Heat Capacity of Hydrocarbons in the Normal Liquid Range STUART T. HADDEN Management Information & Systems Department, Monsanto Co., 800 N. Lindbergh Blvd., St. Louis, Mo.
63166
Stemming from a dimensionless equation for heat capacity, reliable empirical equations are presented for the C, of pure liquid hydrocarbons up to the normal boiling point. The correlating equations are for members of specific homologous series. The homologous series covered are: n-alkanes, isoalkanes, 1-alkenes, alkylcyclohexanes, alkylcyclopentanes, and alkylbenzenes. A separate correlation i s presented for the light alkanes from methane through n-butane. The hydrocarbons isobutane, ethene, propene, and 1-butene have individual equations. Using the concept of conformal solutions, the heat capacity of mixtures of n-alkanes can be calculated. The excess heat capacity of such mixtures can also be computed with reasonable accuracy.
I N F O R M A T I O N on the heat capacity of substances is frequently required. A nomogram published in 1966 ( 1 2 ) presented t h e heat capacity of liquid hydrocarbons with five o r more carbon atoms in t h e molecule. The nomogram was incorporated into the A P I “Technical Data Book Petroleum Refining” (I), which presents correlations primarily directed toward desk use. Since the nomogram was constructed, additional precision heat capacity d a t a f o r liquid hydrocarbons have been published. I t i s now desirable to include these more recent data in correlations which cover a broader range of hydrocarbon types, including the lighter members, and to have computer-oriented methods. The earlier correlation ( 1 2 ) depended on the empirical observation t h a t a simple relation existed between the heat capacities of t h e successive members of a homologous series a t equal reduced temperatures. A reduced temperature is used here to present, in so f a r a s possible, general equations for several homologous series. The concept also gives a reasonable means for interpolating heat capacities f o r hydrocarbons t h a t have no experimental d a t a and extrapolating the temperature range on hydrocarbons w i t h data for only p a r t of the temperature range between freezing and boiling points. Currently, the experimental heat capacity of liquid hydrocarbons a t temperatures below their normal boili n g points a r e usually presented a s quadratic or cubic functions of temperature ( 1 8 ) f o r the individual hydrocarbons. The available experimental data a r e f o r different temperature ranges between freezing and boiling points f o r the successive hydrocarbons of a given type. As a result, the coefficients in t h e least squares heat capacity equations change erratically as the number of carbon atoms increases f o r the members of a homologous series. Consequently, the use of such equations is limited and should not be extrapolated beyond the temperature range on which they a r e based. Estimation methods f o r heat capacities primarily a t temperatures above the normal boiling point have been reviewed by Reid and Sherwood ( 2 7 ) . Thus, there is a need f o r a systematic correlation which covers the temperature range from the freezing point to the normal boiling point for hydrocarbons.
perature range of t h e heat capacity information and t h e normal liquid range (freezing and normal boiling point temperatures) is given f o r comparison. Whenever possible, only heat capacity data more recent than 1945 have been used, because the best calorimetry work has been done since 1945. P r i o r to about 1930, calorimetry work was largely unsatisfactory by present standards. No pre-1930 data have been used. The d a t a f o r the following hydrocarbons a r e in the 1930 to 1945 period: methane through butane, n-pentacosane, isobutane (2methylpropane) , isopentane (2-methylbutane) , ethene, propene, 1-butene, and cyclohexane. With the exception of one data point for n-undecane and one f o r n-hexadecane, all of the data f o r each hydrocarbon were used in fitting the least squares equations. Timmermans’ ( 3 5 ) valuable guide and listing of reliable physicochemical constants was a substantial aid in locating useful data. However, he includes some old heat capacity data, prior to 1930, which should not now be used. More recent data for cyclopentane and cyclohexane a r e indicated in Table 11. None of these later data were used. They confirm the older data’s reliability for cyclopentane, but the later data for cyclohexane a r e higher and seem to be out of line with the homologous series data. The most extensive studies on the heat capacity f o r any hydrocarbon have been made f o r n-heptane. Consequently, i t was the base hydrocarbon used in deriving the correlation f o r pentane and heavier n-alkanes. The literature indicated in Tables I and I1 presents the heat capacity a t saturation pressure, C,. Other than f o r n-heptane, virtually all C , values a r e for circumstances below the normal boiling point. A t pressures of 1 atm. o r less, the values of C, can be considered identical to c,, and this has been done here. CORRELATION
I t is desirable to have a dimensionless equation f o r the constant pressure heat capacity of the members of a homologous series of hydrocarbons. The following equation symbolizes such a relation
EXPERIMENTAL DATA
The literature sources f o r experimental data used in the correlation a r e shown in Tables I and 11. The tem92
Journal of Chemical and Engineering Data, Vol. 15,
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Multiplying both sides of Equation 1 by R gives a relation f o r molar heat capacity. If each side of this
Table I. Heat Capacity Data for Liquid Acyclic Hydrocarbons
%-Alkanes
2-Methylalkanes (Isoalkanes)
Liquid Range, Heat K. (1) Capacity Temp. No. of C Th Atoms Ref. Ranre. OK. - Tt 90.67 111.66 98 111 1 89.88 184.52 90 180 2 85.46 231.08 90 230 3 134.80 272.65 140 268 4 143.43 309.22 149 303 5 177.80 341.89 177 296 6 182 37w 182.54 371.58 7 278 318 183 367 216.35 398.81 220 300 8 219.63 423.95 220 320 9 243.49 447.27 250 320 10 247.56 469.04 11 250 300 263.56 489.43 270 320 12 267.76 508.58 270 310 13 279.01 526.67 14 280 300 283.08 543.77 15 290 310 291.32 559.94 290 320 16 295.13 575.30 302 384 17 301.33 589.86 304 379 18 305.05 603.76 19 309.6 616.9 20 313.6 629.6 21 317.5 641.8 22 320.8 653.4 23 324.0 664.4 24 326.9 675.0 333 373 25 O
a
b
Heat Capacity Temp. Ref. Range, K. O
( 2) (11) ( 7)
117 115 200
257 300 300
(22)
283
308
(25)
200
290
Liquid Range, " K . (1) T,
Tb
113.55 113.25 119.48 154.87
261.42 301.00 333.42 363.20
164.11 192.75 198.50
390.80 416.41 440.15
n-1-A1kenes
Liquid Range, Heat Capacity Temp. K. (1) Ref. Range, K. - T, Tb
150
169 223 300 29 5 310 300
104.00 87.90 87.80 107.93 133.33 154.27
169.44 225.45 266.89 303.12 336.63 366.79
171
310
207 224 247
360 310 310
278
310
171.41 191.78 206.84 223.97 237.92 250.08 260.30 269.42 277.27
394.43 420.02 443.72 465.82 486.51 505.93 524.25 541.54 558.00
107 94 150 135 133
Experimental data for saturated liquid extended to critical temperature. Data from 5 separate studies reported. All sets included in least square fit. Table II. Heat Capacity Data for Liquid Alkyl Cyclic Hydrocarbons
n-A1kylcyclohexanes
n-A1 kylbenzenes
Liquid Range, Heat K. ( 1 ) Capacity Temp. Ref. Range, OK. T, Tb
Heat Liquid Range, Capacity Temp. K. ( I ) Ref. Range, K. T, T b
n-Alkylcyclopentanes Heat No. of C Capacity Temp. Atoms Ref. Range, OK. 5 6 7 8 9
(6)(34)" ( 6) (10) (19) (19)
180 150 140 162 168
300 300 302 365 368
10 11
12 13 14 15 16 17 18 a
(19)
258
321
Liquid Range, K. (1)
T,
179.28 130.69 134.70 155.79 163.18 190.2 200.4 220.4 229.3 244.3 251.02
Tb
322.41 344.96 (23)(4,29)" 280 376.62 ( 6) 155 404.10 (14) 170 429.75 ( 9) 186 463.6 ( 9) 208 476.0 497.0 (24) 240 516.7 535.2 552.53 ( 9) 274 (24)
290
295 286 310 373 365
279.70 146.56 161.83 178.25 198.43 215.6 230.0 242.6 253.5 263.0 271.41 278.90 285.6
300 300 300
353.89 374.08 404.93 429.87 454.10 476.82 497.9 518.0 536.8 554.6 570.74 586.4 601.0
(21) (32) (30) (20) (20)
279 160 178 181 194
353 371 300 371 370
278.68 178.16 178.17 173.65 185.18
353.25 383.77 409.34 432.37 456.42
Auxiliary data on heat capacity, not used in obtaining correlation. ~
relation i s divided by t h e molecular w e i g h t of t h e hydrocarbon w i t h n carbon atoms, t h e h e a t capacity, cp, i n cal./(g. K.) i s obtained. I n the following, both relations a r e used. A linear relation between h e a t capacity, c p , a n d t h e number of carbon a t o m s in t h e molecule at a constant reduced t e m p e r a t u r e w a s found earlier (12) f o r memb e r s of several homologous series. These relations were f o r a u n i t w e i g h t of t h e hydrocarbons w i t h five o r more carbon a t o m s in t h e molecule a n d w i t h t h e reduced temp e r a t u r e calculated w i t h respect t o t h e normal boiling
~
~
point temperature, T , = T/T,,. The present s t u d y largely confirms t h e earlier one. However, t h e more recent d a t a now available result in modifications f o r some homologous series as described below. A correlation i s given f o r t h e light %-alkanes, as well as equations f o r o t h e r individual l i g h t hydrocarbons w i t h less t h a n five carbon a t o m s p e r molecule. T h e h e a t capacity of a liquid hydrocarbon can be represented reliably i n a limited t e m p e r a t u r e r a n g e by a q u a d r a t i c o r cubic equation in temperature. T h e choice f o r t h e t e m p e r a t u r e scale is usually a r b i t r a r y . F o r Journal of Chemical a n d Engineering Data, Vol. 15, No. 1, 1970
93
present purposes, the choice i s a reduced temperature for each hydrocarbon a s defined above. Thus, the heat capacity per unit weight is represented by c, = a
+ bT, + c T , ~+ dT,3
(2)
>
a
temperature and the number of carbon atoms is represented by /In ni8\
where a, b, c, and d a r e constants and T, is reduced temperature. For some homologous series, the constants in Equation 2 vary linearly with the number of carbon atoms in the molecule. For normal alkanes with general formula C n H Z n + with % n 5, t h e heat capacity a s a function of n and T, i s given by c, =
C,, of light n-alkanes, methane through n-butane, i s also represented by Equation 4. For t h e n-alkylcyclopentanes, the relation of reduced
+ bT, + cTr2+ dT,3 + n(Aa + AbT, + ACT,^ + AdTT3) (3)
where Aa, etc., a r e t h e changes in the constant per carbon atom. A linear least squares computer program was used to determine the constants in Equation 3 f o r the homologous series n-alkanes, isoalkanes (2-methylalkanes) , 1-alkenes, and t h e n-alkylcyclohexanes. T h e results given f o r the constants in Table I11 a r e the least square fits, using the experimental data from the sources listed in Tables I and 11. The heat capacities of t h e two homologous series, n-alkylcyclopentanes and n-alkylbenzenes, a r e not adequately represented by Equation 3. For each hydrocarbon of these two series, c, can be represented by a cubic relation, a s in Equation 2. However, the heat capacity changes systematically with t h e number of carbon atoms in the molecule a t constant reduced temperature, and t h e number of carbon atoms is represented by
where ai,etc., a r e constants; n = number of carbon atoms in t h e hydrocarbon f o r which c p is to be calculated ; n, and n2 = number of carbon atoms in first and second reference hydrocarbons. The molar heat capacity,
cp . = ( a l
+ blT, +
( 1 , ) c ~ T , ~d1Tr3) X
+
/ In
sin\
where a ratio of logarithms in the exponents is involved. A nonlinear least squares computer program was used to determine the constants in Equations 4 and 5 f o r t h e indicated hydrocarbon series. The results given for the constants in Table IV a r e the least square fits, using t h e experimental data from the sources listed in Tables I and 11. T h e light hydrocarbons not covered in the preceding correlations f o r t h e homologous series included here a r e ethene, propene, 1-butene, and isobutane. The expErimental data from the sources indicated in Table I were used to determine t h e coefficients in Equation 2 for quadratic or cubic equations giving the best fit, a s presented in Table V. I n the lower portion of Tables 111, IV, and V, statistical information is presented about the data and fit of the equations. The first item is the mean of t h e heat capacity data used. T h e second item i s the root mean square deviation of the d a t a about t h a t mean. T h e third item is the standard deviation of the experimental d a t a about the regression equation. T h e fourth item gives the ratio (expressed in per cent) of the standard deviation to the mean. I n all the tables, this ratio reflects several things. One is the reliability of the selected equation to represent the data. Another may be the quality of the data t h a t a r e being fitted. The fifth and sixth items give the maximum deviation between computed and experimental values and the number of data points used. ACCURACY OF CORRELATIONS
The heat capacity correlation equations presented f o r the homologous series represent the experimental d a t a
Table Ill. Constants in Equation 3 for Heat Capacity of Liquid Hydrocarbons with Five or More Carbon Atoms per Molecule
Homologous Series n-AI kanes
Constant
Isoalkanes
1-Alkenes
n-Alkylcyclohexanes
-0.03083 1.77589 -2.50661 1.20202
0.14510 1.26233 -2.03776 1.06953
0.26796 0.04311 -0.06164
d
0.84167 -1.47040 1.67165 -0.59198
Aa Ab Ac Ad
-0.003826 -0.000747 0.041 126 -0.013950
0.079670 -0.336664 0.499756 -0.221740
0.067176 -0.304272 0.478026 0.221610
0.00433 1 -0.003439 +0.039338
0.5184 0.0304 0.0012 0.24 0.0051 289
0.4752 0.0394 0.0015 0.31 0.0044
0.4844 0.0289 0.0021 0.43 0.0065 127
0.4309 0.0522 0.0026 0.60 0.0075 98
a b C
0.0
0.0
Statistical Information 1. Mean c,
2. RMS dev. of c, 3. Std. dev. of exp. c, about Eq. 3 4. Mean percentage std. dev. of cp (lOOyo X item 3/item 1) 5. Max. dev. of cp from Eq. 2 6. No. of exptl. pts. 94
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76
Table IV. Constants in Equations 4 and 5 for Heat Capacity of Liquid Hydrocarbons
Table VI. Comparison of Molar Heat Capacity Values for n-Heptane
Temp., ’ K.
Computed from Eq. 3 and Table I11
182.55 190 200 210 230 250 270 290 310 330 350 370
48.21 48.14 48.17 48.33 48.97 50.00 51.37 53.00 54.83 56.80 58.83 60.88
Homologous Series 4
Eq. 5
n-Alkyln-Alkyl Light benezenes cyclopentanes alkanes 0.55158 0.32010 14.7486 - 9.8198 -1.12912 -0.01800 2.00806 $0.03292 12.9505 - 4.8349 -0.91821 0.12834
Constant a1 bi c1
di
0.53865 -1.11510 2.12851 -1.01959
0.44974 -0.53041 0.96861 -0.31487
1 4
9 10
.. ..
20.835 5.788
0.4071 0.0378
0.4203 0.0510
0.178
0.0024
0.0027
0.85
0.59
0.64
0.0072 90
0.0096 116
13.1830 54.3924 -72.8410 37.1562
a2 b2
C2
d2 nl n2
Statistical Information 1. Mean c, 2. RMS dev. of cp 3. Std. dev. of exptl. c, about eq. 4. Mean % std. dev. of c, (100% X item 3/item 1) 5. Max. dev. of cp from eq. 6. No. of exptl. pts.
-0.403 58
very reliably. Only f o r t h e light alkanes i s t h e mean percentage s t a n d a r d deviation g r e a t e r t h a n 0.67%. As a group, the light alkane d a t a (methane through butane) a r e t h e oldest d a t a used. Consequently, t h e i r accuracy and consistency a r e suspect. The correlation equations reproduce the minimum in heat capacity, which is strongly exhibited in t h e alkenes. T h e n-alkanes have a slight minimum in t h e i r heat capacity curves. Table VI shows how this character is represented by the general equation f o r n-alkanes by comparing i t s values f o r n-heptane w i t h “selected” values tabulated in t h e Bureau of Mines report ( 1 7 ) on n-heptane. I n evaluating the constants of the general equation, the “selected values” were not used. For the temperature range 240” to 370” K., a n equation i s given ( 2 7 ) f o r t h e “selected values” of a n nheptane, which has a maximum deviation of 0.04 cal./ (g.-mole K.). T h e deviations of the general equation given in Table VI a r e about t h e same, except a t one t e m p e r a t u r e in t h e same range.
Bur. Mines Deviation, Selected Cal./ Values (17) (G-Mole K.) 48.52 48.27 48.15 48.23 48.88 49.98 51.39 53.04 54.85 56.75 58.79 61.04
-0.31 -0.13 $0.02 0.10 0.09 0.02 -0.02 -0.04 -0.02 +0.05 0.04 -0.16
There has been a definite improvement in calorimetry in recent years. But, of necessity, d a t a a s f a r back as 1930 have had to be used. T h e discrepancies of “good” data prior to 1930 with modern heat capacity information can be large-Le., over 1 5 % . Some evaluation of the improvement in calorimetry since 1930 can be seen by examining t h e bias between experimental and correlation values vs. the year of publication of t h e data f o r the individual hydrocarbons used in the correlations. The bias, E?, f o r each hydrocarbon used in deriving a n equation i s defined as
where C,, = experimental heat capacity, C, = computed heat capacity, and m = number of points f o r the hydrocarbon. To p u t the d a t a f o r the different correlations on a common basis, t h e bias has been divided by the standard deviation of the points about the correlation (Tables I11 and I V ) and is tabulated with the year of publication in Tables VI1 and V I I I . Ignoring sign, larger numerical values occur up t o 1 9 4 6 ; t h e r e a f t e r there is a marked decrease in the magnitude of t h e largest values. I n addition to the a g e of t h e data, there seems to be some influence of t h e number of carbon atoms on t h e bias ratio. This can be seen in the ratios f o r n-octane through n-hexadecane, based on the d a t a from one group of experimenters ( 8 ) . For other hydrocarbons, there is a confounding of age with number of carbon atoms
Table V. Constants in Equation 2 for Heat Capacity of Some Individual Liquid Hydrocarbons
Hydrocarbon a b C
Ethene
Propene
1-Butene
Isobutane
0.71418 - 0.29525 0.15407
0.64766 -0.43638 0.31228
0.63192 - 0.46311 0.33869
0.32575 0.22823 -.0.14714 0.12786
0.5788 0.0063 0.0007
0.5059 0.0092 0.0020 0.04 0.0039 18
0.4967 0.0240 0.0004 0.07 0.0010 18
0.4666 0.0356 0.0015 0.32 0.0026 25
d Statistical Information 1. Mean c,
2. 3. 4. 5. 6.
RMS dev. of c, Std dev. of exp. c, about eq. Mean % std. dev. of c, (100% X item 3/item 1) Max. dev. of c, from Eq. 2 No. of exptl. pts.
0.01
0.0011 12
Journal of Chemical and Engineering Data, Vol. 15, No. 1, 1970
95
Table VII. Bias Ratio for Heat Capacities of Liquid Hydrocarbons from Equation 3
n-Alkanes B
No. of C Atoms
5 6 7 8
9 10 11 12 13
14 15 16 17 18 25
Isoalkanes B
std. dev.
Year of publication
-0.13 $0.77 0.10 0.44 -0.38 -0.44 -0.73 -0.88 -0.90 -0.78 -0.86 -0.73 -0.06 +0.25 -1.90
1967 1946 1954 to 1961 1954 1954 1954 1954 1954 1954 1954 1954 1954 1967 1967 1932
1-Alkenes
std. dev.
Year of publication
-0.15 +0.70
1943 1946
+0.41
1947
-0.30
1941
~
B
n-Alkylcyclohexanes
std. dev.
Year of publication
-0.97 -0.41 +1.21 0.40
1937 1939 1944 1947
-0.33 -0.39
1957 1957
B
std. dev.
Year of publication
-0.09 +0.28 1.09 - 0.44 -0.51
1930 1946 1949 1965 1965
+
(Excluded only 3 points) 10.80
1965
+0.18
1949
Table VIII. Bias Ratio for Heat Capacities of Liquid Hydrocarbons from Equation 4 or 5
Equation 4 Light Alkanes No. of C Atoms 1 2 3
4 5 6 7
B
n-Alkylbenzenes
std. dev.
Year of publication
-0.51 -0.38 +1.21 -0.49
1930 1937 1938 1940
~
Equation 5 B
n-Alkylcyclopentanes
std. dev.
Year of publication
-1.23 +0.82 +0.19 -0.41
1948 1946 1945 1965 1965
8
9 10 15
-0.11
B
Year of publication
std. dev.
+0.89 -1.58 +0.23 +0.43
1946 1946 1953 1965 1965
-0.29
1965
+0.07
Table IX. Calculated Results" for Excess Heat Capacity in Binary n-Alkane Systems at x1 = x 2 = 0.5
Calculated C, using Eq. 3 and Table I11
Temp., O
c.
45 60 68.74 55.5 68.74 63.5 86 101 a
nl
6
n2 = 16
n
48.515 49.801 50.545 nl = 6 49.417 50.545 nl = 8 64.787 67.412 69.208
122.448 124.875 126.421 n2 = 24 186.930 190.214 n2 = 24 188.874 194.990 199.572
84.599 86.562 87.785 n = 15 116.376 118.582 n = 16 125.482 129.743 132.897
=
Molecular weights based on atomic weights for C Mean value over temperature range. Linearly interpolated values.
=
12.010 and H
which cannot be separated without making questionable assumptions. COMMENTS
T h e correlation equations give reliable heat capacities between t h e freezing and normal boiling points f o r liquid hydrocarbons if t h e reduced t e m p e r a t u r e i s calculated using t h e boiling points listed in Tables I a n d 11. T h e hydrocarbons are members of commonly occurr i n g homologous series. Excepting t h e light alkanes, t h e 96
C PE
Journal of Chernicol and Engineering Data, Vol. 15, No. 1, 1970
=
=
11
From Eq. 8
MMdata ( 1 3 ) in Eq. 10
-0.88 -0.78 -0.70
1
-1.80 -1.80
i -1.12*
-1.35 -1.46 -1.49
-0.73' -0.57 -0.48
1
-0.j5*
1.008.
heat capacities a r e expressed in cal./(g. K.) . F o r comp u t e r applications these units can be converted to molar heat capacities by multiplying t h e equation by molecular weight expressed as a function of n. A t e s t f o r t h e soundness of t h e statistical method of correlating t h e d a t a over a broad r a n g e of molecular weights is a s follows. T h e temperature derivative of experimental heats of mixing-Le., identical w i t h excess enthalpy of mixtures-is t h e excess h e a t capacity of t h e mixture. T h e excess h e a t capacity of a m i x t u r e c a n b e calculated using t h e concept of conformal solutions. F o r
t h i s concept, t h e h e a t capacity of a m i x t u r e of components f r o m a homologous series would be t h e s a m e as t h e h e a t capacity of a member of t h e series w i t h t h e number of carbon atoms p e r molecule t h e same as t h e molar a v e r a g e number f o r t h e constituents. T h i s c a n b e symbolized by n
=
n,r1
+ n*x2
(7)
T h e molar excess h e a t capacity of t h e m i x t u r e is then
111) when t h e h e a t capacity of members of a homologous series i s represented by E q u a t i o n 3.
ACKNOWLEDGMENT The. a u t h o r t h a n k s t h e Monsanto Go. f o r permission to publish a n d f o r computer t i m e f o r t h e calculations, a n d those individuals w i t h i n a n d outside of Monsanto who expressed i n t e r e s t a n d aided in t h e work.
NOMENCLATURE where CP’)l= t h e molar h e a t capacity of t h e m i x t u r e a n d x , = mole fraction of component i. E q u a t i o n 8 is t h e t e m p e r a t u r e derivative of t h e relation f o r t h e h e a t of mixing, HJI,
where H = enthalpy p e r mole, M indicates mixing, E indicates excess, a n d nz indicates mixture. H e a t s of mixing f o r t h e b i n a r y m i x t u r e s n-hexanen-hexadecane, n-hexane-n-tetracosane, a n d n-octane-ntetracosane at t h r e e o r more t e m p e r a t u r e s were determined by Holleman ( 1 3 ) .T h e mean excess h e a t capacity over a t e m p e r a t u r e r a n g e f r o m t , to t , is given by
a , b, c, d = constants in Equations 2, 3, 4, and 5,
cal./ (g. K.) Aa, A b , A c , A d = constants in Equations 3, increment in a , etc., per carbon atom in a molecule c p = heat capacity a t constant pressure per unit weight, cal./ (g. ’ K.) C, = molal heat capacity a t constant pressure cal./ (g.-mole K.) C, = molal heat capacity a t saturation pressure f = symbol for a function of unspecified form H = enthalpy, cal./g.-mole m = number of points n = number of C atoms per molecule, or molal average in a mixture ni = number of C atoms in ith hydrocarbon R = ideal gas constant, cal./ (g.-mole K.) t = temperature, ’ C. T =t 273.15, absolute temperature, K. T , = absolute temperature a t boiling point of a hydrocarbon with n C atoms, K. T , = absolute temperature a t freezing point, K. T , = T / T , , reduced temperature xi = mole fraction of ith component in a mixture O
+
O
O
A s C P Ei s a small difference between numbers of comparable size whether calculated by E q u a t i o n 8 o r 10, i t s accuracy is not particularly good i n e i t h e r case. T h e maximum absolute value of CPE f o r a b i n a r y system might b e expected t o occur i n a n equal-molar mixture. W i t h that in mind, calculations were made w i t h t h e h e a t capacity relations to compare with t h e results calculated using Holleman’s d a t a f o r such mixtures. A n approximation to t h e e r r o r a r i s i n g f r o m calculations implied by E q u a t i o n 8 can be ascertained f r o m t h e statistical d a t a in Table 111. F o r t h e n-hexane-n-hexadecane system t h e e r r o r i n CPEcould be l a r g e r t h a n kO.12 cal./(g.-mole O K . ) . I n view of t h e unevenness of t h e mean values of dH-’I/dT f o r successive t e m p e r a t u r e r a n g e s i n Holleman’s results, it would seem that t h e e r r o r s in his smoothed HJf values a r e l a r g e r t h a n h i s estimated errors. T h e results of some calculations f o r CPEa r e summarized in Table IX. I n t h e hexane-hexadecane a n d hexanetetracosane systems, Holleman’s d a t a do not lead to a well-defined relation of CPE w i t h t e m p e r a t u r e ; consequently, only a mean value f o r t h e t e m p e r a t u r e r a n g e i s given i n Table I X . A reasonable value f o r t h e e r r o r in CPEf r o m Holleman’s d a t a f o r t h e hexane-hexadecane system m i g h t be g r e a t e r t h a n C0.32 cal./(mole K . ) . T h e maximum difference between t h e two values f o r C,E i n t h i s system is 0.33 cal./ (mole O K . ) . T h i s value approximates t h e e r r o r f o r a difference [(0.12)2 i- (0.32)2]1/2 = w i t h t h e cited errors-Le., 0.34. I n t h e two systems containing tetracosane, t h e differences a r e l a r g e r t h a n m i g h t be expected. T h e source of t h e l a r g e r differences m a y b e a slight systematic overestimation of t h e C, f o r tetracosane, as well as l a r g e r statistical e r r o r s . T h e reasonableness of using t h e conformal solution concept can only b e tested f o r t h e excess h e a t capacity of m i x t u r e s of t h e heavier n-alkanes. It seems plausible t h a t conformal solution t h e o r y can b e applied (Table
SUPERSCRIPTS E = excess m = mixture M = mixing LITERATURE CITED
American Petroleum Institute, New York, “Technical Data Book, Petroleum Refining,” Chap. 7 , p. 143, Chap. 1, pp. 52-6, 62-70, 76, 92, 94, 96, 1966. Aston, J. G., Kennedy, R. M., Schuman, S. C., J . Am. Chem. SOC.62, 2059 (1940). Aston, J. G., Messerly, G. H., Ibid., 62, 1917 (1940). Aston, J. G., Szasz, G. J., Finke, H. L., Ibid., 65, 1035 (1943). Douglas, T. B., Furukawa, G. T., McCoskey, R. E., Ball, A . F., J . Res. Natl. Bur. Std. 53, 139-53 (1954). Douslin, D. R., Huffman, H. M., J . Am. Chem. SOC.68, 173-6 (1946). Ibid., pp. 1704-8. Finke, H. L., Gross, M. E., Waddington, G., Huffman, H. M., Ibid., 76, 333-41 (1954). Finke, H. L., Messerly, J. F., Todd, S. S., J . Phys. Chem. 69, 2094-100 (1965). Gross, M. E., Oliver, G., Huffman, H. M., J . Am. Chem. SOC.75, 2801 (1953). Guthrie, G. B., Jr., Huffman, H. M., Ibid., 65, 1139-43 (1943). Hadden, S. T., H y d r o c a r b o n Process. Petrol. Refiner 45 ( 7 ) , 137-42 (1966). Holleman, T., Ph.D. thesis, University of Amsterdam, Amsterdam, Holland, 1964. Huffman, H. M., Todd, E. S., Oliver, G. D., J . Am. Chem. soc. 71, 584 (1949). Kemp, J. D., Egan, C. J., Ibid., 60, 1521 (1938). Journal of Chemical and Engineering Dato, Val. 15, No. 1, 1970
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McCullough, J. F., Finke, H. L., Gross, M. E., Messerly, J. F., Waddington, G., J . Phys. Chem. 61, 289301 (1957). McCullough, J. P., Messerly, J. F., U. S. Bur. Mines Bull. 596 (1961). Messerly, J. F., Guthrie, G. B., Todd, S. S., Finke, H. L., J. CHEM.ENG.DATA12, 338-46 (1967). Messerly, J. F., Todd, S. S., Finke, H. L., J . Phys. Chem. 69, 353-9 (1965). Zbid., pp. 4304-11. Oliver, G. D., Eaton, M., Huffman, H. M., J . Am. Chem. SOC.70, 1502-5 (1948). Osborne, N. S., Ginnings, D. C., J. Res. Natl. Bur. Stds. 39, 453-77 (1947). Parks, G. S., Huffman, H. M., Thomas, S. B., J . Am. Chem. SOC.52, 1032 (1930). Parks, G. S., Moore, G. E,, Renquist, M. L., Naylor, B. F., McClaine, L. A,, Fuji, P. S., Hatton, J. A,, Zbid., 71, 3386-9 (1949). Parks, G. S., West, T. J., Moore, G. E., Zbid., 63, 1133 (1941). Powell, T. M., Giauque, W. F., Zbid., 61, 2366 (1939). Reid, R. C., Sherwood, T. K., “Properties of Gases and Liquids,” 2nd ed., pp. 286-94, McGraw-Hill, New York, 1966.
Rossini, F. D., Knowlton, J. W., J . Res. Natl. Bur. Stds. 19, 249 (1937). Ruehrwein, R’. A., Huffman, H. M., J . Am. Chem. SOC. 65,1620-5 (1943). Scott, R. B., Brickwedde, F. G., J . Res. Natl. Bur. Stds. 35, 501 (1945). Scott, R. B., Ferguson, W. J., Brickwedde, F. G., Zbid., 33, 1 (1944). Scott, D. W., Guthrie, G. B., Messerly, J. F., Todd, S. S., Berg, W. T., Hossenlopp, I. A., McCullough, J. F., J . Phys. Chem. 66, 911-4 (1962). Spaght, M. E., Thomas, S. B., Parks, G. S., Zbid., 36, 882 (1932). Szasz, G. J., Morrison, J. A., Pace, E. L., Aston, J. G., J . Chem. Phys. 15, 562-4 (1947). Timmermans, J., “Physico-Chemical Constants of Pure Organic Compounds,” Vol. I, pp. 21-201, 1950, Vol. 11, pp. 4-88, Elsevier, New York, 1965. Todd, S. S., Oliver, G. D., Huffman, P. M., J. Am. Chem. SOC.69, 1519 (1947). Wiebe, R., Brevoort, M. J., Zbid., 52, 622-33 (1930). Witt, R. K., Kemp, J. D., Zbid., 59, 273-9 (1937). RECEIVED f o r review February 7, 1969. Accepted July 31, 1969.
Experimental Data and Procedures for Predicting Thermal Conductivity of Multicomponent Mixtures of Nonpolar Gases S. C. SAXENA Department of Energy Engineering, University of Illinois a t Chicago Circle, Chicago, Ill.
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G. P. GUPTA’ Department of Physics, University of Rajasthan, Jaipur, Rajasthan, India A hot-wire cell is used to determine the thermal conductivity of Ne, Ar, Kr, Xe, N,, 0,, H,, and D, gases at 40°, 6 5 O , and 9 3 O C. At these temperatures, the thermal conductivity i s also measured as a function of composition for the systems H,-N,, H,-0,, N,-0,, D,-N,, H,-0,-Ne, H,-N,-0,, H,-Ar-Kr-Xe, and H,-D,-N,-Ar. These data are compared with the predictions of the theory due to Hirschfelder, procedures due to Mason and Saxena, Lindsay and Bromley, Mathur and Saxena, and Ulybin, Bugrov, and ll’in, and a few simple methods. These detailed calculations and their interpretation in the light of experimental data lend valuable infor. mation concerning the status of the art of predicting thermal conductivity of multicomponent mixtures. The thermal conductivity data on binary systems are employed in conjunction with kinetic theory to obtain diffusion and viscosity coefiicients.
A N U M B E R O F M E T H O D S have been developed f o r measuring t h e thermal conductivity of gases a n d gaseous m i x t u r e s [Saxena (ZQ),Saxena and Gandhi ( 3 2 ) ] . The choice of a particular method depends primarily upon environmental conditions ( t e m p e r a t u r e a n d press u r e ) of t h e t e s t gas, and t h e accuracy desired i n t h e measurement. T h e r e is a considerable engineering intere s t in t h e thermal conductivity of multicomponent mixt u r e s involving monatomic and polyatomic gases. F u r Present address, Department of Physics, BSA College, Mathura, India
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ther, because i t is impossible to have d a t a f o r all possible choices, i t i s useful to have a scheme t o achieve t h i s goal. W e have consequently measured t h e t h e r m a l conductivity of t h e b i n a r y systems H,-N2, H2-02, N2-02, a n d D,-N, ; t h e t e r n a r y systems H,-02-Ne a n d H2-N2-02 ; a n d t h e q u a t e r n a r y systems H,-Ar-Kr-Xe a n d H,-D,-N,-Ar. The thermal conductivity of t h e related e i g h t p u r e gases w a s also measured. T h e measurements were made at 40”, 6 5 ” , a n d 93” C., at a pressure less t h a n 1 atm., a n d i n t h e case of m i x t u r e s a s a function of composition. Several almost equally good methods can be adopted f o r measurement under these conditions.
