Q
= Hamaker constant, ergs
8,
=
X
saturation of phase i (volume fraction pore space occupied), dimensionless = distance into porous medium, cm
GREEKLETTERS ff
A I.($
= = =
fiber volume fraction, dimensionless filter coefficient, cm-1 viscosity of phase i, g/cm-sec
SUBSCRIPTS 1 2 C
0
L
refers to aqueous phase refers t o coalesced oil phase refers to limiting saturation for flow of a given phase (except for p c ) = refers to inlet face (except for ICo) = refers to outlet face = = =
Friedlander, S. K., Wang, C. S., J . Colloid Interface Sci. 22, 126 (1966). Kruyt, H. R., “Colloid Science,” Vol. I, Elsevier, New York, N. Y . , 1950. Parker, J., Ind. Eng. Cham. 52, 247 (1960). Rose, P. hl., “Mechanisms of Operation of a Fibrous Bed Coalescer,” M.S. Thesis, Illinois Institute of Technology, Chicago, Ill., 1963. Spielman, L. A., “Separation of Finely Dispersed LiquidLiquid Suspensions by Flow through Fibrous Media,” Ph.D. Dissertation, University of California, Berkeley, Calif., 1968. Spielman, L. A., Goren, S. L., Environ. Sci. Technol. 2, 279 (1968). Spielman, L. A., Goren, S. L., Environ. Sci. Technol. 4, 135 (1970a). Spielman, L. A., Goren, S. L., Tnd. Eng. Chem. 62, 10 (1970b). Spielman, L. A,, Goren, S. L., IND.ENG.CHEM.,FUNDAM. 11, 66 (1972). RECEIVED for review April 30, 1971 ACCEPTED September 24, 1971
literature Cited Dawson, S. V., “Theory of Collection of Airborne Particles by
Fibrous Filters,” Ph.D. Dissertation, Harvard University, Cambridge, Mass., 1969.
This work was supported by the Water Resources Center, University of Californa.
Prediction of Molar Volumes and Thermal Pressure Coefficients from Carbon Number and Temperature for Homologous Series of Unbranched Hydrocarbons in the Liquid State Jan Meisner Shell Research N . V., Koninklijlce/Shell-Laboratorium,Amsterdam, The Netherlands
Based on the principle of corresponding states for chain-molecule liquids, a set of equations has been derived to predict molar volumes and thermal pressure coefficients from carbon number and temperature for homologous unbranched hydrocarbons in the liquid state. The procedure for determining the adjustable parameters occurring in the equations and for checking the consistency of the results is described. The equations should be particularly valuable in phase equilibrium studies, where, with an appropriate computer program, they should provide a rapid means of calculating data of this type.
M a n y calculations associated with the phase equilibria of mixtures of hydrocarbons need the numerical values of the molar volumes of t h e components in the liquid state at various temperatures. For normal paraffins, monoolefins, alkylbenzenes, alkylcyclopentanes, and alkylcyclohexanes u p to CZZthese values can be readily derived from t h e liquid density data recorded, for various temperatures at atmospheric pressure, in the tables of t h e American Petroleum Institute, Research Project 44 (Rossini, et al., 1953). However, although tables of this kind can nowadays easily be stored in computer memories or on magnetic tapes, i t is much more efficient t o have the data expressed as a set of equations which predict t h e molar volumes as a function of carbon number and temperature. We therefore undertook t o derive
such a set of equations, using the formulation of the principle of the corresponding states for chain-molecule liquids given by Holleman and Hijmans (1965). I n this paper the methods of determining the adjustable parameters occurring in the equations and of checking the consistency of the results are described. Equations Resulting from the Corresponding State Principle
According to Holleman and Hijmans (1965), the molar volume V ( n , T ) of a n y chain-molecule liquid with a total carbon number n at zero pressure and absolute temperature T can be described a s
V ( n ,T )
=
VO(~)V*[T/TO(~)I
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
(1) 83
moid.4, md.mim 00000
im
CD
z tt EI 0
9 W M OQ, O d i a d m m *d. DIM M M m m
mm
d d
d d
d.m
0
i
4,
3
. . . . . .MM 9
*,
w W o m ( 9 W M c o i m mr-
I-+Q,wwwcimd.m w a o o + y y m m . .m .m . 0?” ++d
i
o w o m p‘:? mm
r-*
UJm
ai?
i m
Ii - w m
M M
o m
34)
OI-ow
0
5 M
%O
3
0
d.
m
w
oo
m
o
‘990” 0 0 0 0
2
13 bf
m
%E s
WOO Md.
. . c?“
ii0 %
W M 0 0 0 1 0
d.mmmd.
90009
wd.
M O
5 3 2% 8 mm
* I-w m 0. - .
r-m d.m m. m.
m m &mi
0
. .
3 3 i
n i
d
i d .
m 0
md.
a. -. i m
d .. w.
mm
CCO
m m wQ,a 0 mI- Q , O m m m m m M *M i i
i r i
Im- 0w I m - m3 *+- I - a mw m a (w9i m* ow wQ I, - m mdi C U J m d.W m. M. * w . r .- w. O.- m. m. m. W. I -. O .O . . 0. - . . . m. a. mm
M - m O W
0 0 0 0 0
M M 0
i i
C D a
d.
01
Q , I - O m + M I - M a iQ, d i m I-0 MOO m d . I - m d i r - 0 D I m I - O m m r - O M mI- 0 y 03 0 y m, ? W. I-. ? ? i . m. 4 “ W r - 0
mmmmmm
p‘:
0 0
DIN
mm
m o
0 0 0 0
8
0
m M
00
0,-
m o
8
I-wmmI-
mwmmIu s oooc
335
0
00008 00000
8 c; E0
0 0 0 0
00000
84 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
where Vo(n)and To(%)are the so-called reduction parameters for volume and temperature, respectively, and V * is a coiitinuous, monotonically increasing function of the reduced temperature. The same authors furthermore suggested t h a t within a homologous series the reduction parameters Vo(n) and TO(%) can be described approximatively by
Vo(n) =
ao(n
+
5 30
5 20
5 10
(2)
ad
500
and
+ + Pz/n)/fo(n + f d
Tab) = Po(n
PI
(3)
where al,Pi, P 2 , and t1are parameters specific t o a particular homologous series, while the parameters ao,Po, and todescribe the asymptotic behavior in cases where n is very large, all three being the same for all the homologous series considered here.
4 90
480 000
10000 20000 30000 40000
50000 60000 70000
80003
(n)1r8481 Plot of In [V(n, t)/V,(n)] vs. [T/To(n)]i.84s1for [TIT,
Figure 1. normal paraffins from
C1to Czo relative to normal heptane
Determination of the Reduction Parameters for Volume and Temperature
T o calculate the adjustable reduction parameters Vo(n) and To(%)we needed an analytical expression for the function V * in eq 1. Plots of V ( n , T ) us. T for constant values of n suggested an approximation of V * of the type
PI
v*[T/T~(~ =:) el I exp[e,( T / T ~ ( ~ )
(4)
and as a result we were able to write eq 1 in the form In ~ ( nT,)
= 111
+
[e,~o(n)I ez[T/To(n)IBa
(5)
Arbitrarily choosing n-heptane as a general reference compound, so t h a t for normal paraffins V0(7)= 1 and To(7) 1, we fitted eq 5 to the XPI liquid density data by means of a simultaneous generalized nonlinear least-squares procedure, which gave us t h e following numerical values for the adjustable parameters el through e3
el
=
e3 =
[TIT,
(n)~”~~’
Figure 2. Plot of In [V(n, t)/V,(n)] vs. [T/TO(n)]1.8481 for normal monoolefins from Czto Czorelative to normal heptane
119.60
1.8481
and the sets of adjusted reduction parameters Vo(n) and To(%)shown in Table I . I n Figures 1-5, ln [V(n, T)/Vo(n)] is plotted against [T/To(n) 11.8461, showing the very close h e a r correlation between these two quantities, from which it may be inferred that the approximation used for V * is quite appropriate and that the corresponding state principle for chain-molecule liquids holds good. The “lack of fit,” i.e., t h e set of relative deviations of the observed values of V ( n , T ) from those predicted with eq 5 , can be characterized for the various homologous series by the relative standard deviations given a t the bottom of Table I. The order of magnitude of the lack of fit is what might be expected from t h e inaccuracy of t h e liquid density data specified in the introduction to the A P I tables. The precision of t h e adjusted values of Vo(n) and TO(%)for the various compounds can be characterized by the standard errors, also given a t the bottom of Table I. Dependency of the Reduction Parameters on Carbon Number
By substituting into eq 5 the respective adjusted values for the parameters el through e3 and using eq 2 aiid 3 for Vo(n)and To(n), respectively, we arrive a t
Omitting the first member of each homologous series we tried t o fit eq 7 to t h e A P I liquid density data by means of a generalized nonlinear least-squares procedure in order t o obtain numerical values for the adjustable parametes ao, C Y I , PO/&, PI, Pz, and f l for each homologous series. However, for some series we were unable t o arrive a t reliable adjusted values for Po/.$o, P1, Pz, and f l . Some of these in fact appeared to be redundant, which meant t h a t our data did not contain sufficient information to allow them t o be determined. T o obviate these difficulties we reduced eq 3 by a simple division t o
TO(%)= POU
+
+ w / n 2 - t 1 d n 2 ( n+ fd1
F I / ~
(8)
where po = P o / ~ op1 , PI- &, and pz P2 - f l p l . We then discarded the last term of the right-hand side of eq 8 whenever it was found that it did not make a significant contribution to the accuracy of fit, readjusted the values of t h e other parameters by fitting the reduced equation t o the data, and finally repeated t h e procedure for the second-last term. The adjusted parameters thus obtained, together with their standard errors and the relative standard deviations characterizing lack of fit, are given in Table 11, while in Figures Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
85
01
0 V,(n) I T,(n)
02
I
I
I
I
I
t/n 05
04
03 I
I
1
I
I
2 8 + 14
26- 13 24-12 22- lt 20-10 18-09
16--08
O W
10000 20000 30000 40000 50000 60000 70000 8000;, [TIT,
14 -07
(n)j18481
Figure 3. Plot of In [V(n, t ) / V o ( n ) ] vs. [ T / T o ( ~ ) ]for ~ . ~ ~ ~ 12 ~ -*06 normal alkylbenzenes from Cg to Czz relative to normal 1.0-.0.5 heptane 0.8 -04
0.6 -.0.3 0.4 -.0.2
0.2
-.0.1 0
2
4
6
8
10
12
16
14
18
20
22
24 n
Figure 6. Dependency of Vo(n)and To(n)on carbon number; normal paraffins Cz to Czo 1 /n
01
0 I I V,(n) To(n) 28$14
[TIT,
02 I
I
I
04
03 I
I
I
I
05 I
i
2 6 - - 13
(n)1'8481
Figure 4. Plot of In [ V ( n , t ) / V o ( n ) ] vs. [ T / T o ( ~ ) ] ' for .~~~~ normal alkylcyclopentanes from Cg to Czl relative to normal heptane
24-- I 2 22- t t
1
2 0 - - i0 5x0
. .
~,. , ... , ,
,
,
,
.T.
. ,. .
. .. , .
. , .-
, . . ~ .. .,. ..,
7 '
'I
18-09
i6-.08 14-*07 12--06
10- 0 5 08--04
06--03
[T/T,
6-10, Vo(n)is plotted against n, with the appropriate straight line according to eq 2, and To(n)is plotted against the reciprocal of n, with the appropriate curve according to eq 8. Comparing the standard deviations characterizing the lack of fit obtained with eq 5 and 7 in its modified form (Tables I and 11,respectively) it will be seen that for all the homologous series the lack of fit increases when we substitute in eq 5 the Ind. Eng. Chem. Fundam.,
Vol. 1 1 , No. 1 , 1972
02--01
(n1]'8481
Figure 5. Plot of In [(V(n, t ) / V o ( n ) vs. for normal alkylcyclohexanes from C6 to CzZ relative to'normal heptane
86
1
04s-02
'
4 8 0 ' x l I I I I I I I I I I I I I 000 10000 20000 30000 40000 50000 60000 70000 80003
0
2
4
6
8
10
12
14
16
18
20
22
24 n
Figure 7. Dependency of Vo(n)and To(n)on carbon number; normal monoolefins C3to Czo
explicit expressions (2) and (8) for Vo(n)and To(n),respectively. The increase is only slight and of minor significance in the case of the normal paraffins and normal alkylpentanes and moderate in the case of the normal alkylbenzenes and normal alkylhexanes, but quite distinct with the normal monoolefins. It will also be seen from Table I1 that the values obtained
1 /n
0 I
I
I
03
02
01 I
I
I
I
I
1/n
01
0
05
04
To (n)
V ,, (n) T,(n)
28
1.4
28
1.3
26
2.4 1.2
24
2.2 t,t
22
2.0- 1.0
20
1.8-- 0.9
18
1 .6-- 0.8
16
1.4-- 0.7
14
1
9
1.2--0.6
1
t
06403
14
12 l3
\!
10
B
0.8 0.4
05
12
B
1.0 0.5
04
03
I
I
V,(n)
26
02
-
I
I
I
I
I
1
I
I
I
/
I
I
I
0
2
4
6
8
10
12
14
16
18
20
22
24
08 06
03
04
02
02
01
1
0
I
I
I
I
I
I
I
I
I
I
I
2
4
6
8
10
12
14
16
18
20
22
l
l
24 n
n
Figure 8. Dependency of VO(n)and To(n) on carbon number; normal alkylbenzenes C7 to Czz
for both a0 and PO are still slightly dependent on the type of homologous series, the adjusted value of a. for the normal alkylcyclopentanes being significantly lower than the rest, which do not differ significantly, while the adjusted value of PO for the normal paraffins is distinctly higher than the others, which again show differences of only minor significance. These are indications either that eq 2 and/or 8 are only approximative or, equally likely in our opinion, the liquid density data of the various compounds given in the API tables are not quite consistent, due to impurities in the compounds. Figures 6 to 10 would suggest that this is indeed the case for all the homologous series except the normal paraffins (Figure 6).
Figure 9. Dependency of Vo(n) and To(n)on carbon number; normal alkylcyclopentanes Cg to Cp1 0
02
01
I
I
I
I
I
I
I
1 /n 05
04
03
I
I
I
1 168 / 00 8 9
-1
1 -1
Consistency of the Results Obtained for Normal Paraffins
To test the consistency of the results obtained for normal paraffins we made use of the experimental thermal pressure coefficient data for normal paraffins with n = 6, 8, 16, 22, and 36 and for two types of polymethylene a t various temperatures published by Orwoll and Flory (1967). The thermal pressure coefficient is the temperature derivative of the pressure a t constant volume evaluated a t zero pressure. Following a similar procedure to that described for the molar volumes we arrived a t this universal equation describing the thermal pressure coefficient y ( n , T ) as a function of carbon number and temperature y ( n , T ) = ro(n)r*[T/To(n)l (9) where y * is a continuous, monotonically decreasing function and the reduction parameter yo(%)is given by
:I:
1
1 1
0 2 01 0
2
4
6
8
10
12
14
16
18
20
22
24 n
Figure 10. Dependency of Vo(n) and To(n) on carbon number; normal alkylcyclohexanes C7 to C22 y * ( n , T ) = 01 exp[-m{ T / T ~ ( ~ ) J ~ ~ J (11) ] and as a result we were able t o write eq 9 in the form
On the basis of the experimental data we arrived a t an approximation of y * of the type Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 1 , 1972
87
Table 111. Comparison of Adjusted Values of a l , t1,pl, and p 2 Obtained from Thermal Pressure Coefficient Data and Molar Volume Data I
Parameter ffl
51
Pl M2
Adjusted volue on the basis o f molar volume
0,7983 4.49 -4.548 20.96
Std error
Adjusted volue on the bosis of thermal pressure coefficients
Std error
0.0037 0.11 0.043 0.60
1.1 3.7 -3.55 14
2.3 3.3 0.67 11
Table IV. Improved Adjusted Values of the Parameters Occurring in Eq 5, 11, 2, 8, and 10 for the Normal Paraffins as Derived from Pooled liquid Density and Thermal Pressure Coefficient Data Adjusted value
Parometers
el
119.6 5.57 x 10-6 1.8481
03
Standard deviation characterizing lack of fit (2) Specific to thermal pressure coefficient:
0.25%
12 73
Standard deviation characterizing lack of fit
Temp, Compound
n-C6H14
n-C&
...
n-C16H34
, , ,
... ...
...
26,80 107.0 X 10-6 1.6284
11
=
Std error
(1) Specific to molar volume: 82
Table V. Comparison of Measured Thermal Pressure Coefficients. of Normal Paraffins with n 6, 8, 16, 22, and 36 and Polymethylene at Various Temperatures with Those Predicted on the Basis of Eq 12
, , .
I
,
n-C&46
.
1.5%
(3) Common to both: 0.12816. 0.8026 0.0043 0.08106* 40 5.34 0.11 4'1 0.1340. PO -4.872 0.045 P1 25.98 0.69 P2 a Derived from the condition that for n-heptane Vo(7) = ao(7 a , ) = 1. b Derived from the condition that for n-heptane yo(7) = [&(7 El)]/[a0(7 G I ) ] = 1. Derived from the condition that for n-heptane To(7) = ( p 0 / [ 0 ) [ 1 ~ 1 / 7 rd49 E1M?/49(7 €,)I = 1. ff0
ff1
+
+
+
+
+
+
n-C3';H74
Marlex-50 Poly methylene Marlex-6050 Polymethylene
By fitting eq 12 to the thermal pressure coefficient data by means of a nonlinear least-squares procedure we obtained an independent set of adjusted values of o(1, &, PI, and ~ 2 . I n Table I11 these adjusted values are directly compared with those obtained from the molar volume data. It will be seen that the differences between corresponding adjusted values are insignificant relative to the standard errors. Hence the results obtained from the thermal pressure coefficient data, although much less precise, are consistent with those obtained from the liquid density data. Our next step was to fit eq 7, in the modified form, and eq 12 simultaneously to the pooled sets of liquid density and thermal pressure coefficient data in order to obtain improved adjusted values of the parameters common to both equations. To this end we assigned a n appropriate weight coefficient (Le., equal to the reciprocal value of the squared relative standard deviation of the lack of fit obtained in the separate fitting procedure) to each set of data and again performed a generalized nonlinear least-squares procedure. The results 88 Ind.
Eng. Chem. Fundom., Val. 11, No. 1, 1972
a
O C
19.34 45.99 76.54 100.53 123.46 149.02 20.90 34.00 48.48 70.99 100.62 120.80 149.25 20.39 20.95 33.40 55.81 57.42 73.36 79.41 101.10 125.23 49.85 73,77 101.00 124.19 149.83 177.21 81.86 88.20 97.11 103.71 124.41 147.25 176.38 200.81 139.42 154.30 173.20 191.88 139.17 157.57 173.02
Thermal pressure coefficient, bors/deg Measd Pred
8.453 7.073 5.690 4.753 3,920 3,052 9.094 8.428 7.719 6.738 5.574 4.870 3.978 10.556 10.557 9.946 8.748 8.571 8,002 7.745 6.909 6.085 9.532 8.425 7.338 6.595 5.761 5.030 8.697 8,312 8,045 7.844 6.937 6.129 5.521 4.899 7.177 6.755 6.224 5.851 7.231 6.733 6.269
~
8.318 6.903 5 506 4,569 3.795 3.061 9.117 8.449 7.748 6.739 5.559 4.848 3.967 10.157 10.133 9.614 8.716 8.663 8,047 7.824 7,056 6.262 9.203 8.347 7,429 6.699 5.948 5.212 8,360 8.161 7.886 7.686 7,077 6.441 5.709 5.098 7.194 6.837 6,391 5.983 7.200 6.760 6.403
Re1 dev,
% 1.6 2.4 3.3 4.0 3.2 -0.3 -0.3 -0.2 -0.4 0.0 0.3
0.5 0.3 3.9 4.1 3.4 0.4 -1.0 -0.6 -1.0 -2.1 -2.9 3.5 0.9 -1.2 -1.6 -3.2 -3.6 4.0 1.8 2.0 2.0 -2.0 -5.0 -3.4 -4.0 -0.2 -1.2 -2.7 -2.8 0.4 -0.4 -2.1
From Orwoll and Flory (1967).
are given in Table IV, while Table V shows the agreement of the measured thermal pressure coefficients with those predicted with eq 12 using the improved parameter values. The maximum error is 5y0 relative, and the standard error is 1.5Y0 relative. Taking into account the internal precision of the measurements (*0.&1,5% relative), the agreement is quite satisfactory. The adequacy of the relations thus derived for normal paraffins was subsequently checked using a n independent set of data relating to the liquid density a t various temperatures of n-C28Hjs, n-Cs6H7a,and n-C74H130 published by Doolittle and Peterson (1951), these compounds being far outside the range of normal paraffins considered so far in our analysis. The measured molar volumes and those predicted with the
= 28, 36, and 64
Table VI. Comparison of Measured Molar Volumes. of Normal Paraffins with n at Various Temperatures with Those Predicted by Eq 2, 8, and 5 n-C%sHss Temp, OC
Molar vol, ml/mole Measd Pred
50 501.2 501.7 522,5 521.6 100 545.6 544.8 150 571.3 571. a 200 600.1 602.7 250 634.0 638.2 300 From Doolittle and Peterson (1951).
n-C36H74 Re1 dev,
n-CerHlao
Molar vol, ml/mole Measd Pred
%
-0.1 0.2 0.1 -0.1 -0.4 -0.7
661.2 689.0 719.3 753.3 792.3
Re1 dev,
M o l a r vol, ml/mole Measd Pred
%
660.0 687.7 719.6 756,3 798.1
1142 1187 1235 1289 1347
0.2 0.2 0.0 -0.4 -0.7
Re1 dev,
%
1145 1189 1239 1296 1361
-0.3 -0.2 -0.3 -0.6 -1.0
Table VII. Readjusted Values of the Parameters in Eq 2 and 8 for the Homologous Series Other Than Normal Paraffins
Homologous series
(YO
Adjusted value
1c2
PI
a1
Std error
Normal inoiioolefiiis 'r 0.4519 0.0037 Normal alkylbenzenes } 0.12816. - 1 ,1996 0,0058 Sornial alkylcyclopentaiies 0.0057 0.0199 Xormal alkylcyclohexanes -0.1072 0.0096 a Parameter fixed a t value obtained for normal paraffins.
1
E1
Adjusted value
Std error
Adjusted value
Std error
-5.155 - 3,634 -3.601 -3.502
0.043 0,024 0.046 0.042
30.87 8.95 7.89 7.05
1.00 0.18 0.22 0.25
Adjusted value
6.20
~
Std error
0.19
Re1 std dev characterizing lack of flt, %
0.29 0.36 0.35 0.37
Table VIII. Adjusted Values of Vo(n)and To(n)for the lsoparaffins C4 to Cs and the Branched Alkylbenzenes Cs and Cg Compound
Vo(n)
Tdn)
Compound
0.776 0 .895 0 . a04 0.948 0.955 0.942 0.960 1.012 1.014 0.981 0.962 1.022 0.985 1.028 1.019 1.068 1.065 1.058 1.067 1.039 1.068 1.043 1.034 1.067 1.076 1.067 1.106 1.086 1.007 1.106 1.076 ...
1,2-DimethyIbenzene lJ3-Dimethylbeiizene lJ4-Dimethylbenzene Isopropylbenzene 1-Xethyl-2-ethylbeiizene I-Methyl-3-ethylbenzene 1-hlethyl-4-ethylbenzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene lJ3,5-Trimethylbenzene
0 . a71 0.881 0.886 1.006 0.986 1,001 1,009 0.977 0.997 1,005
1.192 1,165 1.168 1,185 1,193 1,179 1.196 1,219 1.218 1.195
Standard error
0.001
0.004
2,2,3,3-Tetramethylbutane
0.630 0.758 0.754 0.883 0,871 0.886 0.876 1.013 1.001 0.972 0.999 0.992 1.012 0.997 0,999 1.144 1.130 1.129 1.118 1.138 1.121 1.131 1.138 1.124 1.113 1.110 1,110 1.121 1.128 1,112 1.113 ...
Standard error
0.0015
0.0035
2-Methylpropane 2-3Iethylbutane 2,2-Dimethylpropane 2-Methylpentane 3-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane 2-Methylhexane 3-Methylhesane 3-Ethylpeiitane 2,2-Dimethylpentane 2,3-Diniethylpentane 2,4-Dimethylpentaiie 3,3-Dimethylpentane 2,2,3-Trimethylbutane 2-Methylheptane 3-hIethylheptane 4-Methylheptane 3-E t h ylhexane 2 ,2-D imethylhexane 2,3-Dimethylhexane 2,4-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 3,4-Dimethylhexane 2-Methyl-3-ethylpentane 3-Xethyl-3-ethylpentane 2,2,3-Trimethylpentane 2,2,4-Triniethylpent ane 2,3,3-Trimethylpentane 2,3,4-Triniethylpentane
Vdn)
Ind. Eng. Chem. Fundam., Vol. 11, No. 1, 1972
89
aid of eq 2, 8, and 5 are given in Table VI, together with the relevant relative deviations. The agreement is very satisfactory, the maximum deviation being 1% relative. Recalculation of the Adjustable Parameters for the Other Homologous Series
The results described in the preceding section led us to believe that the adjusted values of the common parameters a0 and po = P o / € o obtained for the normal paraffins were more reliable than those obtained for the other homologous series. So we recalculated the specific adjustable parameters a1 and p1, p2, and t1 by fitting eq 7 in its modified form to the liquid density data of the other series, with the common parameters a0 and po ‘fixed at the values finally obtained for the normal paraffins. Table VI1 shows these readjusted parameters together with their standard errors and the standard deviation characterizing lack of fit. From this it will be seen that with these parameters i t is possible to predict the molar volume of a homolog a t a given temperature from its carbon number with a standard error of about o.3570 relative, the maximum error observed being 1.4y0 relative. On the whole, the lack of fit of the set of equations thus obtained is slightly increased, but we believe that their consistency is considerably improved. Values of Reduction Parameters for the lsoparaffins Cd to CS and the Branched Alkylbenzenes Cs and CS
Liquid density data of the isoparaffins C4 to CS and the branched alkylbenzenes Cs and C9 are also recorded in the API tables. I n order t o calculate the molar volume as a function of absolute temperature for these species we finally calculated Vo(n)and To(n)for these compounds as well. For this we fitted eq 5 , with the values of through e3 given in (6), t o the liquid density data of each species using a n ordinary linear least-squares procedure. The resulting adjusted values of the reduction parameters for volume and temperature of these compounds are listed in Table VIII. Their precision is characterized by the standard errors a t the bottom of the table. The lack of fit was about the same as that found for the normal paraffins. Conclusions
Using the principle of corresponding states for chainmolecule liquids, we have derived an equation which will predict the molar volume of a compound once its reduction parameters for volume and temperature are known. The reduction parameters for volume and temperature for the normal paraffins, monoolefins, alkylbenzenes, alkylcyclopentanes, and alkylcyclohexanes up to C22, the isoparaffins u p to CS, and the branched alkylbenzenes CS and CS have been determined from the liquid densities a t atmospheric pressure and a t various temperatures taken from API, Research Project 44 (Rossini, et al., 1953). The dependency of the reduction parameters on carbon number having been derived for the five homologous series of unbranched hydrocarbons mentioned above, it proved possible t o predict the molar volume of a homolog at a given temperature from its carbon number with a standard error of about 0.35Yc relative. The maximum error observed amounted t o 1.4y0 relative.
90 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
When the adequacy of the relation between molar volume, carbon number, and temperature obtained for the normal paraffins was checked with an independent set of liquid density data for n-C28H@,n-C38H7a,and ni-C64H130 a t various temperatures, the agreement between the measured molar volumes and those predicted with the aid of the derived relation was very satisfactory; the maximum deviation is 1% relative. I n another consistency test, making use of thermal pressure coefficient data for normal paraffins with n = 6, 8, 16, 22, and 36 and two types of polymethylene a t various temperatures, with an equation describing the coefficient as a function of carbon derived in the same way and containing the same characteristic parameters as the molar volume equation, the agreement between measured and predicted values was again quite satisfactory, the standard error being 1.5% relative and the maximum error 5y0 relative; the internal precision of the original data is given as 10.5-1.5y0relative. Acknowledgment
The author wishes to thank Jacob de Swaan Arons for posing the problem, Jaap Hijmans for stimulating and helpful discussions, and Ank C. P. hl. Fels for numerous computer calculations. Nomenclature
parameters of eq 2 and 10 parameters of eq 3 r(n, T ) = thermal pressure coefficient written as a function of n and T , bars/deg d n ) = reduction parameter for thermal pressure coefficient written as a function of n y* [ T / T o ( n ) = ] reduced thermal pressure coefficient (relative to n-heptane) written as a function of reduced temperature = parameters of eq 11 71, 72, 73 po, pl, p2 = parameters of eq 8 where PO = P o / ~ o p1 , = PI - €1, and p2 = Pn - Ew1 n = total carbon number of a chain-molecule liquid = parameters of eq 4 el, e2, e3 T = temperature, O K = reduction parameter for temperature written To(%) as a function of n = molar volume written as a function of n and V(n,T ) T , ml/mole = reduction parameter for volume written as a Vo(n) function of n V *[ T j T o ( n )=] reduced molar volume (relative to n-heptane) written as a function of reduced temperature (relative to n-heptane) = parameters of eq 3 and 10 €0, €1 ao, a1
Po, PI,p2
=
=
literature Cited
Doolittle, A. K., Peterson, R. H., J . Amer. Chem. SOC.73, 2145 (1951).
Holleman, J. Th., Hijmans, J., Physica 31, 64 (1965). Orwoll, R. A , , Flory, P. J., J . Amer. Chem. Soc: 89, 6814 (1967). Rossini, F. D., et al., “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” API Research Project 44, Tables Id to 3d, jd,20d to 24d, Pittsburgh, Pa., 1953. RECEIVED for review January 7, 1971 ACCEPTED September 27, 1971
