Ind. Eng. Chem. Res. 1998, 37, 2565-2570
2565
CORRELATIONS Correlation of the Density of the Liquid Phase of Pure n-Alkanes with Temperature and Vapor Pressure Frederick C. Tobler† 21 Winant Road, Princeton, New Jersey 08540
The density of the liquid phase of normal alkanes at the normal boiling point is represented by an empirical equation based on two known variables: the temperature of the normal boiling point and the ratio of the mole weight to the sum of the atomic numbers of all the atoms in the molecule (the “alkane number”). This equation is also valid, when applied to boiling points of different but constant pressures, but with specific coefficients for each pressure. The correlation between these coefficients and vapor pressure leads to one equation for the density of the liquid phase of the first 17 n-alkanes at vapor pressures from 1.333 to 101.325 kPa. Molecular weight and alkane number are also shown to give accurate correlations of the critical density, the liquid density at constant temperature and pressure, and the refractive index at constant temperature and pressure. Introduction The temperatures of the normal boiling points (Tnbp) of the n-alkane series from methane to heptadecane are readily available from various handbooks with little difference in the reported values. The pressure (P) is by definition 101.325 kPa. On the other hand, the molar volume of the liquid phase at the normal boiling point (Vlnbp) or the density (dlnbp) are rarely, if ever, given. They have to be found by interpolation of the pertinent data given in tables of thermodynamic properties of individual substances. This determination for the nalkanes revealed that Vlnbp increases as the chain grows in length, but the increment for each addition of one ethylene group also increases. The increment is not a constant additive property. It is 17.30 g/cm3 between methane and ethane, and it reaches 28.40 g/cm3 between hexadecane and heptadecane. Translating Vlnbp into dlnbp produces an interesting phenomenon: dlnbp increases rapidly to a maximum at hexane and then decreases steadily. The determination of dlnbp for the range from methane to heptadecane has been based on data in Thermodynamic Data for Pure Compounds by Smith and Srivastava (1986) (S&S), in Handbook of Physical Properties of Liquids and Gases by Vargaftik (1983) (VAR), and in TRC Thermodynamic TablessHydrocarbons by the Thermodynamics Research Center, Texas A&M University (1994) (TRC). Table 1 lists the results. The curve at 760 mm in Figure 1 illustrates the rapid rise of dlnbp from methane to hexane, where the liquid density reaches 0.613 56 g/cm3, and the smooth decrease from that point to heptadecane. The figure also shows the observed densities of the liquid phase at five other constant pressures below the normal boiling point (nbp). Anselme et al. (1990) published their experimental results of the critical density (dc) from pentane to †
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octadecane. Their Figure 3, p 324, bears a striking resemblance to the curve of dl at the nbp. Accurate knowledge of the properties of the n-alkane series is of importance, because many other important series of organic compounds are derived from it by branching, by addition of double or triple C-C bonds, or by substitution of atoms other than carbon and hydrogen. This paper therefore presents accurate empirical equations representing the density of the liquid phase in the coexistence system at equal pressures. Deduction of the Equation for Density of the Liquid Phase of the n-Alkanes at the Normal Boiling Point of 101.325 kPa Inasmuch as pressure at the nbp is a constant, it cannot be a variable in the analytical representation of dlnbp of the n-alkane series. I found that dl at this constant pressure can be represented by
M dl ) A + B + CT2 Zm
(1)
The variable M/Zm is a pure, relative number, without dimension. Since the same number stands for all alkanes having the same number of carbon and hydrogen atoms in the molecule, i.e., the isomers, I shall refer to it as the “alkane number”. It is derived from Mr/ NAZm, where Mr is the relative molar mass, NA Avogadro’s number, and Zm the sum of the atomic numbers of the atoms composing the molecule. Zm is also the number of protons in the molecule. Since NA is a universal constant, it is not required in this context. Mr is numerically equal to M. Zm is not a constant in the n-alkane series but a specific number for each substance. M and Zm separately are linear functions of the number of carbons (n) in the n-alkane series, but the ratio M/Zm is not a linear function of n. The
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2566 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 Table 1. Database of n-Alkanes at 101.325 kPa and Calculated Density of Liquid substance
source
Tnbp, K
dlnbp source, g/mL
M/Zm
methane ethane propane butane pentane hexane heptane octane nonane decane undecane dodecane tridecane tetradecane pentadecane hexadecane heptadecane
S&S S&S S&S S&S S&S S&S S&S S&S S&S VAR VAR, TRC VAR VAR VAR VAR VAR VAR
111.633 184.550 231.100 272.650 309.220 341.886 371.568 398.823 423.968 447.270 469.13 489.43 508.62 526.74 543.89 560.20 575.71
0.422 49 0.544 07 0.582 52 0.601 34 0.609 94 0.613 56 0.613 40 0.611 14 0.608 11 0.603 28 0.598 44 0.593 03 0.587 1 0.581 4 0.576 3 0.571 7 0.566 5
1.6042 1.6705 1.6960 1.7095 1.7178 1.7235 1.7276 1.7307 1.7332 1.7351 1.7367 1.7381 1.7392 1.7402 1.7411 1.7418 1.7425
octadecane nonadecane eicosane n-C28H58 n-C36H74 n-C64H130
VAR VAR VAR K&Z K&Z K&Z
590.62 604.91 618.63 704.8 771 907
N.A. N.A. N.A. N.A. N.A. N.A.
1.7431 1.7437 1.7441 1.7467 1.7482 1.7505
dlnbp eq 2, g/mL
difference percent
difference absolute percent
0.422 57 0.542 59 0.584 43 0.602 40 0.610 26 0.612 94 0.612 62 0.610 44 0.607 07 0.602 93 0.598 22 0.593 22 0.587 94 0.582 51 0.576 98 0.571 38 0.565 57 mean 0.560 11 0.554 45 0.548 80 0.509 19 0.473 74 0.388 70
0.020 -0.270 0.329 0.176 0.052 -0.100 -0.126 -0.114 -0.170 -0.052 -0.043 0.032 0.144 0.191 0.118 -0.054 -0.131 0.000 10
0.020 0.270 0.329 0.176 0.052 0.100 0.126 0.114 0.170 0.052 0.043 0.032 0.144 0.191 0.118 0.054 0.131 0.125
Figure 1. Liquid densities of 17 n-alkanes from methane to heptadecane at six different boiling points.
effectiveness of the ratio M/Zm as a known variable in eq 1 of the n-alkane series may be due to the fact that it represents a relationship between the mass and the electrical configuration of the different molecules of the series. M and Zm are very basic properties of molecules. There is no uncertainty in their values. The first test of eq 1 was made with data at the nbp. The variable Tnbp is based on careful observations and is known accurately to five or six significant figures to heptadecane. Tnbp is also a nonlinear function of n. The specific constant A and the coefficients B and C for the n-alkanes at the nbp have been determined by regression. The database has been compiled from the three sources mentioned previously in connection with
Figure 1. The result of the regression was
M dlnbp ) -2.6832 + 1.93900 - 3.9168 × 10-7T2nbp Zm (2) r2 of the regression was 0.999 606. Table 1 contains details of the database at the nbp, dlnbp calculated by eq 2, and the difference in percent between dlnbp in the database and calculated dlnbp. The mean of the differences was 0.000 10%, and the absolute mean was 0.125%. The maximum difference was 0.329%. The result is excellent. The maximum dlnbp of the n-alkane series is at hexane with 0.612 94 g/cm3.
Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2567
Figure 2. Observed liquid densities of 17 n-alkanes from methane to heptadecane and calculated with the Rackett method and the GCVOL Method by Elbro et al.
Table 1 also has the results of extrapolations to six higher n-alkanes. Data for T in the calculations have been taken from VAR and from projections of Tnbp by Kudchadker and Zwolinski (1966) (K&Z). The results with octadecane, nonadecane, and eicosane represent reasonable extrapolations of data given in VAR. Doolittle and Peterson (1951) reported liquid densities of C28H58, C36H74, and C64H130 ending with 573.15 K. The projection of dlnbp for the three substances is also given. The sources of dlnb and Tnbp are identified in Table 1, and dlnb is a linear interpolation from densities given in the sources. Performance of Equation 2 Compared to Methods Given in the Literature The results of the correlation of dlnbp with eq 2 has been compared with three methods from the literature. The Properties of Gases and Liquids by Reid et al. (1987) describes several methods. The “additive methods” by Schroeder and by Le Bas fail to account for the drop in dlnbp beginning with heptane (pp 52-53). The method by Tyn and Calus (1975) (pp 53-54), with the recent selections of the critical density (dc) by Ambrose and Tsonopoulos (1995), produced a fair representation of dlnbp with an absolute mean deviation of 2.84% from methane to heptadecane and a maximum of 7.74%. The “Modified Rackett Technique” is given by Reid et al., p 61, as
RTc 2/7 Vl ) ZRA[1+(1-Tr) ] Pc
(3)
where ZRA is derived from the Pitzer acentric factor (ω) by
ZRA ) 0 29056 - 0.08775ω
(4)
It is based on the critical constants and the distinct constant ZRA for each substance. The method applied in this manner to the n-alkane series, using the critical constants of Ambrose and Tsonopoulos (1995) and ω from the Reid et al. “Property Data Bank”, failed to represent the downturn in dl after hexane. Figure 2 is a graphic image of the result. The accidental “good” dl at tetradecane is the result of a misprint of ω in the Reid et al. “Property Data Bank” on p 727. The deviation is clearly systematic. Basing the calculation on the alternate approach of using the values of ZRA given in Tables 3-10 of the Reid et al. handbook with a known reference density and temperature resulted in a mean absolute deviation of 0.224% with a maximum of 0.467%. This is a fair result but not equal to the one given by eq 2. Elbro et al. (1991) proposed a new group contribution method called GCVOL. It was designed to predict dl as a function of T. I have used the applicable group contributions given by them in Table III, p 2579, to calculate dl of the first 17 n-alkanes at the nbp. The result was only fair for ethane and then deteriorated rapidly to heptadecane. As also shown in Figure 2, dl reached a maximum of 0.6333 g/cm3 for octane and ended with 0.6167 g/cm3 for heptadecane. GCVOL fails to represent the n-alkane series, the prototype for many other hydrocarbon series. Alkane Number and Correlation of Other n-Alkane Properties The correlation of dl at the nbp required T as a variable since the temperature of the nbp is not a constant. I have found that sets of dl of the n-alkane series given at constant temperatures and pressures can be correlated very acccurately by regression of dl on the molecular weight and the alkane number, for example,
2568 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998
the liquid density from pentane to heptadecane at 293.15 K and 1.013 25 bar, as given in TRC Thermodynamic TablessHydrocarbons, pp a-1010 and 1011. Regression of dl on M and M/Zm gave
M dl ) 5.8826 × 10-5 M + 5.76861 - 9.28769 Zm
(5)
r2 of the regression was 0.999 972. The absolute mean of the difference in percent was 0.029% and the maximum 0.052%. A second and more interesting example may be found in Doolittle and Peterson (1951), who on p 2147 provided liquid densities, corrected to vacuum, for 10 n-alkanes from C5H12 to C64H130 at eight constant temperatures from 263.15 to 573.15 K. Regressions of dl on M and M/Zm at the eight different temperatures had r2 between 0.999 855 and 0.999 992. The absolute mean deviation of the 50 data points was 0.036% and the maximum 0.132%. Since M and M/Zm are known for all nalkanes, the accurate results of the regressions permit reliable estimation of dl at the eight constant T values for the n-alkanes missing in their data set. Critical Density. At the critical point, neither Tc nor Pc is constant and a third known variable in addition to M and M/Zm is required. Anselme et al. (1990) published their experimental results of dc from pentane to octadecane. Their Figure 3, p 324, bears a striking resemblance to the curve of dlnbp, suggesting dlnbp as this variable. Ambrose and Tsonopoulos (1995) used their dc data, when they reviewed the critical properties of the n-alkanes for IUPAC. Regression of dc selected by them on M, M/Zm, and dlnbp gave
M + dc ) 0.45136 + 1.02 × 10-4M - 0.32133 Zm 0.532l9dlnbp (6) The liquid densities at the nbp, a known variable, are given in Table 1. r2 of this regression was 0.998 000. The mean deviation was 0.2285% and 0.00051 g/cm3 and the maximum (at decane) 0.908% and 0.0021 g/cm3. This result is well within the limits of uncertainty set by the authors in their Table 1. Expressing eq 6 in general terms and then substituting general equation (1) for dlnbp gave
M + DTnbp2 dc ) A + BM + C Zm
(7)
Regression on the same database gave
M dc ) -0.97416 + 2.1164 × 10-5M + 0.70945 Zm 1.527 × 10-7Tnbp2 (8) r2 of the regression was 0.997 005. The mean of the absolute deviation was 0.000 68 g/cm3 and the maximum 0.0018 g/cm3. All results are well within the limits of uncertainty for dc set by Ambrose and Tsonopoulos (1995). Extrapolated to 100 carbons with Tnbp by Kudchadker and Zwolinski (1966), the curve of dc is a
close match to the one marked “Teja et al.” in Figure 3, p 134, of Tsonopoulos and Tan (1993). Refractive index of the sodium D line (nD) at 293.15 K and 1.013 25 bar from pentane to heptadecane, as given in the TRC database, p a-1010 and 1011, is a virtually linear function of M/Zm (r2 ) 0.999 535). Adding M as a second variable improved the correlation. Regression of nD on M and M/Zm gave
M nD ) 2.8682 × 10-5M + 3.03268 - 3.85432 Zm
(9)
r2 of the regression was 0.999 956. The absolute mean of the difference was 0.0096% and the maximum 0.169%. Deduction of the Equation for Density of the Liquid Phase of the n-Alkanes at Equal Pressures below 101.325 kPa The collections of data by S&S, VAR, and TRC used to establish the database at 101.325 kPa (760 mmHg) have also provided the data at a number of other equal pressures between 10 and 760 mmHg. Data for the pressure range from 101.325 (760 mmHg) to 13.3322 kPa (100 mmHg) cover the full range from methane to heptadecane, while the pressure range below 13.33221.333 22 kPa (10 mmHg) no longer includes methane, because the liquid phase of methane ends just below 13.3322 kPa. P, Vl, and T from methane to nonane have been taken from S&S, TRC-provided decane, and undecane and VAR dodecane to heptadecane. Some density data for butane not available in S&S were taken from TRC. The values of the constant and the coefficients were clearly affected by the presence or absence of methane in the data sets and showed a significant difference in constant and coefficients, but not in results, in the range from 760 to 100 mmHg. The uniformity of r2 within each set of regressions is amazing. It not only documents the ability of eq 1 to represent dl at a specific constant pressure by a specific equation but also gives clear evidence of the high quality of the database collected from three different sources. In effect, eq 1 and the databases confirm each other. The results of the regressions with data sets from ethane to heptadecane between 101.325 and 1.333 22 kPa are given in Table 2. Table 2 is the basis for the determination of dl of the n-alkanes within the pressure range of 10-760 mmHg. The constant and coefficients for pressures not listed in the table may be interpolated, and their use with eq 1 will give good densities. The ratio A/B is a virtual constant of 1.3829 in the regressions with methane and 1.3807 without methane. Equation 1 therefore can be given as
(
)
M + CT2 dl ) A 1 + 0.7237 Zm
(10)
Constant A can be represented with sufficient accuracy by
A ) A′ + B′ log P and coefficient C by
(11)
Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2569 Table 2. n-Alkanes, Constant A, and Coefficients B and C of Equation 1 Regression: Ethane to Heptadecane P, mmHg
log P
constant A
coefficient B
coefficient C
r2
760 700 600 500 400 300 250 200 150 100 80 60 50 40 30 20 10
2.880 81 2.845 10 2.778 15 2.698 97 2.602 61 2.477 12 2.397 94 2.301 03 2.176 09 2.000 00 1.903 09 1.778 15 1.698 97 1.602 06 1.477 12 1.301 03 1.000 00
-2.676 85 -2.686 63 -2.691 21 -2.694 63 -2.702 29 -2.724 48 -2.741 64 -2.754 52 -2.770 05 -2.799 64 -2.810 20 -2.832 18 -2.851 33 -2.844 45 -2.872 63 -2.896 14 -2.944 49
1.935 40 1.942 17 1.946 79 1.950 98 1.957 88 1.974 14 1.986 25 1.995 94 2.007 82 2.028 87 2.036 99 2.052 21 2.064 94 2.062 31 2.081 07 2.097 61 2.130 40
-3.9190 × 10-7 -3.9220 × 10-7 -3.9211 × 10-7 -3.9130 × 10-7 -3.9121 × 10-7 -3.9580 × 10-7 -3.9991 × 10-7 -4.0218 × 10-7 -4.0677 × 10-7 -4.1519 × 10-7 -4.1952 × 10-7 -4.2594 × 10-7 -4.3115 × 10-7 -4.3245 × 10-7 -4.4063 × 10-7 -4.4919 × 10-7 -4.6804 × 10-7
0.997 830 0.998 016 0.998 295 0.998 523 0.998 733 0.998 674 0.998 438 0.998 744 0.998 881 0.998 801 0.998 705 0.998 648 0.998 385 0.998 294 0.998 710 0.998 571 0.998 566
C ) A′′ + B′′P + C′′ log P
(12)
Placing eqs 10 and 11 into eq 9 gives in general terms
M M dl ) A + B + CT2 + D log P + E log P + Zm Zm FT2 log P + GT2P (13) Equation 13 has been tested with a database of 135 data points for the n-alkanes from methane to heptadecane taken from S&S for methane to nonane and from VAR for decane and dodecane to heptadecane. The data points for each substance were spaced evenly over the available range of pertinent data. Pressures were in a range from 10.56 to 760 mmHg. r2 of the regression was 0.999 659. The mean absolute deviation of the 135 data points was 0.134% and the maximum 0.680%. The result of the regression was
M dl ) -2.93642 + 2.14920 - 5.5268 × 10-7T2 + Zm M log P + 7.9795 × 0.07437 log P - 0.06497 Zm 10-8T2 log P - 1.1958 × 10-10T2P (14) The result confirmed that eq 14 with the specific constant and coefficients of the six known variables is a valid equation for the calculation of the liquid density of n-alkanes within the parameters set for this specific database. The liquid densities and pressures of the coexistence system of n-alkanes from methane to heptadecane within the pressure range of eq 14 are readily available from various handbooks. There is little eq 14 can do for these substances. However, eq 14 can serve “diagnostic” purposes. It describes a uniform pattern present in the database of the 17 n-alkanes. When eq 14 is applied to a specific data set of one substance, the difference between dl observed and calculated indicates how closely the set conforms to that pattern. For example, there is great disagreement in dl between sets of tetradecane given in TRC and those given in VAR, while T and P are in fair agreement. They cannot both be correct. Equation 14 shows that the VAR set conforms to the general pattern, while the TRC set does not. Examination of p d-1015 (June 30, 1973) of TRC Thermodynamic
TablessHydrocarbons shows that the dl data of tetradecane do not fit the pattern of the other data on that page. Observed data are scarce beyond that point. Equation 14 can be used to estimate dl within the pressure range of 10-760 mmHg for n-alkanes beyond heptadecane, provided T and P are available. For example, Kudchadker and Zwolinski (1966) have proposed the coefficients A, B, and C for the Antoine vapor pressure correlation for C21H44 to C100H202. They will provide an estimate of the vapor pressure in mmHg at a given T. When the use of eq 13 is confined to a set of data from one substance, the alkane number is a constant. Equation 14 may then be given as
dl ) A + BT2 + C log P + DT2 log P + EPT2
(15)
Preliminary calculations have shown that eq 15 gives very accurate correlations to Tr ) 0.85 of the liquid density with vapor pressure and temperature of the coexistence system of polar and nonpolar substances (elements, inorganics, and organics) not encumbered by hydrogen bonding, polymerization, or decomposition. Detailed coverage of this subject is beyond the scope of this paper. Conclusions The alkane number M/Zm has shown to be a very exact variable useful in the correlation of certain properties of the n-alkane series. Either alone or joined by the molecular weight and the square of the temperature, these variables are able to supply smoothed sets of data of the liquid density at various boiling points, the critical density, certain other densities of the liquid phase, and the refractive index. With the n-alkane series as the point of departure for other n-alkyl series, the availability of accurate smoothed sets is important. Symbols A, B, C, ... ) coefficients in several general equations dc ) critical density, g/cm3 dl ) density of the liquid phase, g/cm3 dlnbp ) density of the liquid phase at 101.325 kPa, g/cm3 M ) molar weight, g/mol Mr ) relative molar weight NA ) Avogadro’s number n ) number of carbons in a molecule
2570 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 nD ) refractive index P ) pressure, various units Pc ) critical pressure T ) temperature, K Tnbp ) temperature at the normal boiling point, K Tc ) critical temperature Tr ) reduced temperature V ) molar volume, cm3/mol Vl ) molar volume of the liquid phase, cm3/mol Vlnbp ) molar volume at the normal boiling point, cm3/mol Vc ) molar volume at the critical point, cm3/mol Zm ) sum of the atomic numbers of all the atoms of a molecule ZRA ) constant in the Rackett equation Greek Letter ω ) Pitzer acentric factor
Literature Cited Ambrose, D.; Tsonopoulos, C. Vapor-Liquid Critical Properties of Elements and Compounds, 2. Normal Alkanes. J. Chem. Eng. Data 1995, 40, 532. Anselme, M. J.; Gude, M.; Teja, A. S. The Critical Temperatures and Densities of the N-Alkanes from Pentane to Octadecane. Fluid Phase Equilib. 1990, 57, 317-326. Doolittle, A. K.; Peterson, R. H. Preparation and Physical Properties of a Series of n-Alkanes. J. Am. Chem. Soc. 1951, 73, 2147.
Elbro, H. S.; Fredenslund, A.; Rasmussen, P. Group Contribution Method for the Prediction of Liquid Densities as a Function of Temperature for Solvents, Oligomers and Polymers. Ind. Eng. Chem. Res. 1991, 30, 2576-2582. Kudchadker, A. P.; Zwolinski, B. J. Vapor Pressures and Boiling Points of Normal Alkanes, C21 to C100. J. Chem. Eng. Data 1966, 11, 253. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill Book Company: New York, 1987. Smith, B. D.; Srivastava, R. Thermodynamic Data for Pure Compounds. Elsevier: New York, 1986. TRC Thermodynamic TablessHydrocarbons; Marsh, K. N., Director; Thermodynamics Research Center, The Texas A&M University System: College Park, TX, 1994. Tsonopoulos, C.; Tan, Z. The Critical Constants of Normal Alkanes from Methane to Polyethylene. Fluid Phase Equilib. 1993, 83, 127-138. Tyn, M. T.; Calus, W. F. Estimating Liquid Molal Volume. Processing, 1975, 21 (4), 16. Vargaftik, N. P. Handbook of Physical Properties of Gases and Liquids, 2nd ed.; Hemisphere Publishing Company: Washington, DC, 1983.
Received for review October 10, 1996 Revised manuscript received February 24, 1998 Accepted February 28, 1998 IE9606451
