THERMAL CONDUCTIVITY OF GASES Hydrocarbons at Normal Pressures DlPAK ROY AND GEORGE THODOS h'orthwestern University, Evanston, Ill.
Thermal conductivity measurements, k * , obtained from the literature for gaseous hydrocarbons at normal pressures have been used to obtain the product, k*X, where X = M1/2Tc1/6/Pc2/3. This product represents the sum of the translational, rotational, and vibrational Contributions. The translational contribution (k*X), is identical to k*X for monatomic gases and i s a unique function of TR. The difference k*X - (k*X), = (&*A), (k*X), has been designated asX, and the ratio X / X ~ , = i o a when related to TR produces relationships specific to aliphatics, olefins, acetylenes, naphthenes, and aromatics. Group contributions developed in this study permitted the establishment of XTR=i0o for the calculation of thermal conductivities of 2 7 hydrocarbons. These values, when compared with corresponding experimental measurements, produced an average deviation of 2.1%; for 109 points.
+
HE theoretical development of the thermal conductivity Tof gases has been limited largely because of the inability to account properly for the contributions owing to rotation and vibration. For monatomic gases, which possess only translational motion, the theoretical treatment of thermal conductivity has been well advanced by Hirschfelder, Curtiss, and Bird (1954) with the development of the relationship
where !2(2,2)*[T*] is .the reduced collision integral, T* = T/(e/x), and e and u are the Lennard-Jones force constants. Values calculated with Equation 1 for neon, argon, krypton, and xenon were found to be in good agreement with experimental measurements for these substances (Owens and Thodos, 1957). For the thermal conductivity of polyatomic gases at low densities, Eucken (1 91 3 ) introduced the relationship kp* = k,*
[i2 $1
to account for the exchange of rotational and vibrational energy during a collision. For monatomic gases, C, = 3R/2, which reduces the Eucken factor, 4C,j15R f 3/5, to unity. However, for carbon dioxide (Kennedy and Thodos, 1961), ammonia (Groenier and 'Thodos, 1961), and water (Theiss and Thodos, 1963), the Eucken factor is inadequate for predicting the thermal conductivities of these polyatomic gases. The inability of the Elucken factor to account for the thermal conductivity of polyatomic gases makes its application limited. Furthermore, the h e a capacity data required to establish this factor may not be readily available for such polyatomic gases. Consequently, a new approach was sought that would permit the establishment of this transport property directly from the basic constants of these polyatomic gases and their molecular structure. Treatment of Experimental Data
A dimensional analysis for the thermal conductivity of gases a t normal pressures 1e.ads to the relationship (Mathur and Thodos, 1965) k"h = .zCmTRn
(3)
where the thermal conductivity parameter, X = M1/2Tc1/6/Pcz/a A comprehensive treatment of thermal conductivity measurements for the monatomic gases, neon, argon, krypton, and xenon produced a single relationship of k*X us. TR for all these gases (Roy, 1967). This relationship can be expressed i n equation form as follows: r
The thermal conductivity parameter, A, is unique because o its capability of expressing the thermal conductivity behavior of these monatomic gases in a unified way, irrespective of their molecular weights. This behavior is not unexpected because these monatomic gases possess only translational and no rotational and vibrational motion. Therefore, Equation 4 can be assumed to represent the translational contribution not only of monatomic gases, but of polyatomic gases as well, irrespective of their molecular weights and structures. Thermal conductivity measurements available in the literature for hydrocarbons of all types were used to obtain the product k * h . The basic constants and critical values (Kobe and Lynn, 1953) for these hydrocarbons are presented in Table I along with the sources of thermal conductivities used in this study. When this product was related to T R ,individual relationships were obtained for each hydrocarbon. These relationships for the normal paraffins, when expressed on loglog coordinates, were linear and essentially parallel to each other, with the exception of methane. For normal paraffins up to and through n-octane, these relationships are presented in Figure 1. The general trend of the k*X us. TR relationships resulting directly from the thermal conductivity measurements is linear on log-log coordinates over the limited temperature range for kvhich data are available. An attempt to extrapolate linearly can be misleading because the dependence of k*X on TR actually is known not to be linear over the complete temperature range. To account for the contribution due to the translational, rotational, and vibrational motions of these hydrocarbons, the product, k*X, has been assumed to consist of the three different additive contributions as follows: k*X = (k*X)g
+ (k*X), + (k*X),
(5)
Since Equation 4 has the capability to account for (k*X),, the difference k*X - ( k * X ) , represents the sum, (k*X), (k*X),.
+
vot.
7
NO. 4
NOVEMBER 1968
529
Table 1.
Basic and Related Constants, Sources of Data, and Resulting Average Deviations for Gaseous Hydrocarbons at Normal Pressures
Av.
X a t T R = 1.00 Actual Calcd.
No. of Deu.,
T,, P,, M " K . Atm. 16.04 191.1 45.8
0,694
0.83 X 10-5
0.83 X 10-6
Ethane
30.07 305.5 48.2
1.072
3.10
3.10
Propane
44.09
1.469
6.72
6.72
n-Butane
58.12 425.2 37.5
1.800 10.90
10.90
n-Pentane
72.15 469.8 33.3
2.279 16.57
16.08
n-Hexane
86.17 507.9 29.9
2.717 21.25
21.27
100.2 114.2
540.2 27.0 569.4 24.6
3.165 26.35 3.634 31.64
26.46 31.64
Carmichael et al., 1966; Geier and Schafer, 1961; Hercus and Labv. 1919: Johnston and Griily, 1946; Keyes, 1954, 1955; Lambert et al., 1955; Lenoir and Comings, 1951; Lenoir et al., 1953; Mann and Dickins, 1931 ; Smith, et al., 1960 Geier and Schafer, 1961; 18 Keyes, 1954; Lambert et al., 1955; Leng and Comings, 1957; Lenoir et al., 1953; Mann and Dickens, 1931; Senftleben, 1953; Senftleben and Gladisch, 1949; Vines and Bennett, 1954 Lambert et al., 1955; Leng 13 and Comings, 1957; Mann and Dickins, 1931; Senftleben and Gladish, 1949; Smith et al., 1960; Vines and Bennett, 1954 Kramer and Comings, 1960; 14 Lambert et al., 1955; Mann and Dickins, 1931; Senftleben, 1953; Senftleben and Gladisch, 1949; Smith et al., 1960; Vilfm, 1960 Lambert et al., 1955; Smith 6 et al., 1960 Lambert et al., 1950, 1955; 5 Vines and Bennett, 1954 Lambert et al., 1955 1 Lambert et al., 1955 1
58.12 72.15 72.15 86.17 100.2 114.2
408.1 461.0 433.8 489.4 520.3 544.1
36.0 32.9 31.6 30.7 27.4 25.4
1.896 2.302 2.337 2.658 3.115 3.518
10.86 16.31 14.72 20.79 26.73 32.49
10.86 16.26 14.72 21.38 26.80 31.92
Lambert Lambert Lambert Lambert Lambert Lambert
28.05 282.4 5 0 . 5
0.989
1.75
1.75
Propylene
42.08 365.0 45.6
1.355
5.55
5.95
1-Butene
56.10 419.6 39.7
1.762 10.53
10.13
2-Butene (cis)
56.10 435.0 41.23 1.718 10.57
10.57
2-Butene (trans) Diolefins 1,3-Butadiene
56.10 430.5 39.65 1.769 10.57
10.57
Normal paraffins Methanea
n-Heptane n-Octane Isoparaffins 2-Methylpropane 2-Methylbutane 2,2-Dimethylpropane 2,2-Dimethylbutane 2,4-Dimethylpentane 2,2,4-Trimethylpentane Olefins Ethylene
Acetylenes Ethyne
54.09
370.0 42.0
425.0 42.7
X
Source
et et et et et et
al., al., al., al., al., al.,
Points
1955 1955 1955 1955 1955 1955
Chaikin and Markevich, 1958; Hercus and Laby, 1919; Keyes, 1954; Lambert et al., 1955; Lenoir and Comings, 1951 ; Senftleben, 1953; Senftleben and Gladisch, 1949 Lambert et al., 1955; Senftleben 1953; Senftleben and Gladisch, 1949 Senftleben and Gladisch, 1949; Vilim, 1960 Lambert et al., 1955; Senftleben and Gladisch, 1949; Vilim, 1960 Lambert et al., 1955
yo ...
2.62
1.73
1 .oo
2.87 1.29 1.90 2.82
1 1 1 1 1 1
0.21 0.10 0.55 1.67 1.73 3.02
13
3.29
3
6.41
2
3.29
3
3.42
1
1.46
1.653 10.34
9.94
Lambert et al., 1955; Senftleben and Gladisch, 1949; Vilfm, 1960
3
2.00
26.04 309.2 6 1 . 6
0.853
2.16
2.16
Gardiner and Schafer, 1956; Senftleben and Gladisch,
6
0.78
52.07 464.3 49.4
1.487
9.59
9.19
Senftleben and Gladisch, 1949
1
3.89
42.10 406.7 57.20 1.190
5.60
5.58
3
2.33
70.13 511.8 44.55 1.880 1 1 . 1 84.16 553.0 40 0 2.240 20.1
14.95 20.13
Lambert et al., 1955; Vines and Bennett, 1954 Lambert et al., 1955 Lambert et al., 1950, 1955; Vines and Bennett, 1954
'5
1.17
78.11 562.0 48.6
15.03
5
0.43
..
...
1949
Vinylacetylene Naphthenes Cyclopropane Cyclopentaneb Cyclohexane Aromatics Benzene
1.906 15.00
Abas-Zade, 1949; Lambert et al., 1950, 1955; Vines and Bennett, 1954 Abas-Zade, 1949
Tolueneb 92.13 594.0 41.6 2.318 21.00 21.03 a For methane, its speciic k*X us. T R relationship must be used. Because of its diyerence in behauior, no comparisons can be made. thermal conductivity measurements; no average deviations made.
530
I&EC FUNDAMENTALS
...
Questionable
30 -
30 20
10 Normal Paraffins 0 V
Methane Ethane
+
n-Pentane
A n-Hexane
8 6 5
2 *z
4
e-!!
3
v
+
I L 0.3 0.4 O
2
0.6 0. L81 1I.0
I 3
I 4
I I I I l l 5 6 ' 8 IO
y.
U
I1
x
T A Figure 1 . paraffins
*d
2
Relation between k*X and TR for some normal
1.0 For the temperatures for which experimental thermal conductivities are availablr, the translational contribution, (k*X) ,, constitutes a significant part of the total value k*X and therefore the correlation of the difference, k*X - ( k * X ) , , against T R should present a relationship that is more amenable to extrapolation. This contribution due to the rotational and vibrational motion of the molecules is difficult to resolve into its component parts, largely because of the complex nature of these polyatomic gases. Therefore, the sum, ( k * X ) , @*A),, has been considered collectively as a group and has been designated a5 X . Since k*X and ( k * h ) , are functions of temperature, their difference, X , must also depend on temperature. This difference, X = k*h - @*A),, has been related to TR as shown in Figure 2 for the normal paraffins through n-hexane. The relationships of this figure are not linear, but are essentially parallel, with the exception of methane. This parallel behavior makes them amenable to a treatment for the development of a single correlation that should be unique to this particular class of hydrocarbons. T o accomplish this objective, values of X at T R = 1.00 were obtained directly from the relationships presented in Figure 2 for ethane, propane, and n-butane. These reference values have been designated as XI, and are presented in Table I. The ratio, X / X 1 , for each of these hydrocarbons, with the exception of methane, has been related to TR, using rectilinear coordinates, as shown in Figure 3. The relationship of Figure 3 can be expressed in equation form as follows:
+
r:
NORMAL PARAFFISS.-- = -0.152 TR XI
+ 1.191 TR20.039 TR3 (6)
Values of X I X I , obtained from Figure 3 for the remaining normal paraffins through n-octane and a number of isoparaffins, permitted the establishment of X I values for them, which are also included in Table I. The same procedure was followed for the olefins for which adequate thermal conductivities were available. The X / X l values for ethylene and propylene established a relationship
-
0.8
0.6 0.5 0.4
0.3
0.2 0.3 0.4
0.6 0.8 1.0
2
3
TR
Figure 2. Dependence of rotational and vibrational contributions, (k*X), (k*X),,, on TR for some normal paraffins
+
comparable to that of Figure 3. This relationship can be expressed in equation form as follows: OLEFINS.
X1
TR
= -0.255
+ 1.065 TRz + 0.190 TR3
(7)
Following an approach similar to that used for the paraffins, values of X1 \$'ere established for the different butenes and 1,3butadiene and are listed in Table I along with the values for ethylene and propylene. The thermal conductivity behavior of acetylenes is restricted to that of ethyne, which permitted the establishment of the relationship
X
ACETYLENES. - =
X1
-
0.068 T R
+ 1.251 TR2- 0.183 TR3 (8)
A similar treatment of values for cyclopropane, cyclohexane, and benzene produced for these cyclic hydrocarbons a single relationship which can be expressed as follows : VOL. 7
NO. 4
NOVEMBER 1 9 6 8
531
9t
-
I
I
I
I
I
I
I
I
I
I
I
35x16~
I
0-
-
I
Normal Paraffins
P
x, = -0.152TR+ I.19IT i -0.039Ti
7 6
0
5
+ 0 B
4
0
+
0
3
Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane
-
2
7’ I
-
0
I
I
0
I
I
I
I
I
I
I
I
I
I
2
I Relation between
X/X1
1
2
3
3
4
5
6
7
8
Number of Carbon Atoms
,
T R Figure 3. paraffins
0 0
-
Figure 4. Relation between XI and number o f carbon atoms for normal paraffins
and TR for normal
+
X CYCLICS.-
= -0.354
T,
+ 1.501 TR2- 0.147 TR3 (9)
X 1
The values of XI, for the hydrocarbons for which limited data are available, were obtained from the appropriate relationships. These values, along with those obtained directly from the respective X us. T, relationships, are also presented in Table I. Equations 6 to 9 permit the calculation of X, provided this value a t TR = 1.00 is available for each hydrocarbon. The capability of predicting these reference values, X I , will enhance the applicability of Equations 6 to 9 for the establishment of thermal conductivities of hydrocarbons.
chain lengths. Thus, for n-decane, X I = Il0.90 6(5.185)] X 10-5 = 42.01 X 10-5. T o obtain X at T , = 1.OO for isoparaffins, it becomes necessary to introduce side chains to the longest normal paraffinic structure. This procedure requires identification of the carbon atom on which a replacement is to be made and also of the carbon atoms surrounding it. The number of carbon to carbon bonds associated with each carbon atom has been arbitrarily selected to define the type of carbon atoms involved in these substitutions. Thus, Structure
Group Contributions for Prediction of XI
Type of carbon atom
The values of X I for the normal paraffins presented in Table I, when plotted against the number of carbon atoms, produced a relationship which was linear for the hydrocarbons beyond propane. The behavior shown in Figure 4 suggests that the replacement of hydrogen by a methyl group is constant and equaltoAX1 = (31.64 - 10.90) X 10-5/(8 - 4) = 5.185 X 10-5 for hydrocarbons above n-butane. T o establish the value X I = 10.90 X 10-jfor n-butane, successive replacements of hydrogen by methyl groups are made, using methane as the base substance. The group contributions corresponding to these substitutions are as folloivs:
An analysis of the X1 values already obtained for the isoparaffins has produced the following group contributions to be associated with the replacement of hydrogen by a methyl group :
Base group, methane First methyl substitution Second methyl substitution Third methyl substitution Fourth and successive methyl substitutions
0 . 8 3 x 10-5 2.27 3.62 4.18 5,185
These group contributions permit the calculation of X a t T R = 1.00 for n-butane, as X I = (0.83 2.27 f 3.62 4.18) x 10-5 = 10.90 X 10-5. Successive substitutions by methyl groups beyond n-butane involve the addition of the value AX, = 5.185 X to produce normal paraffins of longer
+
532
l&EC FUNDAMENTALS
+
2
3
1t 2 + l 1t 2 4 2 1e 2 4 3 2t242 1t 3 + l
4 . 1 4 X 10-6 5.36 6.58~ 6.58a 3.86
lt341
5.12
f
AX1
1
4
i lc3+l
4
a
6.38~
Extrapolated values.
The type of carbon atom from which the arrows point away is the one involved in the methyl substitution. These arrows point to the type of carbon atoms surrounding it.
The present list of group contributions is limited and frequently it may become necessary to utilize this information to estimate values for substitutions other than those already presented. In order to avoid any ambiguities concerning the order of substitutions to produce a n isoparaffin, the longest straight chain is to be synthesized a t first, followed by the introduction of side chains, beginning with the carbon atom furthest to the left and proceedinq in a clockwise direction. For example, the value X a t TR = 1.00 for 2,2,4-trimethylpentane becomes
Thus, the contribution due to each double bond in a ring structure becomes
1
AX1 = - (-5.10 3
X 10-j) = -1.70 X (for each double bond in ring)
The available data for benzene and toluene from the work of Abas-Zade (1949) were used to obtain the methyl group contribution associated with the substitution of a hydrogen atom on a benzene ring. This value is as follows: Methyl substitution in benzene ring.
c-c-c-c-c I
n-Pentane (10 90 f 5.185) X 10-6 = 16.085 X 10-6 1t 2 + 2 5.36 2t241 5.36 1 ~ 3 4 2 5__ .12
1
Xi = 31.925 X 10-6
1
The group contributions associated with the removal of two hydrogen atoms from two adjacent carbon atoms for the formation of olefinic boncjs !,rere obtained from the analysis of X1 values for ethylene, propylene, and butenes. The resulting group contributions for the formation of carbon-carbon double bonds are as follows: First Double Band
AX1 - 1 . 3 5 x 10-5 -0.77 -0.33
1-1 1-2 2432
Second Double Bond 2-1
-0.19
X 10-5
Using acetylene as a test substance, for the formation of a triple bond associated with the removal of four hydrogen atoms from two adjacent carbon atoms, the contribution is as fOllO\VS : Acetylenic bond (irrespective of position)
x
A X , = -0.94
10-5
A comparison of the X i values for cyclopropane and cyclohexane with the values of the corresponding normal paraffins indicates the existence of a constant difference in X I as follows: Cyclopropane Propane Cyclohexane n-Hexane
XI
X 10-5 10-5 ul= -1.12 x 10-5 = 5.60
Until a time when more experimental measurements will become available the present group contributions and values presented are recommended. These values are not to be considered as final, but they represent the best compromise resulting from an analysis of the available thermal conductivity measurements of this method. T h e group contributions, AX], along with Equation 4 and Equations 6 to 9 permitted the calculation of k* values for all the hydrocarbons used in this investigation. These values were calculated for the temperatures for which experimental reported and were used t' O b t a i n the average deviations presented in Table I for each of these hydrocarbons. The over-all average deviation for 109 points considered was found to be 2.1%. Application of Method
The present status concerning the thermal conductivity behavior of gaseous hydrocarbons is limited to the analysis of data for the hydrocarbons used in this investigation. However, the information derived from this study can be used to predict the thermal conductivity of hydrocarbons for which measurements are yet to be made. To illustrate the procedure to be followed, which is consistent with the group contributions already developed, the following examples are presented.
Example 1. Calculate the thermal conductivity of 1,3,5trimethylbenzene (mesitylene) in the gaseous state a t atmospheric pressure and 200' C. For this aromatic hydrocarbon (Kobe and Lynn, 1953), T , = 641O K., P, = 33 atm., and M = 120.19. T R = 473.2/641 = 0.738. Substituting this value into Equation 9 produces the following:
-X_-
x1
-0.354(0.738)
AX^
= -1.15
x
10-5
The average of these differences, AXl = - 1.14 X 10-jrepresents the value to be used for the prediction of X1 for the naphthene from the X Ivalue of the corresponding saturated aliphatic hydrocarbon. T h e contribution for the formation of a double bond in a naphthene structure has been established from the X1 value for benzene as follows :
Xl
+ 1.501 (0.738)'
xl = 6.72 x
X1 = 20.10 X 10-5 2'1 = 21.25 X 10-5
A X , = 6.00 X 10-5
- 0.147(0.738)3 = 0.496
The rotational and vibrational contributions a t T E = 1.00 for this hydrocarbon are calculated by producing benzene from n-hexane as follows: Base group, methane Methyl substitutions First Second Third Fourth Fifth n-Hexane
0.83
AX I
x
10-5
2.27 3.62 4.18
For cyclohexane, X I = (21.27 - 1.14) X 10-5 = 20.13 X 10-5 and the corresponding value for benzene becomes X1 = [20.13 - 3(1.70)] X 10-5 = 15.03 X 10-5. Benzene = 15.03 X 10-5 3 Methyl substitutions: 3(6.00 X 10-5) = 18.00
Benzene 15.00 X 10-5 Cyclohexane 20.10 X 10-5 AX1 = - 5 . 1 0 X 10-5
~
Xi
1,3,5-TrimethyIbenzene (three double bonds in ring)
Therefore, at 200' C. and 1 atm. VOL.
X
= 33.03
= @*A),
X 10-6
+ (k*X),
7 NO. 4 N O V E M B E R 1 9 6 8
=
533
0.496(33.03 X 10-j) = 16.38 x 10-5. Substituting T R = 0.738 into Equation 4 yields the translational contribution, ( k * X ) i = 1.97 X 10-5. Thus, for this aromatic hydrocarbon,
k*X = ( k * X ) ,
+ (k*X), + ( k * X ) , = (1.97 + 16.38) X 10-5 = 18.35 X 10-5
Since the thermal conductivity parameter, X = (120.19)1/2 (641)1/S/(33)2/3 = 3.129 X 10-5, the thermal conductivity value for 1,3,5-trimethylbenzene a t 200' C. and 1 atm. becomes k* = 18.35 x 10-5/3.129 = 5.86 cal./sec. cm. ' K . This value cannot be compared with a n experimental value, since no measurements are available for this hydrocarbon. Example 2. Calculate the thermal conductivity of vinylacetylene [CH2=CHC=CH] in the gaseous state a t atmospheric pressure and 70' C. For this hydrocarbon (Kobe and Lynn, 1953), T , = 464.3' K., P, = 49.4 atm., and M = 52.07. TR = 343.2/464.3 = 0.739. Substituting this value into Equations 7 and 8, the following ratios result:
X X1
-
= -0.255(0.739)
+ 1.065(0.739)2f 0.190(0.739)3 = 0.482
r,
A
- -- -0.0684(0.739) f 1.251 (0.739)' -
Xl 0.183(0.739)3 = 0.558 The average of these two ratios should represent the behavior of this unsaturated hydrocarbon which includes both an olefinic and an acetylenic bond. Thus, (X/X1)av= '/2[0.482 0.5581 = 0.520.
+
The rotational and vibrational contribution of this hydrocarbon a t T R = 1.00 is established from n-butane as follows: n-Butane 1 Olefinic double bond 1 Triple bond Vinylacetylene
X1 = 10.90 X 10-5 AX1 = -0.77 A X , = 0.94 X1 = 9.19 X 10-5
(1 -2)
Therefore. a t T R = 0.739 (70' (2.). X = 0.520 (9.19 X 10-5) = 4.78 X 10-5. -T h e translational contribution a t TR = 0.739, according to Eauation 4 becomes (k*XI, = 1.97 x 10-5 to produce a value k*'X = (4.78 f 1.97)' x '1'0-5 = 6.75 X IO-;. T h e thermal conductivity parameter, X = (5 2,07)'/*(464.3)'/6/ (49.4)'j3 = 1.487. Thus at 70' C. and 1 atm., k* = 6.75 X 10-5/1.487= 4.54 X 10-5 cal./sec. cm. OK. Senftleben and Gladisch (1949) report for this substance a t 70' C. the experimental value, k* = 4.37 X 10-5 cal./sec. cm. OK. For this case the deviation is 3.9%. I ,
Acknowledgment
The authors extend their gratitude to the National Aeronautics and Space Administration for the support of this study through Grant NsG-405-14-007-003. Nomenclature
heat capacity a t constant volume, cal./g.-mole OK. thermal conductivity of gas at normal pressures, cal./sec. cm. 'K. exponents, Equation 3 molecular weight critical pressure, atm. gas constant temperature, 'K .
534
I&EC FUNDAMENTALS
T* T,
normalized temperature, T / ( ~ / K ) critical temperature, 'K. reduced temperature, T / T , rotational and vibrational contribution to k*h,
X
= = = =
XI AX,
= (k*X), (k*?), at T R = 1.00 = contribution in replacement of a hydrogen atom
T R
+ + @*XI,
(k*X),
UC
2,
by methyl group or removal of hydrogen atoms to produce unsaturated bonds = critical volume, cc./g.-mole = critical compressibility factor, P,v,jRT,
GREEKLETTERS 0 . = constant, Equation 3 e = maximum energy of attraction between two molecules for Lennard-Jones potential, ergs = Boltzmann constant 1.3805 X 10-'6 erg/' K . K = thermal conductivity parameter, M1/2T2/6/P2/3 x = collision diameter for Lennard-Jones potential, A. 7 I Q ( ~ A[T*] * = collision integral for thermal conductivity SUBSCRIPTS = monatomic = polyatomic P r = rotational t = translational U = vibrational m
literature Cited
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