Property Scaling Relations for Nonpolar
Hydrocarbons
Sai R. Panuganti1, Francisco M. Vargas1, 2,
Walter G. Chapman1
1
Chemical and Biomolecular Engineering Department, Rice University,
Houston, USA
2 Department of Chemical Engineering, The Petroleum Institute,
Abu Dhabi, UAE
February, 2013
1
Outline
• One-Third Rule
• Electronic polarizability
• Dielectric constant
• Critical temperature and pressure
• Surface tension
• Conclusion
2
One-Third Rule
• Specific Refractivity: independent of the temperature and pressure
n, refractive index and ρ, mass density (g/cc)
 n2 1  1
 2
  Constant
n 2 
• For nonpolar hydrocarbons and their mixtures
 n D2  1  1 1
 2
 
 nD  2   3
n2 1
 True volume of the molecules in unit volume
2
n 2

 n 1 
 2

n  2
2
 True density of the molecules
• But strictly speaking, it is a function of the mass density and can be
 n2 1  1
expressed as
2
 2
  0.5054  0.3951  0.2314 
n  2 
L-L Expansion
3
One-Third Rule
Increase
Temperature
Volume occupied by molecules
without considering space
between molecules
 nD 2  1  1 1
 
For nonpolar hydrocarbons  2

 nD  2   3
V increases, ρ decreases
n increases
Vargas FM, Chapman WG; Fluid Phase Equilibria, 2010; 290:103-108
4
Electronic Polarizability
n 2  1 4

N
2
3
n 2
 Lorentz–Lorenz equation:
where, N – Number of molecules per unit volume
α – Polarizability
 Refractive index and Polarizability are independent of the amount
of material
n 2  1 4 N a
n 2
2
where, Na – Avogadro number
v – Molar Volume (v = MW/ρ)

3 v

 With the help of One-Third Rule the above expression simplifies as
 
MW
4N a
 The above equation is independent of the state of the substance and its
polarizability can be computed by just knowing the molecular weight
5
Predicted Polarizability
Mean Electronic Polarizability (cm3 x 10-24)
45
+4%
Deviation
30
25
20
X=Y
15
10
Predicted from L-L Expansion
Predicted from One-Third Rule
40
35
40
30
25
15
10
0
0
•
10
15
20 25 30
Experiment
35
40
0
45
• Using One-Third Rule
Average absolute deviation is 4.16 %
X=Y
20
5
5
+ 2.5 %
Deviation
35
5
0
Mean Electronic Polarizability (cm3 x 10-24)
45
5
•
•
10
15
20 25 30
Experiment
35
40
45
Using Lorentz-Lorenz Expansion
Average absolute deviation is 2.72 %
• Data shown is for 80 different nonpolar
hydrocarbons belonging to different homologues series
6
Dielectric Constant
 It is well established that for weakly magnetic materials
n  r
εr , relative permitivity
 For low-loss materials like nonpolar hydrocarbons,
k, dielectric constant
 r ( )   r (0)  k
 Substituting dielectric constant in the One-Third Rule and solving
for dielectric constant
2  3
k
3 
 The dielectric constant expression can handle operational
variations in temperature and pressure
 It is independent of the knowledge of individual constituents of a
mixture or the composition allowing the use for complex fluids
7
such as crude oils and polydisperse polymers
Predicted Dielectric Constant
2.8
2.8
Dielectric Constant
Predicted from L-L Expansion
2.6
2.6
Predicted from One-Third Rule
Dielectric Constant
2.4
2.4
2.2
2.2
2
X=Y
Series6
1.8
X=Y
Series6
2
1.8
1.6
1.6
+ 1 % Deviation
+ 2 % Deviation
1.4
1.4
1.4
1.6
1.8
2
2.2
Experiment
2.4
2.6
• Using One-Third Rule
• Average absolute deviation is 1.98 %
2.8
1.4
•
•
1.6
1.8
2
2.2
Experiment
2.4
2.6
Using Lorentz-Lorenz Expansion
Average absolute deviation is 1.0 %
• Data shown is for 260 nonpolar hydrocarbons, including polymers,
mixtures with varying temperatures and pressures
8
Panuganti SR, Vargas FM, Chapman WG; IEEE Transactions on Dielectrics and Electrical Insulation, 2013; Submitted
2.8
Critical Temperature and Pressure
 nD 2  1 
a 0 .5
  2.904
   52.04 2

v
 nD  2 
 From literature we have,
Hildebrand and Scott
Buckley et al.
 Thus, the following expression holds good
a
1/ 2
 nD 2  1  MW 
 MW 




 52.042 2

2
.
904
  

n

2
 20 
 D
 20 
Let,
 Applying One-Third Rule
also
 TC TB

 P
 C
f ( MW ,  20 )  MW  0.1674
TC
 function(MW ,  20 )
1/ 2
PC
1/ 2




 function( MW ,  20 )
Hildebrand JH, Scott RL; The Solubility of Nonelectrolytes, 1950
Buckley et al; Petroleum Science and Technology, 1998; 16:251-285
9
MW
 20
Critical Temperature and Pressure
350
300
y = 0.613x + 24.85
R² = 0.9973
y = 0.577x + 11.12
R² = 0.9984
250
(Tb*Tc/Pc)0.5 {K/atm0.5}
Tc/Pc0.5 {K/atm0.5}
300
250
200
150
100
200
150
100
50
50
0
0
100
200
300
f(MW,ρ20)
400
TC
 0.613 f (MW ,  20 )  24.85
1/ 2
PC
500
0
0
100
 TC TB 


 P 
 C 
200
300
f(MW,ρ20)
400
500
1/ 2
 0.577 f ( MW ,  20 )  11.12
Panuganti SR, Vargas FM, Chapman WG; Industrial and Engineering Chemistry Research, 2013; Accepted
10
Predicting Critical Properties
1100
70
Critical Temperature (K)
Critical Pressure (atm)
60
900
700
Predicted
Predicted
50
500
40
30
20
X=Y
300
X=Y
10
100
0
100
300
500
700
Experiment
900
Average absolute deviation
is 2.2 %
1100
0
10
20
30
40
Experiment
50
60
Average absolute deviation
is 4.5 %
• Data shown is for 80 different nonpolar
hydrocarbons belonging to different homologues series. The
applicability to mixtures is limited to nonpolar hydrocarbons
composed of similar sized molecules
11
70
Surface Tension from Hole Theory
 Volume of hole = Volume of liquid - Volume of solid
 Heat of fusion = Energy required for the formation of all the holes
2
2
2

(
P

P

P
4
Pr2 

x
y
z )
3
2  
E  Eq  EP   r ( p  po )  4r   


2m1
2m2 
3
 
 Solving the Schrodinger wave equation for a hole in a liquid,

8/7
 a
 2
V
2/7
 h

1/ 7
 2.4 
 Using the correlation of a/v2 from the previous section, at a given
   0.1674  
temperature we have
where, h(  )  

1/ 8
14
  C1h(  )  C2
 For example at 20oC we have
Furth R; Proc. Phys. Soc., 1940; 52:768-769



 20  34.39h(  20 )  7.509
12
Auluck FC, Rai RN; Journal of Chemical Physics, 1944; 12:321-322
Predicted Surface Tension
 The practical application of equation can improved further by
incorporating the temperature variation of surface tension
  TC  T h(  )
 With reference temperature as 20°C, surface tension at any other
temperature can be calculated as
 Tc  T  h( T ) 
40





34
.
39
h
(

)

7
.
509
n-Xylene
T
20


Tc  293  h(  20 ) 
Ethylbenzene
Methylcyclohexane
30
The parameter of critical temperature
can be eliminated using the equation
obtained in the critical properties
section.
Predicted
Cyclopentane
n-Hexane
20
10
Average absolute deviation is 1.8 %
0
0
10
20
Experiment
30
40
13
Conclusion
Input Parameters
Property
Density
MW
Boiling Point
Function of Mixtures
Temperature
Critical Temperature
Y
Y
Y
-
Y
Critical Pressure
Y
Y
Y
-
Y
Surface Tension
Y
Y
Y
Y
N
Electronic Polarizability
N
Y
N
-
-
Dielectric Constant
Y
N
N
Y
Y
• Polarizability of an asphaltene molecule of molecular weight
750 g/mol will be 99.16x10-24 cc
• Polydispere asphaltene system with density between 1.1 to 1.2 g/cc
at ambient conditions will have a dielectric constant
between 2.737 and 3
Panuganti SR, Vargas FM, Chapman WG; IEEE Transactions on Dielectrics and Electrical Insulation, 2013; Submitted
Panuganti SR, Vargas FM, Chapman WG; Industrial and Engineering Chemistry Research, 2013; Accepted
14