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Ind. Eng. Chem. Res. 1996, 35, 2408-2413
Sublimation Pressures of n-Alkanes from C20H42 to C35H72 in the Temperature Range 308-348 K Frederic L. L. Pouillot, Karen Chandler, and Charles A. Eckert* School of Chemical Engineering and Center for Specialty Separations, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
This paper presents a new predictive method for the sublimation pressures of heavy n-alkanes from C20H42 to C35H72 in the temperature range 308-348 K. The peculiar behavior of solid n-alkanes in terms of their heats of solid transitions is reviewed briefly. Also, the behavior of heat capacities of the n-alkanes through the solid-solid transitions is discussed, and the relative importance of this behavior is evaluated. Finally, the new expression for the sublimation pressures of n-alkanes is compared with other available correlations. Introduction
∆Hsub(T) ) ∆Hfus(Tm) + ∆Hvap(Tm) +
Sublimation pressures of solid phases are often needed to compute the fugacities and solubilities of solutes and for the correlation of solubility data. However, experimentally measured sublimation pressures of heavy n-alkanes are very limited in the literature, and thus, correlations must often be used to predict the sublimation pressures of these compounds. We present a predictive method for the sublimation pressures which is based on available thermodynamic data on solid-solid transitions, melting points, solid heat capacities, and ideal gas heat capacities. From this predictive method, sublimation pressures were generated and correlated as a function of temperature and carbon number, and this correlation is also presented. Sublimation Pressure Computation The latent heat accompanying a phase change, between phase R and phase β, is given by the thermodynamic relationship ∆H
d ln P R 98 β ) dT RT2
(1)
where P is the pressure in the vapor phase. Integration of eq 1 from the melting temperature to the temperature of interest for the system gives
Psub(T) ln sub ) P (Tm)
∫T
∆H
T
R 98 β
m
RT2
dT
(2)
The sublimation pressure of the n-alkanes can therefore be computed according to the exact thermodynamic relation
ln Psub(T) ) ln Psub(Tm) +
∫TT
m
∆Hsub(T) RT2
dT
(3)
Due to the unavailability of the heat of sublimation at the melting temperature, it is taken as the sum of the heat of fusion and the heat of vaporization at the melting temperature. Also, several solid-solid transitions can exist below the melting temperature, and a maximum of two phase transitions has been reported experimentally for each of the n-alkanes. The temperature dependence of the heat of sublimation is therefore defined as follows: S0888-5885(95)00590-2 CCC: $12.00
∫TT
first transition
m
gas (Cideal (T) - Csolid (T)) dT + p p
∆Hfirst transition +
∫TT
second transition
first transition
gas (Cideal (T) p
Csolid (T)) dT + ∆Hsecond transition + p
∫TT
second transition
gas (Cideal (T) - Csolid (T)) dT (4) p p
This computation requires the heat capacities of both the solid and gas phases, the heat of sublimation at the melting temperature, and the heats of transitions. Heats of Transitions of the n-Alkanes In the case of n-alkanes, complications arise due to solid-solid transitions and corresponding discontinuities in heat capacities. Generally, the individual members of the n-alkane series exhibit irregular solid state behavior. An interpretation of this behavior is complicated because there are four distinct crystal structures (hexagonal, orthorhombic, monoclinic, triclinic) which describe the solid phases of odd-numbered n-alkanes with more than nine carbon atoms and even n-alkanes with more than four carbon atoms (Muller, 1928; Muller, 1930; Muller and Lonsdale, 1948; Hoffman and Decker, 1953; Shearer and Vand, 1956; Broadhurst, 1962). In the hexagonal phase, molecules have a high degree of rotational freedom about their chain axes, and the chains are arranged perpendicularly to the plane formed by the methyl end groups. This form is stable just below the melting point in the odd alkanes from C9H20 to C43H88 and in the even alkanes from C22H46 to C44H90. The remaining three phases are the low-temperature, nonrotating solid phases. The orthorhombic phase, which is the stable low-temperature phase of the odd alkanes above C9H20 and for all the even alkanes above about C40H82, also consists of chains perpendicular to the end group planes. Therefore, this form transforms easily from the hexagonal phase when chain rotation stops upon cooling. In the remaining two nonrotating solid phases, monoclinic and triclinic, the chains are tilted with respect to the end group planes. Therefore, the transition from the liquid or hexagonal phase to one of these phases is more complex. The triclinic phase is the stable low-temperature phase for the even alkanes below C26H54, and the monoclinic phase is the stable low-temperature phase for the even alkanes from C26H54 to C40H82 (Broadhurst, 1962). © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2409
The packing of chains in the low-temperature, nonrotating solid phases is very similar for all three structures. The only difference is the relative displacement of adjacent chains in the direction of the chain axes; none for orthorhombic, one C-C unit for triclinic, and two C-C units for monoclinic. Because the packing of the chains is similar, the relative stability of the different phases is determined mainly by end group packing. This implies that the difference in free energies between the phases becomes small for the longer alkanes as the ratio of end to chain groups diminishes (Broadhurst, 1962). Differences in even and odd alkanes result only when a tilted phase is involved, because the differences are mainly due to packing differences in the end group layers. The end group packing is the same for the evenand odd-numbered chains if the alkane chains are packed vertically. However, when the alkane chains are tilted only the even alkanes have the symmetry for equivalent packing of both end groups, and this type of equivalent packing is apparently low-energy packing. If an odd-numbered alkane is tilted, one end will assume a low-energy position and the other end will assume a different high-energy position. Therefore, the even alkanes can have a lower energy in the tilted form, but the odd alkanes cannot (Broadhurst, 1962). Another problem in accurately describing the solidsolid transitions of alkanes is that solid phases can be unstable. Hoffman and Decker (1953) reported that, in certain cases, rotational transitions occur as the temperature is lowered, but on rewarming the transitions do not reappear. This irreversibility of the rotational transitions occurs because the nonrotating orthorhombic phase is unstable and spontaneously transforms to the stable monoclinic or triclinic phases. These more stable nonrotating phases melt directly to a liquid and do not go through the rotating hexagonal phase. Phase behavior of the solid alkanes can be drastically altered due to defects of the long-chain crystals caused by impurities. For alkanes, the most common impurities are other chain compounds of similar length. For example, if a small amount of C19H40 is present in C20H42, the C20H42 freezes into an orthorhombic form and then goes through a transition into the triclinic form rather than forming the triclinic form directly. Most of the even alkanes below C20H42 have also shown this stable orthorhombic phase just below the melting point and an orthorhombic to triclinic transition. The general effect of impurities is to stabilize the vertical hexagonal and orthorhombic structures by removing rotational constraints and by interfering with the orderly end group packing needed by the monoclinic and triclinic phases. Therefore, impurities can actually cause the occurrence of solid-solid phase transitions (Broadhurst, 1962; Takamizawa et al., 1982). The heats and temperatures of the solid-solid and solid-liquid transitions of the n-alkanes from CH4 to C100H202 were reported by Broadhurst (1962) and are shown in Table 1. Due to the many complications which were discussed, the heats of the transitions seem very difficult to correlate. However, when one considers the sum of the heats of the different transitions, both solidsolid and solid-liquid, a more regular picture emerges as shown in Figure 1. For the even n-alkanes, the sum of the heats of transitions includes the βT f l transition for 10 < N < 22 and the added β’s f RH and RH f l transitions for N g 22. For the odd n-alkanes, the sum of the heats of
transitions includes the added βo f RH and RH f l transitions for 9 < N < 37. The data were regressed as
∆H′fusion ) -10.99 + 4.02 × N
(5)
for the even n-alkanes and
∆H′fusion ) -13.02 + 3.78 × N
(6)
for the odd n-alkanes, where ∆H′fusion is the heat of fusion at the melting temperature plus any heats of solid transitions which occur. Heat Capacities of the n-Alkanes The dependence of the heat capacity on temperature seems continuous through solid-solid transitions for even-numbered n-alkanes and discontinuous for oddnumbered n-alkanes. Figure 2 shows heat capacities for C17H36 as an ideal gas phase, liquid phase, and solid hexagonal and orthorhombic phases. Jin and Wunderlich (1991) have reduced the heat capacity of n-alkanes in the solid state to the following form:
CpCNH2N+2(T) ) 2CpCH3(T) + (N - 2)CpCH2(T)
(7)
However, considering the discontinuity in behavior through the solid-solid transitions, they did not correlate the heat capacities of odd n-alkanes above 240 K, and they did not correlate the heat capacities of even n-alkanes above 300 K. There are very limited data on the heat capacities of odd n-alkanes in their unstable solid phase (only available up to C17H36), but if one plots the available experimental data, there seems to be a regularity in terms of temperature dependence regardless of the carbon number of the solute, as shown in Figure 3. Instead of trying to rationalize and correlate this behavior, we evaluated the relative importance of the heat capacities in the computation of the sublimation pressures. We computed the sublimation pressure of C17H36 with three different sets of assumptions. The first consisted of the strict application of the exact formulas given above (2Cp). The second assumed that the heat capacity of the unstable solid phase is well represented by the correlation of the stable phase (1Cp), which corresponded to the following
∆Hsub(T) ) ∆H′fusion(Tm) + ∆Hvap(Tm) +
gas (T) - Csolid (T)) dT ∫TT (Cideal p p m
(8)
Finally, we computed the sublimation pressure neglecting the integrated difference in heat capacities, and still including the solid-solid heats of transitions in the heat of fusion (no Cp). For these calculations, the solid and liquid heat capacities were provided by Messerly et al. (1967), and the ideal gas heat capacities were computed from Thinh et al. (1971) and Thinh and Trong (1976). The results are shown in Figure 4 for the temperature range 260-275 K where the differences are the most noticeable. In the case of higher alkanes, the differences are likely to be smaller because the difference in temperature between the solid-solid and the solid-liquid transitions gets smaller. Also, for the even n-alkanes, the heat capacity varies continuously through the transition, and the difference between the solid and ideal gas heat capacity is fairly small at all temperatures. For ex-
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Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996
Table 1. Heats (kJ/mol) and Temperatures (K) of Transition of the n-Alkanes from CH4 to C100H202a carbon number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 46 50 52 54 60 62 64 66 67 70 82 94 100 a
Tm
∆Hfus
RH f l 15.48 247.6
22.19
267.8
26.43
283.1 295.1 305.2 313.4 317.2 320.7 323.8 326.7 329.5 332.0 334.4 336.6 338.6 340.9 342.5 344.3 345.9 347.7 349.1 350.6 352.2 353.5 354.7 356.1 357.3 358.5
Ttransition
∆Hfus
90.67 89.88 85.46 134.8 143.4 177.8 182.5 216.4 219.7 243.5
βT f l 0.9420 3.615 3.526 4.664 8.399 13.04 14.04 20.75 6.284 28.73
263.6
36.86
279.0
45.10
291.3
53.39
301.3
62.01
309.8
69.92
34.62 40.51 45.85 47.73 48.99 54.01 54.93 57.78 59.54 60.46 64.69 66.15 68.87
Ttransition
∆Hfus
Ttransition
∆Hfus
316.2
βT f R H 28.22
217.2 236.6
βo f R H 6.862
255.0
7.666
270.9
9.173
283.7
11.07
295.2
13.82
305.7
15.49
313.7
21.77
320.2
26.08
326.2
28.97
331.4
31.57
335.7
28.05
321.3
βo f l 100.1 105.5
76.62
326.5
31.32 βM f R H 34.25
331.2
35.46
335.2
37.51
338.7
80.01 86.46 88.89
118.5
131.9 143.6
340.6 342.6 345.1 347.0 348.7 350.6 352.3 353.9 355.5 357.0 358.4
342.2 30.56 30.56
48.06 βM f β o 9.923
345.3 348.1
359.6 361.2 365.3 367.2 368.2 372.4 373.7 375.3 376.8 377.3 378.5 383.5 387.0 388.4
Broadhurst (1967).
ample, the heat capacity reported is of the order 1298 J/mol/K for C36H74 at 340 K and 921 J/mol/K for C36H74 at 340 K as an ideal gas. The difference of temperature between the two transitions of this alkane in going from the most stable solid phase to the liquid is 4 °C. The integrated difference is therefore of the order of 1507 J/mol. The heat of fusion at the melting point is 88.89 kJ/mol, and the heat of vaporization is 170.3 kJ/mol. Hence the integrated heat capacities account for a change of about 0.6% as compared to the heat of sublimation and thus, were assumed to be negligible. On the other hand, the heats of transitions added to 40.49 kJ/mol which account for a change of almost 16%
as compared to that of the heat of sublimation and thus, were not neglected in the calculations. Correlation of Sublimation Pressures By neglecting the integrated heat capacities and using eqs 5 and 6 for the heats of fusion, eq 3 is reduced to
Psub(T) ) Pvap(Tm) × ∆Hvap(Tm) + (A + B × N) 1 1 exp Tm T (R/1000)
([
][
])
(9)
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2411
Figure 4. Predicted sublimation pressures of C17H36 with different assumptions. 2: 2Cp. 0: 1Cp. b: no Cp. Figure 1. Heats of transition of the n-alkanes as a function of carbon number (data from Table 1). b: βT f l even alkanes. 9: β’s f RH even alkanes. 2: β’s f RH odd alkanes. 1: RH f l. 4: sum of the heats of transitions of odd alkanes (bottom line). 0: sum of the heats of transitions of even alkanes (top line).
Table 2. Sublimation Pressures and Latent Heats of Vaporization, Sublimation, and Fusion for C16H34, C17H36, and C18H38a compd
temp state range, K
n-C16H34 liquid solid n-C17H36 liquid solid n-C18H38 liquid solid a
293-308 288-290 298-313 288-293 303-313 288-298
A 4189 6579 4374 6866 4730 7995
B
∆Hvap
∆Hsub
∆Hfus
10.260 80 261 18.466 126 023 45 762 10.333 83 820 18.738 131 549 47 730 11.036 90 602 21.831 153 153 62 551
Bradley and Shellard (1949).
Figure 2. Heat capacity of C17H36 as a function of temperature (Messerly et al., 1967; Thinh et al., 1971; Thinh and Trong, 1976). b: ideal gas phase. O: liquid phase. 4: hexagonal phase. 2: orthorhombic phase.
Figure 5. Comparison of sublimation pressures for C17H36 computed from the proposed correlation and experimentally measured values. O: Bradley and Shellard (1949).
vaporization were computed using correlations of Morgan and Kobayashi (1994). In order to evaluate the applicability of this approach to the computation a priori of the sublimation pressures of n-alkanes, we use the data of Bradley and Shellard (1949) who report the sublimation pressures and latent heats of vaporization and fusion and sublimation for C16H34, C17H36, and C18H38 as shown in Table 2. The constants A and B are given for use in the equation
Figure 3. Odd-numbered n-alkane heat capacities of the unstable hexagonal phase as a function of temperature (Messerly et al., 1967). O: C9H20. b: C11H24. 0: C13H28. 9: C15H32. 4: C17H36.
where for even-numbered n-alkanes A ) -10.99 and B ) 4.02 and for odd-numbered n-alkanes A ) -13.02 and B ) 3.78. The heat of vaporization is in kJ/mol and R ) 8.314 J/mol/K. The vapor pressures and the heats of
log10(P0) ) -
A +B-C T
(10)
where P0 is in bar, the latent heats are in J/mol, and T is in K. The constant C is used to convert the unit of pressure from the original unit to bar and is equal to 1.875. The results are shown in Figure 5 for C17H36, and we see that there is good agreement between the sublimation pressures computed with this correlation
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Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996
Figure 6. Comparison of sublimation pressures from various correlations for C25H52 as a function of temperature.
and the experimentally measured sublimation pressures. Similar results were obtained for C18H38.
Figure 7. Comparison of sublimation pressures from various correlations for C26H54 as a function of temperature.
Comparison with Available Correlations To our knowledge, there are only two correlations that have been developed and can be used for the prediction of sublimation pressures of n-alkanes. One is by Moradinia (1986) and the other by Drake (1993). “Drake” correlation is as follows:
(
)
Tm T
(11)
C ) 11.85 + 0.101vs + 14.34
(12)
ln Psub ) ln Pm + C × 1 where for the n-alkanes
The melting pressure is taken as the saturation pressure at the melting point. “Moradinia” correlation is as follows:
Figure 8. Sublimation pressures of n-alkanes as a function of carbon number at 308 K computed from the final proposed correlation (eq 18). 2: odd-numbered n-alkanes. 1: evennumbered n-alkanes.
A ) 6.56653 + 1.76232 × N
(16)
alkanes, and the overprediction increases as the carbon number increases. However, the mathematical formulation of Moradinia’s correlation did seem to represent the trends in data fairly well. Thus, sublimation pressure data were generated as described with eq 9, and these data were regressed in the same form to provide a new correlation for sublimation pressures as a function of temperature and carbon number. Final Proposed Correlation. The final result for the n-alkanes from C20H42 to C35H72 and for T ) 308 K to T ) Tm is
B ) -244.831 - 915.569 × N
(17)
ln Psub ) A + B/T
where the units of P and T are bar and K, respectively. The constant C is used to convert the unit of pressure from the original unit to bar and is equal to 2.303. In Figures 6 and 7, we compare the sublimation pressures predicted by our correlation to the sublimation pressures predicted by the “Drake” correlation and the “Moradinia” correlation for C25H52 and C26H54. Drake’s correlation was developed from corresponding states principles and regressed on many data, which however did not include the n-alkanes of interest in the range of temperatures studied here. Also, we note that the correlation developed by Moradinia failed to require that the sublimation and the vapor pressures be equal at the triple point (assumed to be the same as the melting point). Accordingly, this correlation overpredicts systematically the sublimation pressures of odd
where for even-numbered n-alkanes
ln(Psub) ) A +
B +C T
(13)
where for even-numbered n-alkanes
A ) 3.04549 + 2.23175 × N
(14)
B ) 2080.07 - 1129.73 × N
(15)
and for odd-numbered n-alkanes
(18)
A ) 12.31 + 1.87 × N
(19)
B ) -67.54 - 1013.7 × N
(20)
and for odd-numbered n-alkanes
A ) 11.35 + 1.78 × N
(21)
B ) 34.63 - 978.3 × N
(22)
where the units of P and T are bar and K, respectively. The sublimation pressures computed from this correlation for C20H42 to C35H72 at 308 K are shown in Figure 8.
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2413
Conclusions A new predictive method for the sublimation pressures which is based on thermodynamic data was presented for n-alkanes from C20H42 to C35H72 in the temperature range 308-348 K. The sublimation pressures computed with this predictive method were compared to experimentally measured sublimation pressures and were found to be in good agreement. Also, from this predictive method, sublimation pressures were generated and correlated as a function of temperature and carbon number. This correlation was also presented and was compared to other available correlations. Acknowledgment We gratefully acknowledge the financial support from the GRI (Gas Research Institute) and the GPA (Gas Processors Association). List of Symbols R ) physical phase RΗ ) hexagonal phase β ) physical phase βM ) monoclinic phase βo ) orthorhombic phase βT ) triclinic phase ∆H ) change in enthalpy ∆Hfirst transition ) heat of first solid-solid transition ∆Hfus ) heat of fusion ∆H′fusion ) heat of fusion at the melting point plus heats of solid-solid transitions ∆Hsecond transition ) heat of second solid-solid transition ∆Hsub ) heat of sublimation ∆Hvap ) heat of vaporization A ) constant B ) constant Cp ) heat capacity Cpideal gas ) ideal gas heat capacity Cpsolid ) solid heat capacity l ) liquid phase N ) carbon number of the n-alkanes P ) pressure Pm ) melting pressure P0 ) sublimation pressure from Bradley and Shellard (1949) Psub ) sublimation pressure Pvap ) vapor pressure R ) universal gas constant T ) temperature Tfirst transition ) temperature at first solid-solid transition Tm ) melting temperature
Tsecond transition ) temperature at second solid-solid transition vs ) solid molar volume
Literature Cited Bradley, R. S.; Shellard, A. D. The Rate of Evaporation of Droplets. III. Vapour Pressures and Rates of Evaporation of StraightChain Paraffin Hydrocarbons. Proc. R. Soc. London, Ser. A 1949, 198, 239-251. Broadhurst, M. G. An Analysis of the Solid Phase Behavior of the Normal Paraffins. J. Res. Natl. Bur. Stand. 1962, 66 A (3), 241249. Drake, B. D. Prediction of Sublimation Pressures of Low Volatility Solids. Ph.D. Thesis, University of South Carolina, 1993. Hoffman, J. D.; Decker, B. F. Solid State Phase Changes in Long Chain Compounds. J. Phys. Chem. 1953, 57, 520-529. Jin, Y.; Wunderlich, B. Heat Capacities of Parafins and Polyethylene. J. Phys. Chem. 1991, 95, 9000-9007. Messerly, J. F.; Guthrie, G. B.; Todd, S. S.; Finke, H. L. LowTemperature Thermal Data for n-Pentane, n-Heptadecane, and n-Octadecane. J. Chem. Eng. Data 1967, 12, 338-346. Moradinia, I. The Solubility of High Molecular Weight Solid Hydrocarbons in Supercritical Ethane. Ph.D. Thesis, Georgia Institute of Technology, 1986. Morgan, D. L.; Kobayashi, R. Extension of Pitzer CSP Models for Vapor Pressures and Heats of Vaporization to Long-Chain Hydrocarbons. Fluid Phase Equilib. 1994, 94, 51-87. Muller, A. A Further X-Ray Investigation of Long Chain Compounds (n- Hydrocarbon). Proc. R. Soc. London, Ser. A 1928, 120, 437-459. Muller, A. The Crystal Structure of the Normal Paraffins at Temperatures Ranging from that of Liquid Air to the Melting Points. Proc. R. Soc. London, Ser. A 1930, 127, 417-430. Muller, A.; Lonsdale, K. The Low-Temperature Form of C18H38. Acta Crystallogr. 1948, 1, 129-131. Shearer, H. M. M.; Vand, V. The Crystal Structure of the Monoclinic Form of n-Hexatriacontane. Acta Crystallogr. 1956, 9, 379-384. Takamizawa, K.; Ogawa, Y.; Oyama, T. Thermal Behavior of n-Alkanes from n-C32H66 to n-C80H162, Synthesized with Attention Paid to High Purity. Polym. J. 1982, 14, 441-456. Thinh, T.-P.; Trong, T. K. Estimation of Standard Heats of Formation ∆H*fT, Standard Entropies of Formation ∆S*fT, Standard Free Energies of Formation ∆F*fT and Absolute Entropies S*T of Hydrocarbons from Group Contributions: An Accurate Approach. Can. J. Chem. Eng. 1976, 54, 344-357. Thinh, T.-P.; Duran, J.-L.; Ramalho, R. S. Estimation of Ideal Gas Heat Capacities of Hydrocarbons from Group Contribution Techniques. Ind. Eng. Chem. Process Des. Dev. 1971, 10, 576582.
Received for review September 25, 1995 Accepted January 2, 1996X IE950590N
X Abstract published in Advance ACS Abstracts, May 1, 1996.
