The Melting Temperatures of the n–Paraffins and the Convergence Temperature for Polyethylene

Abstract

The extrapolation of the melting points, T_m, of the n-paraffins to large chain lengths (n → ∞) is reexamined in order to resolve the differences in the proposed values of the convergence temperature $T_0=\underset{n\to \infty }{\text{lim}}{T}_m$. Experimental liquid entropies can be made consistant with a term, R In n, proposed by Flory and Vrij. This term effectively replaces the well-known expression T_m = T₀ (n + a)/(n + b) with an expression T_m = T₀ (n + a)/(n + ln n + 6); thus, slowing the convergence rate and increasing T₀ from 141.1 °C to 144.7 °C. Independent estimates of the parameters in the melting relationship were obtained from thermodynamic data and the least squares estimate of T₀ = 144.7 °C (calculated from 33 melting points with a standard deviation of T_m = 0.3 °C) could not be altered by more than ± 0.5 °C by any reasonable variation of the parameters. A simplified melting expression is obtained for polyethylene which includes both the chain end and fold surface energies, and it is shown that chain end effects partly account for the discrepancy between the 144.7 °C convergence temperature and experimental melting temperatures (~ 139 °C) of extended chain polyethylene crystals.

1. Introduction

In 1962, I published a paper [1]¹ which was intended to establish on thermodynamic grounds an analytical expression T_m = f(n) for the orthorhombic normal paraffin melting temperatures, T_m, as a function of the number of carbon atoms per molecule, n. That work resulted in an expression,

$$T_m=T_0\frac{n+a}{n+b}$$ (1)

of well-known form in use since 1931. The constants were found to be a = −1.5, b = 5.0, and $T_0=\underset{n\to \infty }{\text{lim}}{T}_m(n)=414.3$ °K (141.1 °C) with an estimated error of ± 2.4 °K.

In 1963 Flory and Vrij [2] reexamined the thermodynamic basis for the paraffiin melting equation and modified the assumption [1] of a linear dependance of the liquid entropy on n, by adding a term, R In n, to account for the disordering of the methyl layers during melting. The effect of this additional term was to raise the value of T₀ from 414.3 °K (141.1 °C) as predicted by eq (1) to 418.5 ± 1 °K (145.3 °C). The difference between these two values is quite significant when calculating, for instance, the surface free energy of polyethylene crystals from melting data or crystal growth rates. It is the primary purpose of this paper to investigate this discrepancy and to establish more firmly a value for T₀.

2. Liquid Entropies

The n-parafnn melting relationship, ΔG(T_m, n) = ΔH(T_m, n) − T_mΔS(T_m, n) = 0, can be obtained following Flory and Vrij by equating the Gibbs free energy of fusion (written as a function of temperature and n) to zero. If the dependence of ΔG on n is linear, one obtains equation (1). If not, one obtains a more general form of (1) where a and b are functions of n. The limiting value, T₀, depends on the functional form of ΔG(T_m, n). The solid enthalpies and entropies and the liquid enthalpies are experimentally observed to be linear with n [3], and the problem centers around the functional form of the liquid entropies, S_l(T, n), which experimentally are found to show some non-linearity. This nonlinearity in S_l(T, n) appears in ΔG(T, n) and alters T₀ in a way which depends on the magnitude and functional form of the nonlinear terms.

Since it is the liquid entropy that supplies the nonlinear terms to the melting relationship one can investigate these terms independently of the melting temperatures by looking at the n-dependence of experimental liquid entropies. Experimental liquid entropies for the n-paraffins from C₅H₁₂ through C₁₈H₃₈ at 300 °K are listed in table 1.² If S_l were linear, then differences between consecutive values of S_l = S_cln + S_el would be constant, and if (S_l − R ln n) were linear as indicated by the work of Flory and Vrij then differences between consecutive values of (S_l − R ln n) = S_cln + S_el would be constant. These differences are plotted in figure 1 as a function of n. Figure 1 shows that neither S_l nor S_l − R ln n are linear with n, and that the R ln n term is too strong and overcompensates for the non-linearity in S_l. The magnitude of the R ln n term could be reduced or alternatively a term like 1/n could be added with or without the R ln n term. Included in figure 1 are consecutive differences of the function (S_l − R ln n − 5.6/n) which do not show any increasing or decreasing trend with n.

Table 1.

Experimental values of the absolute liquid entropies in cal/mol · deg at 300 °K for the n-paraffins with from 5 to 18 carbon atoms per molecule

n	Sl(cal/mol/deg)
5	63.21a
6	71.05
7	78.86
8	86.67
9	94.505
10	102.272
11	110.019
12	117.869
13	125.591
14	133.407
15	141.111
16	148.865
17	156.68
18	164.37b

^aData obtained from the U.S. Bureau of Mines, Bartlesville, Oklahoma.

^bThe value for C₁₈ was extrapolated to 300 °K from its melting point, 301.3 °K.

Figure 1. Differences between experimental liquid entropies, S_l, (open circles); S_l − R ln n (triangles); and S_l − R ln n − 5.6/n (closed circles), for consecutive n-paraffins at 300 °K versus average number of carbon atoms.

A more formal analysis of the entropy data is given in table 2 which shows coefficients and standard deviations resulting from least squares fits of the liquid entropy data for several functions. The lowest standard deviation was obtained using S_l = S_cl + S_el + R ln n + c/n, but the fit using S = S_cl + S_el + cR ln n was also quite satisfactory. The inclusion of a 1/n² term seems an unnecessary complication. But we are still not able to select unambiguously a functional form for S_l based on the data alone. In order to clarify this situation we have examined the statistical theory of polymer chains based on the liquid lattice model. This model allows one to calculate an expression for the entropy of a polymer chain in its liquid phase and leads not only to an unalterable R ln n term as pointed out by Flory and Vrij but also predicts terms in powers of 1/n (see appendix A).

Table 2.

Least square fits of the liquid n-paraffins entropies at 300 °K for several functions

Function	Standard deviation of function	Constants and their (standard deviations)
		Sc	Se	c	d
Sl = Scln+Sel,	(0.061)	7.78 (0.004)	24.4 (0.05)
Sl = Scln+Sel,+cR In n	(0.028)	7.71 (0.010)	23.4 (0.14)	0.38 (0.06)
Sl −R In n = Scln + Sel	(0.092)	7.59 (0.006)	21.9 (0.02)
$S_l-R\text{ln}n=S_{cl}n+S_{el}+c\left(\frac{1}{n}\right)$	(0.027)	7.65 (0.005)	20.7 (0.11)	5.61 (0.49)
$S_l-R\text{ln}n=S_{cl}n+S_{el}+c\left(\frac{1}{n}\right)+d{\left(\frac{1}{n}\right)}^2$	(0.028)	7.66 (0.016)	20.4 (0.50)	7.73 (4.61)	−6.04(13.0)

It seems most reasonable to conclude then that the correct expression for the liquid entropy of the n-paraffins as a function of n and T is

$$S_l(n,T)=S_{cl}(T)n+S_{el}(T)+R\text{ln}n+5.6/n$$ (2)

where terms in 1/n² have been ignored as has the temperature dependence of the relatively unimportant 1/n term.

3. Melting Equation

By expanding ΔG = ΔG_cn + ΔG_e − RT ln n − 5.6T/n about T = T₀ (see reference 2 for details) one finds,

$$n\text{Δ}H\frac{\text{Δ}T}{T_0}-n\frac{\text{Δ}{C}_p}{2T_0}{(\text{Δ}T)}^2-T_m\left[R\text{ln}n+5.6\left(\frac{1}{n}\right)\right]=T_m\text{Δ}{S}_e-\text{Δ}{H}_e+\frac{\text{Δ}{C}_{pe}}{2T_0}{(\text{Δ}T)}^2$$ (3)

which differs from Flory and Vrij’s eq (8) only in that it includes the (1/n) term and the ΔC_pe term which they chose to neglect. Direct calculations show that neglected terms remain below 1 percent of the magnitude of the leading terms in eq (3). Here ΔT = T₀ − T_m; − ΔH/T₀ and − ΔC_p/T₀ are the first and second temperature derivatives of ΔG_c at T = T₀; − ΔS_e and − ΔC_pe/T₀ are the first and second temperature derivatives of ΔG_e at T = T₀; and ΔH_e = ΔG_e + T₀ΔS_e. Rewriting (3) in the form of (1) one obtains,

$$T_m=T_0\frac{n+\frac{\text{Δ}{H}_e-T_0\text{Δ}{C}_{pe}/2}{\text{Δ}H-T_0\text{Δ}{C}_p/2}}{n+\frac{T_0}{\text{Δ}H-T_0\text{Δ}{C}_p/2}\left[\left(\text{Δ}{S}_e-\frac{\text{Δ}{C}_{pe}}{2}\right)+R\text{ln}n+5.6\left(\frac{1}{n}\right)-\left(\frac{\text{Δ}{C}_p}{2}n+\frac{\text{Δ}{C}_{pe}}{2}\right)\frac{\text{Δ}T}{T_0}\right]}.$$ (4)

This result is equivalent, except for the ln n and (1/n) terms, to eq (1.6) in reference [1]. Equation (4) can be abbreviated as

$$T_m=T_0\frac{n+a}{n+B(n)}$$

which can now be treated as a generalization of (1). One can calculate from experimental T_m values

$$B(n)=\frac{T_0}{T_m}(n+a)-n$$ (5)

and these B(n) values are shown in figure 2 for various values of T₀ and a. The quantity B(n) can also be expressed from equation (4) in the following form:

$$B(n)=\frac{T_0}{\text{Δ}H-T_0\text{Δ}{C}_p/2}\left[\left(\text{Δ}{S}_e-\frac{\text{Δ}{C}_{pe}}{2}\right)+R\text{ln}n+\frac{5.6}{n}-\left(\frac{\text{Δ}{C}_p}{2}n+\frac{\text{Δ}{C}_{pe}}{2}\right)\left(1-\frac{T_m}{T_0}\right)\right]$$ (6)

Figure 2. B(n), calculated using eq (5), as a function of n for different values of parameters T₀ and a.

The thermodynamic quantities in eq (6) were assigned values as follows. The molar entropies and enthalpies for the liquid and solid n-paraffins (see footnote 2) were plotted as a function of n. The slopes and intercepts of these curves equal respectively S_c(T), H_c(T) and S_e(T), H_e(T) and differences between the liquid and solid values gave ΔS_c(T), ΔH_c(T) and ΔS_e(T), ΔH_e(T) which were then extrapolated to T = T₀ to give ΔH, ΔS, ΔS_e, and ΔH_e. ΔC_p and ΔC_pe were taken as the slopes of the ΔH(T) and ΔH_e(T) curves at T = T₀. (A more detailed description of this procedure can be found in [3].) The results of these calculations are summarized in table 3. Numbers from table 3 were used to evaluate the various terms in eq (6) and these terms are shown in figure 3.

Table 3.

The values of several quantities at 418 °K determined from thermodynamic data on the n-paraffins

Quantity	Value	Estimated possible errora
H	1000	±25 cal/mol
ΔCp	0.5	±0.5 cal/mol tlep
ΔCe	−3400	±500 cal/mol
ΔHe	−3000	±500 cal/mol
ΔSe	1.0	±0.5 cal/mol deg
ΔCpe	−8.0	±4.0 cal/mol deg

^aThe estimates of error are intentionally pessimistic and serve only as a guide in examining the sensitivity of eq (4) to extreme variations of these quantities.

Figure 3. A. B(n), calculated from eq (6) using different values for the thermodynamic quantities. B. Values of the four variable terms in eq (6) and their sum using the values in table 3 and T₀ = 418 °K.

The curves are (1) b₀R In n (where b₀ = T₀/(ΔH − T₀ΔC_p/2) ≃ 0.47); (2) 5.6 b₀(1/n), (3) b₀(ΔC_p/2)n(ΔT/T₀), ΔC_p = 0.5; (4) b₀(ΔC_pe/2)(1 − T_m/T₀), ΔC_pe = − 8; (5) sum of curves 1–4, (6) same as (5) with b₀(ΔS_e − ΔC_pe/2) = 5 b₀ added, (7) same as (6) with ΔC_pe = − 12, (8) same as (6) with ΔC_p = 1.0, (9) same as (6) with ΔC_p = 2.0.

Figures 2 and 3 are useful for getting a feeling for how the various terms in B(n) affect T₀ and a. In figure 2 the change in shape of B(n) at small n is associated with a change in a, whereas a change in the slope of B(n) at large n is associated with a change in T₀. Figure 3 shows that the term which accounts mostly for the shape of B(n) at large n is the In n term and that the remaining three n-dependent terms in B(n) mostly affect the shape of B(n) at small n. The quantity (ΔS_e − ΔC_pe/2) is treated here as an adjustable parameter and is used to regulate the vertical position of B(n).

When B (n) was assumed constant as was done in [1], curve fitting of the melting equation gave a = − 1.5 and T₀ = 414.3 °K. These results can be anticipated from figure 2 which shows that straightest curve is the a = − 1.5 curve and the flattest curve would be for T₀ a little less than 415. Including a In n term and a partially compensating ΔC_p term in B (n) as was done by Flory and Vrij gives B (n) a shape similar to the a = − 3.0 curve and a large-n slope similar to the 419 °K curve. Thus one can anticipate their results of T₀ = 418.5 °K and a = − 2.7. In this present paper we are adding to B (n) the 1/n and ΔC_pe terms and from figure 3 we can anticipate that these will raise the small-n end of the curve and thus significantly reduce the magnitude of a but only slightly lower the large-n slope and thus only slightly reduce the T₀ value found by Flory and Vrij.

In other words, the In n term is the crucial one in B (n) as far as establishing T₀ is concerned, and the constants in the other terms can be varied within the generous range of uncertainties given in table 3 without appreciably affecting T₀. In the following section the results of fitting the melting temperatures to eq (4) will be shown to verify these anticipated results.

4. Fitting the Melting Data

Previously, eq (1) was fit to 14 values of T_m for the orthorhombic-liquid transition of the n-paraffins with 44 ⩽ n ⩽ 100 [1]. Flory and Vrij calculated 19 additional values for 11 ⩽ n < 44 from data on the enthalpies and temperatures of the orthorhombic-hexagonal and hexagonal-liquid transitions. The resulting 33 values are listed in their paper. The values for the five shortest paraffins were found to be smaller by 0.1, 0.3, 0.2, 0.2, and 0.1 °C respectively than those given by Flory and Vrij, based on quadratic extrapolations of experimental free energies.

These data were fit by least squares using an Omnitab program to eq (4) in the form:

$$T_m(n)=T_0\left(\frac{n}{n+B(n)}\right)+T_{0}a\left(\frac{1}{n+B(n)}\right)$$

where B(n) is given by eq (6).

Such a fit yields least squares estimates of T₀ and T₀a with the assumption that the bracketted terms are known functions of n and that all experimental uncertainty is contained in T_m(n). Thus all constants in B(n) had to be assigned values before fitting. To adapt to these restrictions we assigned values to ΔC_p, ΔC_pe and the factor T₀/(ΔH − T₀ΔC_p/2) from table 3 and then did the least squares fitting for each of a series of closely spaced values of (ΔS_e − ΔC_pe/2). The best value for (ΔS_e − ΔC_pe/2) was taken as that which gave a minimum standard deviation in T_m for fixed values of ΔC_p, ΔC_pe, and T₀/(ΔH − T₀ΔC_p/2). This method yields a pseudo-least-squares value for (ΔS_e − ΔC_pe/2), and the standard deviations were adjusted to account for this third constant. The other constants in B(n) could have been handled in the same way except that the fit was fairly insensitive to them. Instead, the values of ΔC_p, ΔC_pe, and T₀/(ΔH − T₀ΔC_p/2) were varied either together or singly within the range allowed by table 3 and the fit was repeated each time yielding new values for T₀, T₀a and (ΔS_e − ΔC_pe/2). In this way the effect of variations in the assigned constants on the values of the derived constants could be observed directly. To eliminate the difficulty of having the ratio ΔT/T₀ appear in B(n), calculated values for this ratio were fed back into B(n) until the calculated T₀ became constant. The results were quite insensitive to the initially assumed value of ΔT/T₀, and T₀ converged to a constant value to better than 4 figures after 2 iterations.

Using T₀/(ΔH − T₀ΔC_p/2) = 0.47, ΔC_p = 0.5 and ΔC_pe = − 8.0 gave best fit values of T₀ = 417.9 °K, a = − 1.5 and (ΔS_e − ΔC_pe/2) = 5. The values in table 3 give independently a = − 1.5 and (ΔS_e − ΔC_pe/2) = 5, which is encouraging agreement. The standard deviations for T_m and T₀ based on 33 minus 3 degrees of freedom were calculated to be 0.30 and 0.12 °K respectively.³ Changing the assumed values of ΔC_p, ΔC_pe, and T₀/(ΔH − T₀ΔC_p/2) within the limits allowed in table 3 did not alter the calculated value of T₀ by more than ± 0.5 °K. The results are as anticipated by the arguments of the previous section.

The results lead strongly to the conclusion that T₀ is very nearly equal to 417.9 °K. To apply an uncertainty based on three standard deviations (3 × 0.12 °K) is misleading since it does not take account of uncertainties in the functional form of (4), but it is difficult to see how T₀ could be altered by more than 0.5 °K by any reasonable means. Hence, the results essentially confirm those of Flory and Vrij with a slight but significant reduction in their estimate of T₀ = 418.5 ± 1.0 °K because of two relatively small terms which they neglected.

5. Discussion

The highest experimentally observed melting temperature for polyethylene is 138.7 °C, and for poly-methylene, 141.4 °C [6, 7]. Both of these were extended-chain pressure-crystallized specimens. Brown and Eby [8] used a form of eq (1) to extrapolate the melting temperatures of polyethlene to infinite molecular weight and found T₀ = 143.5 °C. By the same procedure Fujiwara and Yoshida [9] found T₀ = 144.8 °C. Weeks [10] used a plot of crystallization versus melting temperatures and extrapolated the data to the line T_c = T_m to find T₀ = 145.5 °C. Thus, a value of T₀ (paraffins) ≃ 145 °C is not out of line with extrapolated values for T₀ (polyethylene), but is significantly higher than experimentally observed melting points for extended chain polyethylene where there should be no effect from chain folded surfaces. An overall explanation of the melting data for polyethylene seems still to be lacking.

It is of interest to examine eq (3) in the limit of large n,

$$T_m(\text{ paraffins, }n\gg 1)=T_0\left(1+\frac{\text{Δ}{G}_e-R{T}_0\text{ln}n}{n\text{Δ}H}\right)$$ (7)

For a polymer, one would observe some contribution from the R ln n term if the chain ends were ordered in the crystal. Given a narrow length-fraction of polyethylene, one might find a high degree of chain end ordering in extended chain crystals, and very little chain end ordering in the folded chain crystals. Assume that ΔG_e is the same for any type of crystal and that a term RT₀ ln α n takes care of end group ordering effects where α has a value from 1/n (completely disordered) to 1 (completely ordered), depending on n, the distribution in length and the mode of crystallization. Assume further that ΔG_f is the free energy contributed to the crystal by a mole of folds in excess of that attributable to the associated CH₂-groups if in a nonfolded configuration, and that the folds, when present in the crystal, occur in regular planes separated by n_f carbon atoms. We can now modify eq (7) as follows,

$$T_m(\text{ polyethylene, }n\gg 1)=T_0\left[1+\frac{\text{Δ}{G}_e-T_{0}R\text{ln}\alpha n}{n\text{Δ}H}-\frac{\text{Δ}{G}_f}{n_f\text{Δ}H}\right]$$ (8)

in order to explicitly account for both chain ends and folds.

Equation (8) can be compared to the well-known equation,

$$T_m=T_0^{\prime }\left[1-\frac{2{\sigma }_e}{l\text{Δ}{h}_f}\right]$$ (9)

Here σ_e = − (C/2A)ΔG_f erg/cm², A = 18.5 × 10⁻¹⁶ cm² = the effective cross-sectional area of a CH₂ chain, C = 6.95 × 1O⁻¹⁷ ergs-moles/cal-molecule is a dimensional conversion constant, l = 1.27 n_f × 10⁻⁸ cm and Δh_f = (C/1.27 × 10⁻⁸ A) ΔH = 2.96 × 10⁹ ergs/cm³ is the heat of fusion of a CH₂ chain crystal.

One can now write (8) in the form of (9) to a good approximation by letting

$$T_0^{\prime }=T_0\left[1+\frac{\text{Δ}{G}_e-T_{0}R\text{ln}\alpha n}{n\text{Δ}H}\right]$$ (10)

where $T_0^{\prime }$ is the effective convergence temperature for a polymer containing end groups. $T_0^{\prime }$ was calculated using T₀ = 417.9, ΔG_e = − 3,400 cal/mole, ΔH =1000 cal/mole for the two limiting values of α₀, and the results are shown in figure 4, plotted as a function of the number of chain ends per 1000 carbon atoms.

Figure 4. Effective T₀ for polyethylene fractions as a function of the number of chain ends (calculated from eq (10)).

The α = 1 line represents the case of extended chain crystals with ordered methyl layers and the α = 1/n line applies to chain folded crystals where the chain ends are supposed to be randomly ordered in the crystal.

Notice that the apparent convergence temperature for our hypothetical polyethylene is lower when the chain ends are ordered (α = 1) than when the chain ends are disordered (α = 1/n). This result is of interest primarily because the experimental results mentioned at the beginning of this section give higher values for $T_0^{\prime }$ for folded crystals of polyethylene than the actual measured values for extended chain polyethylene. (The former would presumably exhibit more randomization of ends than the latter.) Also notice that the depression in $T_0^{\prime }$ is not insignificant for typical polyethylene where the weight average molecule has about 1000 carbons (2 ends per 1000 carbons) [7]. We shall not procede further to include the additional effects of length distribution and side branching but merely emphasize the significant point here that chain ends play an important part in calculating thermodynamic quantities for polyethylene and one can use the T₀ = 417.9 °K calculated in this paper only if one includes ΔG_e − RT₀ ln n explicity in polyethylene melting expressions like (8).

Finally, it should be mentioned that the increase in T₀ to 144.7 °C does not greatly alter the results in reference [3] where the thermodynamic properties of an infinite CH₂ chain crystal and liquid were obtained from paraffin data. Also the introduction of the R In n term does not affect those results since the nonlinearity, although not understood, was recognized and taken into account empirically in that paper.

6. Appendix A

In order to examine the theoretical form of the liquid-glass entropy difference, ΔS_lg, as predicted by the liquid lattice theory (for a general discussion of the theory see Miller [4]), it is convenient to start with eq 20 in reference [5].⁴ This equation includes the important effects of chain stiffness. Letting their z, the primary valance of the backbone chain atoms, equal 4 and writing ΔS_lg as a molar quantity, eq 20 [5] is,

$$\text{Δ}{S}_{lg}/R=n\text{ln}\left(V_0/S_0\right)+np\text{ln}\left(V_0/S_0^2\right)+\text{ln}3(n-1)+(n-3)A$$ (A.1)

where R = molar gas constant, n = number of carbon atoms in the chain, p = n₀/n n_x = fraction of vacant, n₀, to occupied, n n_x, lattice sites (n_x = number of n − mers), V₀ = n₀/(n n_x + n₀) = p/(1 + p) = volume fraction of vacant to total lattice sites, S₀ = 2n₀/[(n + 1)n_x + 2n₀] and

$$A=\ln [1+2\text{exp}(-\text{Δ}\beta )]+\frac{2\text{Δ}\beta \text{exp}(-\text{Δ}\beta )}{1+2\text{exp}(-\text{Δ}\beta )}$$

where Δβ = (ϵ₂ − ϵ₁)/RT and (ϵ₂ − ϵ₁) is the energy difference between the trans and gauche configurations for a carbon-carbon bond.

Ignoring p compared to 1 (the derivation was carried out without simplications and gave essentially the same results), writing V₀/S₀ = (1/2)(1 + 1/n) and dropping the second term in (A.1) gives,

$$\text{Δ}{S}_{lg}/R=n\text{ln}(1/2)(1+1/n)+\text{ln}3n(1+1/n)+(n-3)A=n(-\text{ln}2+A)+(\text{ln}3-3A)+\text{ln}n+(n+1)\text{ln}(1+1/n)$$

Expanding,

$$\text{ln}(1+1/n)=\left(1/n-1/2n^2+1/3n^3\dots \right)$$

and

$$\text{Δ}{S}_{lg}/R=n(-\text{ln}2+A)+(1+\text{ln}3-3A)+\text{ln}n+1/2(1/n)-(1/6)\left(1/n^2\right)$$ (A-2)

Thus we have in addition to the linear and constant terms an unalterable In n term and a power series in (1/n).

Of some further interest are the predicted magnitudes of the coefficients. Letting (ϵ₂ − ϵ₁) = 500 cal/mole, then A = 1.0 at 300 °K and we have,

$$\left(\text{Δ}{S}_{lg}/R\right)=0.3n-0.9+\text{ln}n+(1/2)(1/n)-(1/6)\left(1/n^2\right)$$

Experimentally, the liquid crystal entropy differences at 300 °K for the n-paraffins fit roughly,

$$\left(\text{Δ}{S}_{lc}/R\right)=1n+0.5+\text{ln}n+2.6(1/n)-3\left(1/n^2\right)$$

The agreement is not good but we can improve things somewhat by taking account of the extra volume of a CH₃ group compared to a CH₂ group. From the x-ray length of the orthorhombic phase l = (1.27 n + 2) × 10⁻⁸ cm indicating an effective CH₃ length of roughly twice a CH₂ length. Replacing n by n + 2 gives,

$$\left(\text{Δ}{S}_{lg}/R\right)=0.3n-0.3+\text{ln}n+2.5(1/n)-3.2\left(1/n^2\right)+\dots$$

Even the discrepancy in the constant term is now not too disturbing, and the numerical agreement lends some support to the adoption of the above form for ΔS_lc.