Extrapolation of the Orthorhombic nParaffin Melting Properties to Very LongChain LengthsMartin G. Broadhurst Citation: The Journal of Chemical Physics 36, 2578 (1962); doi: 10.1063/1.1732337 View online: http://dx.doi.org/10.1063/1.1732337 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/36/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Calculation of Physicochemical Properties for Short- and Medium-Chain Chlorinated Paraffins J. Phys. Chem. Ref. Data 42, 023103 (2013); 10.1063/1.4802693 Disorder in paraffin chains of urea adducts and nparaffins J. Chem. Phys. 92, 6867 (1990); 10.1063/1.458274 Theory of phase transitions in crystals of longchain molecules and the application to nparaffins J. Chem. Phys. 72, 353 (1980); 10.1063/1.438856 Infrared Spectra of Polyethylene and Long Chain nParaffins J. Chem. Phys. 24, 1115 (1956); 10.1063/1.1742709 Premelting Anomalies of Some Long Chain Normal Paraffin Hydrocarbons J. Chem. Phys. 10, 686 (1942); 10.1063/1.1723645

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 36, NUMBER 10 MAY 15,1962

Extrapolation of the Orthorhombic n-Paraffin Melting Properties to Very Long Chain Lengths

MARTIN G. BROADHURST

National Bureau of Standards, Washington, D. C.

(Received December 26, 1961)

An analysis of the equation T M = To (n+a)/(n+b) is made in order to determine the ~onveq~ence te:n­perature To of the melting temperatures of the n-paraffins from C44

0

to ClOD. A g~od fit IS obtamed u~mg To=414.3°K, a= -1.5, and b=5.0. This value of To (414.3°K= 141.1 C) has an estImated total uncertamty of ±2.4°K and is proposed as the correct value of the equilibrium melting t.emperature of polyethylene. It is shown that presently available data do not permit accurate extrapolatIOn of the n-paraffin heats and entropies of fusion to the polymer limit.

1. INTRODUCTION

CORRELATIONS of the behavior and properties of polyethylene with those of the n-paraffins are

common in current literature and it is generally agreed that the change in physical properties accompanying an increase in chain length from moderate to very long chains is reasonably smooth and free of serious anom­alies. This conclusion is a natural consequence of the view that a -CH2- unit experiences the effects of its immediate surroundings only and hence its contribution to the properties of a -CH2- chain molecule is reasonably insensitive to the length of the chain. Chain ends small non-CH2 groups in the chain, and chain folds (in the case of polyethylene lamellas) might well be handled as perturbations on the basic structure and behavior patterns established by the -CH2-units.

This paper reports part of a study that is aimed at a better understanding of the basic nature of the -CH2-chain. After reviewing some previous work on melting point equations we shall establish within narrow limits the validity of the equation TM=To(n+a)/(n+b), where T M and n are the melting temperatures and corresponding numbers of carbon atoms in the paraffins, and To, a, and b are constants. This relationship is fitted to the relevant n-paraffin data and the results discussed. In particular, To (commonly called the convergence temperature) is determined for the orthorhombic n-paraffins and this value is proposed as the equilibrium melting temperature of polyethylene, which also has an orthorhombic structure. The heats and entropies of fusion for the n-paraffins, and the difficulties in extending these quantities to the polymer limit are discussed. Mathematical considerations of the melting point equation are included in the Appendixes at the end of the paper.

II. MELTING TEMPERATURES

have been proposed. For most of these equations there exists a value To defined as

It is this constant which represents the equilibrium melting point of polyethylene, i.e., the melting point of a large crystal (of long -CH2- chains) that contains an equilibrium number of defects.

By far the most popular and successful of the proposed melting-point equations1- s can be put in the following form:

TM= To(n+a)/(n+b), (1)

where a and b are constants. This relationship was first derived by Garner, Van Bibber, and King l on the basis of the proposed linearity of the heats and entropies of fusion of four even numbered paraffins. Grey2 arrived at (1) by way of a melting theory which equated the vibrational energy of the chains to the lattice energy at TM • Meyer and Van der Wyk3 found empirically that a form of Eq. (1) provided a good fit of the melting points of several paraffins. StaJlberg et al.4

found that the melting points of their C82 and ClOO

were best predicted by the empirical equation of Etessam and Sawyer.s The equation of these latter authors [derived from the apparently ad hoc assumption that dM/dT= (aM+b)2, where M=14n+2 is the molecular weight and T is the corresponding melting temperature] has not only the form of Eq. (1) but nearly the same values for the constants as those found in this current study. None of the above references gives a very convincing derivation of Eq. (1) and hence its justification is one of the necessary concerns of this paper.

I W. E. Garner, K. Van Bibber, and A. M. King, J. Chern. Soc. 1931, 1533.

2 C. G. Gray, J. lnst. Petrol. 29, 226 (1943). 3 K. H. Meyer and A. Van der Wyk, Helv. Chim. Acta 20,

Many relationships TM= TM(n) between the melting 1313 (1937). temperatures (TM ) of the n-paraffins and the corre- 4 G. StiilIberg, S. StiilIberg-Stenhagen, and E. Stenhagen, Acta.

Chern. Scand. 6, 313 (1952). sponding numbers (n) of carbon atoms in the molecule 5 A. H. Etessam and M. F. Sawyer, J. lnst. Petrol. 25, 253 (or equivalently the molecular length or weight) (1939).

2578

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n-PARAFFIN MELTING PROPERTIES 2579

There are in fact several important aspects to the problem of calculating a significant value for To, which must be considered. (1) The validity of the functional form of Eq. (1) must be established not only over the range of chain lengths covered by experimental data but also up to the limit of infinite chain length. (2) The n-paraffin melting points used in fitting the function TM(n) and correctly evaluating the constants therein must themselves be accurate temperatures of equilibrium between the liquid and crystal phases. (3) These liquid and crystal phases must be the ones that persist up to the very long chain limit. That is, the extrapolated melting points of one phase would obviously not predict the melting points of some different phase.

The proposed validity of Eq. (1) is based on the following argument: In Appendix I, a general thermo­dynamic treatment based on the linearity of the enthalpy and entropy with number of carbons in any given n-paraffin phase at any given temperature, leads to

T M = To(n+a)/\ (n+b)-k[(nal+i3l) In(To/TM )

p q

+ L1naT/(r-1) ](TMT-L Tor-I) + LU3T/(r-1)] r=2 r=2

The term in boldface brackets converges to a constant as n becomes large for any values of p, q, and the a's and i3's. Hence if Eq. (1) is satisfactory at short chain lengths, it becomes even a better approximation at longer chain lengths. It will be shown below that Eq. (1) does fit the appropriate experimental data within the limits of experimental error.

Before selecting a series of data with which to fit Eq. (1), one must be certain that the solid and liquid structures of the chosen paraffins are the same structures which exist in very long -CH2- chains. Polyethylene is known to crystallize with an orthorhombic subcell which is apparently stable at all temperatures at which the material is crystalline.6 •7 Thus it seems reasonable that the large, long chain length, ideal crystal would similarly melt from the same ortho­rhombic structure, and we are thus limited to con­sidering only those n-paraffins which also melt from the orthorhombic crystalline phase. Whereas the paraffins below C44 melt from a hexagonal or triclinic phase, it is known that above C44 the orthorhombic subcell structure (i.e., either the orthorhombic or monoclinic phase) is stable at and below the melting point.8

The question of the similarity of the liquid structures of the paraffins and very long -CH2- chain liquids is

6 C. W. Bunn, Trans. Faraday Soc. 35, 482 (1945). 7 E. A. Cole and D. R. Holmes, J. Polymer Sci. 46,147 (1960). 8 M. G. Broadhurst, J. Research Nat!. Bur. Standards 66A,

No.3 (1962).

TABLE 1. Experimental and calculated orthorhombic melting points of the longer n-paraffins. The calculated points are based on Eq. (3). References to the data have been given previously.8

Number of carbons in Experimental Calculated tJ.TM

chain TM(OK) TM(OK) (Exp-Calc) 1

44 359.6 359.3 +0.3

46 361.2 361.5 -0.3

50 365.3 365.3 0.0

52 367.2 367.0 +0.2

54 368.2 368.6 -0.4

60 372.4 372.9 -0.5

62 373.7 374.1 -0.4

64 375.3 375.3 0.0

66 376.8 376.4 +0.4

67 377.3 376.9 +0.4

70 378.5 378.4 +0.1

82 383.5 383.3 +0.2

94 387.0 387.1 -0.1

100 388.4 388.6 -0.2

obviously difficult to resolve. However, if one considers accurate values for the molar free energy, enthalpy, and entropy of the shorter paraffins9 one finds these quantities to be quite accurately linear with number of carbon atoms at any constant temperature. It is difficult to see why this linearity should not continue or what conditions could be expected to alter the nature of the liquid state at some longer chain length if such conditions are not operative at shorter chain lengths. Even so, we shall here treat the "constancy" of the liquid state at all chain lengths as an assumption and depend on the results for its justification.

In a recent critical survey8 the melting points of fourteen n-paraffins from C44 to ClOO were collected. These experimental temperatures (all for the ortho­rhombic-liquid transition) are listed in Table 1. Unfortunately, data on these longer paraffins are both scarce and relatively inaccurate because of difficulties in synthesizing and purifying samples. It will be seen, however, that the data scatter is not unreasonably large.

Using Eq. (1) in the form

TM = To(n/n+b) +Toa(1/n+b) ,

least-squares calculations with the data in Table I were made for several assigned values of the constant b. To, a, and the standard deviation from each calculation were plotted as a function of the corresponding value

9 Private communication from J. P. McCullough, Bureau of Mines, Bartlesville, Oklahoma; data based ~n study reported earlier by H. 1.. Finke, M. E. Gross, G. Waddmgton, and H. M. Huffman, J. Am. Chern. Soc. 76, 333 (1954).

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2580 MARTIN G. BROADHURST

390

TM = 414.3(n - 1.5)1(n +5.0)

t 385

~ 380

w a: 375 ::> I-<t a: w 370 "-::;: W I-

365 C> Z

5 w :IE

100 110

ATOMS. n

FIG. 1. The orthorhombic-liquid melting temperatures of the n-paraffins from C44 to CIOO• The curve is calculated from Eq. (3) and the circles represent data collected from the literature. The calculated convergence temperature is 414.3°K±2.4°K.

of b, and the correct values of the constants were chosen as those which corresponded to a minimum in the standard deviation. The resulting calculation gives

TM(OK) =414.3(n-1.S)/(n+S.0). (3)

The standard deviation of TM is 0.3°K. Assuming no variation in b, the standard error of To for these data is calculated directly to be 0.4 oK. The possible variation in b was assumed to be ±2.0 which leads to a corre­sponding uncertainty in To of ±1.2°K. Combining the uncertainty in To due to the uncertainty in b with three times the standard error of To at constant b, one arrives at a total uncertainty in To of ±2.4°K. The calculated value of To is especially sensitive to errors in TM for the longer paraffins. For instance, To would be increased by about 1.2°K if the experimental value for either ClOO or C94 were increased by O.soK. It should be remembered that the above relationship applies only to the melting temperatures of the longer n-paraffins (C44 and above), which melt from an orthorhombic subcell structure.

The convergence temperature found here (To= 414.3°K= 141.1 DC) is somewhat higher than previously reported equilibrium melting temperatures for poly­ethylene. This is a natural and expected discrepancy as discussed in considerable detail recently by Hoffman and Weeks. lO There is now little doubt that crystalline polyethylene, grown either from the melt or from solution, consists of platelets of -CH2- chains, with the -CH2- units arranged in an orthorhombic subcell and the chain axes perpendicular to the lamellar surfaces. These lamellar surfaces consist of chain

10 J. D. Hoffman and J. J. Weeks, J. Research Nat!. Bur. Standards 66A, 13 (1962).

foldsY·12 If the folds were replaced by CH3 surfaces. the above description fits crystalline paraffins. Hence the change in the melting temperature of polyethylene with lamellar thickness (which depends on the crystalli­zation temperature and thermal treatment) is no more surprising than the change in melting temperature of the paraffins with chain length. The commonly reported melting temperature of carefully crystallized linear polyethylene is about 138°C or about three degrees below the value of To calculated here. However, from Eq. (3) one finds that the n-paraffin which would give the same TM = 138°C is around CS60, which would have a lamellar thickness (1= 1.27n) of about 1100 A. This thickness is in line with the largest commonly observed polyethylene lamellas (around 1000 A). Of course, since the fold surface energy in a polyethylene crystal is probably different from the CH3 surface energy in a paraffin crystal, one would not expect that a given lamellar thickness would melt at the same temperature in both cases. That is, the melting points of the two structures would asymptotically approach To along different paths because of the difference in the constants a and b in Eq. (1). An approximate treatment of the explicit relevance of the end-surface energy in Eq. (1) is given in Appendix II.

III. ENTHALPY AND ENTROPY OF FUSION

One of the original derivations of Eq. (1) was based on the linearity of the enthalpy f:..Hr and entropy f:..Sr of fusion of the paraffins as a function of the number of carbons n.l While these quantities are nearly linear, a careful look at available data shows a definite non­linearity.s One might well expect the enthalpy f:..H and entropy f:..S difference between the liquid and a given solid phase to be linear with n at constant temperature, but f:..Hj and f:..Sr (defined by AHj=f:..H IT~TM and ASj=f:..S IT=TM) involve different temperatures for different n's. Even in the simplest general case where f:..H is linear with temperature [f:..H = (aon+J3o) + (aln+J3l) T, see Appendix I], one finds with the aid of Eq. (1) that f:..Hj= (aln2+a2n+a3)/(n+b), where the a's are constants. A similar treatment could be given for f:..Sj, resulting in the conclusion that f:..Hj and f:..Sj are not in general linear with n. Unfortunately, the available f:..Hj data do not cover enough longer paraffins with enough accuracy to permit a very dependable calculation of what the heat (and entropy) of fusion of long -CH2- chains would be, except to place the values somewhere around 0.9-1.0 kcaI/mole for AH, and 2.15 to 2.4 caI/mole-oC for f:..Sj. It should again be remembered that one must consider only the ortho­rhombic-liquid transition when extrapolating the above quantities to very long chain lengths, since (contrary to one reportl3) the increment increase in AHj and

11 J. r. Lauritzen, Jr., and J. D. Hoffman, J. Research Natl. Bur. Standards 64A, 73 (1960).

12 J. D. Hoffman and J. r. Lauritzen, Jr., J. Research Nat!. Bur. Standards 65A, 297 (1961).

13 F. W. Billmeyer, Jr., J. App!. Phys. 28, 1114 (1957).

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n-PARAFFIN MELTING PROPERTIES 2581

!lSI per -CH2- group for the shorter paraffins is The melting temperature is given by definitely different for the orthorhombic and triclinic structures.8 TM=!lH(TM, n*)/!lS(TM, n*). (1.2)

IV. CONCLUSIONS

A reasonable argument can be made for the validity of Eq. (1) for describing the melting temperatures of the longer n-paraffins as a function of the number of repeating -CH2- units in the chain. Using Eq. (1) and data on the melting temperatures of the n-paraffins above C44 , one can obtain, within narrow limits, a: value for the melting temperature of an ideal n-paraffin crystal in the infinite chain length limit. The value thus obtained is higher than the highest measured melting temperature for polyethylene, but this ap­parent discrepancy can be explained as a natural con­sequence of the chain-folded nature of polyethylene lamellas.

APPENDIX I. GENERAL DERIVATION OF MELTING TEMPERATURE EQUATION

Assume that at constant pressure there exists a general function !lH(T, n) of temperature T and number of carbon atoms n, which describes the enthalpy difference !lH between two n-paraffin phases at any temperature or chain length. For some particular paraffin n* one can write the particular enthalpy difference of this compound as !lH*(T)=.!lH(T, n*). Corresponding to !lH* there is an entropy difference !lS*(T)=.!lS(T, n*) which will be assumed to exist for all T and n*. Starting with

(where the last term is the change in !lS* going from T to T M) , and adding and subtracting the term o[!lS*( T) ]TM-+To, one arrives at the familiar equation

(Toa[!:J.H(T n*)] dT !lS(T, n*) =!lS(To, n*) - J

T a; T·(1.1)

Equation (1.6) is a general equation, restricted only by the apparently accurate assumption (1.3). Hence (1.6) is applicable to any transition, either solid-solid or solid-liquid, and to any lamellar crystalline substance. It is easily seen that (1.6) is equivalent to Eq. (1) if the term in brackets is constant. That is, either the constant terms are dominant or the variable terms change in such a way that their sum is constant. Without a knowledge of the a's and i3's we cannot proceed further except in the very long chain-length limit where it can be shown that all terms in the

Assume further that at constant temperature and pressure !lS and !lH are linear functions of n. Recent accurate data9 on the solid and liquid forms up to n-C16 demonstrate quite precise linearity and this linearity will be assumed to hold at all longer chain lengths for which the phases involved do not change. Hence one can write,

!lH = n!lhc (T) + !lhe ( T) ,

!lS=n!lsc(T) +!lse(T) , (1.3)

where !lh and !ls are the enthalpy and entropy dif­ferences associated with the CH2 unit (subscript c) and the chain ends (subscript e). Expressing the !lh as power series in temperature we have

p q

!lH=nLaTT'+ Li3rT', (1.4) r=0 T=O

where the two series are terminated after a sufficient number of terms to adequately describe !lH.

Equation (1.4) can be used to evaluate Eq. (1.1) giving

n*!lhc(To) To !lS(T, n*) = +!ls.(To) - (n*ai+i3i) In-

To T

p r ~ r - Ln*ar-(Tri- P-i) - Li3r-

,....2 r-l r=2 r-l

• (TOr-LT'-l). (1.5)

!ls.(To) is the end group entropy difference at To, and

To= limTM(n)

is the melting temperature of a very large, ideal, CH2-chain crystal.

The ratio of (1.4) to (1.5) can be put in the form (replacing n* by n),

(1.6)

brackets converge to a constant as n becomes large, provided that T M has the general form of ( 1). This latter provision can be verified independently for large n as shown in Appendix II.

APPENDIX n. EXPLICIT DEPENDENCE OF T M ON THE SURFACE ENERGY

We begin withiO

(2.1)

where T M is the observed melting point of a lamella of

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2582 MARTIN G. BROADHURST

thickness 1 and surface free energy u-e' Letting 1 = 1.27n+1.98 AS we have

Equation (2.2) is a rough approximation to Eq. (1.1) but serves the purpose of showing that the change with n of the melting point of the n-paraffins (and equiva­lently polyethylene lamellas) can be explained by the influence of the lamellar CH3 (or fold) surfaces which decrease in number as the lamellar thickness increases.

THE JOURNAL OF CHEMICAL PHYSICS

In this respect, Eq. (1) is an expression of crystal-size effects and is thus applicable to any lamellar crystal (one and only one small dimension) where n is taken as the number of repeating units (e.g., molecules, atoms) between the lamellar surfaces.

Even with the simple form of Eq. (2.2) one can fit the orthorhombic melting points with a standard deviation in T M of 0.3 oK. If this is done one obtains To=412.3°K and u-./ .:lhr-3.7, although these particular numerical results are less appropriate than those in the body of this paper.

VOLUME 36, NUMBER 10 MAY IS, 1962

Shock-Tube Study of the Kinetics of Nitric Oxide at High Temperatures*

KURT L. WRAY AND J. DEREK TEARE

Avco-Everett Research LaboraJory, Everett, Massachusetts

(Received December 14, 1961)

A shock-tube program was carried out in which the NO concentration was followed as a function of time behind the shock front by absorption of 1270 A radiation, where ground vibrational state O2 and N2 are essentially transparent. The absorption coefficients of the species NO, O2, and N2 as functions of the re­spective vibrational temperatures were determined by measuring the absorption by the shock-heated gas at a point in the time history corresponding to complete vibrational relaxation but before the onset of dis­sociation.

Time history analyses were made on a total of 42 shock-tube runs covering a temperature range of 3000°-80000 K on the following six mixtures: !% NO, !% NO+i% O2, 10% NO, 50% NO, 20% air, and 100% air-the diluent in all cases being argon.

An IBM 704 computer was programmed to integrate the vibrational and chemical rate equations as a function of time behind the shock front, subject to the constraints of the conservation equations. The perti­nent rate constants were varied in a systematic trial-and-error manner in order to get the best fit to all the data.

I. INTRODUCTION

THE chemistry occurring in the relaxation region behind normal shock waves in air involves both

the formation and removal of nitric oxide. Several reactions contribute to this process, and the relevant rate constants have not been measured at temperatures above 43000 K. This paper presents the results of a shock-tube investigation of the kinetics of nitric oxide by means of an ultraviolet absorption technique. A companion paper' discusses the vibrational relaxation of nitric oxide. Both studies form part of an investiga­tion at the Avco-Everett Research Laboratory of the relaxation processes behind shock waves in air and its component gases.2

* This work was supported jointly by Headquarters, Ballistic Systems Division, Air Force Systems Command, U. S. Air Force and Advanced Research Projects Agency monitored by the Army Rocket and Guided Missile Agency, Army Ordnance Missile Command, U. S. Army.

1 K. L. Wray, J. Chern. Phys. 36, 2597 (1962), following paper. 2 K. L. Wray, J. D. Teare, B. Kivel, and P. Hammerling,

"Relaxation Processes and Reaction Rates behind Shock Fronts in Air and Component Gases," Avco-Everett Research Laboratory, Research Rept. 83, (December 1959). To be published in "Eighth Symposium (International) on Combustion."

The present experiments include studies on pure air N2, O2, NO, and various mixtures of these with each other and with argon, and cover a temperature range 3000° to 8000 oK. As we shall see, such systems at high temperatures are fairly involved, and the analysis of the experimental data requires the aid of an elec­tronic computer. To this end, computer programs using an IBM 704 have been written which permit the calcu­lation of the density, temperature, and concentration time histories for any given set of pertinent rate con­stants. That is, the rate equations for all the pertinent reactions occurring during the approach to full equilib­rium are integrated as a function of time from the shock front, within the constraints of the conservation equations. These programs include a postulated cou­pling of vibrational and dissociative processes.

Along with the experimental program, a theoretical program has also been carried out at this Laboratory. Pertinent to this paper should be mentioned the three­body recombination theory proposed by Keck3 which contributed early estimates of the dissociation rates of oxygen, nitrogen, and nitric oxide, and a theory by

3 J. C. Keck, J. App!. Phys. 32, 1035 (1960).

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