7%
Compares

77. Solution 7 (Cont.) At 100 bar Fig. 3.16 Exp. value
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Solution 7 (Cont.)
At 100 bar
Fig. 3.16
Exp. = cm3 mol–1
→ differs = 6.5%
Exp. value

78. Tutorial 5 Tutorial 6 Tutorial 7
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Tutorial 5
The specific volume of isobutane liquid at 300 K and 4 bar is cm3 g–1. Estimate the specific volume at 415 k and 75 bar.
Tutorial 6
The density of liquid n-pentane is g cm–3 at 18 C and 1 bar. Estimate its density at 140 C and 120 bar.
Tutorial 7
Estimate the volume change of vaporization for ammonia at 20 C. At this temperature the vapor pressure of ammonia is 857 kPa.