# Pippard Relations Close to the Melting Point in the Premelting Region of Hexadecane

Authors: Hamit Yurtseven; Özcan Tilki; Mustafa Kurt
Journal: Journal of Thermodynamics
 (2012)

---

## Abstract

We examine here the validity of the Pippard relations close to the melting point in the premelting region of hexadecane. For this verification, we analyze the observed data for the thermal expansivityαp
obtained for various pressures at constant temperatures of 302 and 325 K in this system. By calculating the isothermal compressibility κT and the specific heat Cp in the same pressure region of the premelting region of hexadecane, we obtain that Cp varies linearly with αp and also that αp varies linearly with κT for this system. This indicat es that some molecular organic compounds, such as solid hexadecane studied here, can exhibit λ-phase transitions prior to melting which are expected to verify the Pippard relations.


---

## Body

## 1. Introduction

Phase behaviour of pure n-alkanes is important for industrial and scientific applications. The controlled precipitation of solid phases of these mixtures is widely applied in pharmaceutical, cosmetic, and food industries as a separation process, as also pointed out previously [1]. Those paraffin waxes are also used in latent thermal energy storage applications, and additionally for cool storage applications in regard to the phase transition processes a binary mixture of tetradecane and hexadecane is widely used [2].In the literature, various sources of experimental data have been reported in regard to the thermodynamic properties of hexadecane. Some recent experimental reports are the following:T-X phase diagrams [1], entropy changes with molality at constant temperature [3], an anomalous heat effect during colloidal phase transitions [4], determination of the melting temperature (6.8°C) and a corresponding heat of fusion (121 kJ/kg) for a mixture of tetradecane (40%) and hexadecane (60%) [2], and also the crystallization temperatures of two binary mixtures: the tetradecane + hexadecane and tetradecane + pentadecane system up to 100 MPa [1].Solid hexadecane among long-chain paraffin solids exhibits considerable precursor effects [5]. It has been shown that there exists an orientational disorder around the long axis of the molecule in solid hexadecane [6]. Experimentally, it has been demonstrated that this is a kinked chain-like disorder rather than a librational disorder [7, 8], as also indicated by Pruzan et al. [9]. This orientational disorder occurring in solid hexadecane gives rise to instability and melting of the solid phase. Instability and melting have been associated with a soft phonon of zero wave vector [10–13].Solid hexadecane melts into the liquid phase as a second-order transition associated with a particular type of orientational disorder which depends on the symmetry [9]. As in many molecular organic compounds, solid hexadecane exhibits orientational disorder in a small range where the thermodynamic quantities such as the thermal expansivity αp, isothermal compressibility κT, and the specific heat Cp show anomalous behaviour close to the melting point [5, 9].The anomalous behaviour ofαp, κT, and Cp can be described by a power-law formula near the melting point in solid hexadecane. In the premelting region of hexadecane close to the melting point, we can assume that αp, κT, and Cp exhibit similar critical behaviour associated with the same critical exponent. Buckingham and Fairbank [14] have indicated that Cp behaves like κT and αp in the vicinity of the critical point. In the precursor zone of the solid hexadecane, Cp, αp, and κT are expected to increase as approaching the melting point. This increase in the thermodynamic quantities considered is similar to the critical behaviour of λ-phase transitions. Pippard has established a linear variation of the specific heat Cp with the thermal expansivity αp in the vicinity of the λ-phase transition in NH4Cl [15]. The Pippard relations have been tested for a variety of materials exhibiting λ-phase transitions.In this study, we also examine the Pippard relations in the premelting region of hexadecane near the melting point. For this examination, we calculate the isothermal compressibilityκT and the specific heat Cp from our analysis of the thermal expansivity αp which was measured experimentally at various pressures near the melting point for constant temperatures of 302 and 325 K in solid hexadecane [9]. By obtaining the values of the critical exponent γ from our analysis, we calculate κT and Cp as functions of pressure in the premelting region of hexadecane. We then establish a linear variation of Cp with αp (first Pippard relation) and also a linear variation of αp with κT (second Pippard relation) close to the melting point in this molecular organic compound.In Section2, we review the method and conditions about experimental study of Pruzan et al. [9]. In Section 3, our calculations and results are presented. In Sections 4 and 5, we present our discussion and conclusions, respectively.


## 2. Review of the Experimental Technique

Using aP-V-T method, the thermal expansivity αp was observed by Pruzan et al. [9] at various pressures close to the melting point. Using a high-pressure calorimetric method, the heat of compression was measured. The measurement of heat, in open-system conditions, was conducted, which flows out of (into) the sample when it is compressed (decompressed) isothermally [9]. The thermal expansivity αp was then determined according to the first Maxwell equation:
(1)αp=-V-1(∂S∂T)P,
where V and S are the molar volume and entropy of the sample, respectively. This technique was referred to as the piezothermal method [9]. For the measurements, the pressure steps were about 20 MPa far from the melting point and they were 0.1–10 MPa close to the melting point [9]. For their measurements, Pruzan et al. [9] reported that their observations were made over different times, sometimes greater than 72 h, during which no change could be observed in the expansivity variation against pressure for hexadecane.


## 3. Calculations and Results

Near the melting point of solid hexadecane, the thermodynamic quantities such as the thermal expansivityαp, isothermal compressibility κT, and the specific heat Cp can exhibit critical behaviour. This critical behaviour of the thermal expansivity can be expressed as
(2)αp=A(P-Pm)-γ,
where γ is the critical exponent for αp and A is the amplitude. Here, Pm denotes the melting pressure of solid hexadecane, which is close to the critical point [5]. Using the thermodynamic relation near the melting point:
(3)αpκT=(dPmdT),
the isothermal compressibility κT can be expressed as a function of pressure
(4)κT=A(P-Pm)-γ(dPm/dT)
by means of (2).We can also obtain the pressure dependence of the specific heatCp using the thermodynamic relation near the melting point as follows:
(5)Cp=TVαp(dPmdT).
In (5), the volume V is a function of pressure, which can be obtained as
(6)VT(P)=Vmexp[A1-γ1(dPm/dT)(P-Pm)1-γ],
where Vm denotes the melting volume of solid hexadecane. Equation (6) can be derived easily using the definition of the isothermal compressibility κT=-(1/V)/(∂V/∂P)T. Thus, the pressure dependence of the molar volume VT(P) has been derived here, as given by (6), in the case when the isothermal compressibility κT is a function of pressure according to (4). We note here that when the isothermal compressibility κT is a function of the molar volume V, the pressure dependence of the molar volume VT(P) can also be derived. But, its functional form should not be the same as (6) since the volume dependence of the isothermal compressibility κT changes the pressure dependence of the molar volume in the premelting region close to the melting point of solid hexadecane. Finally, by means of (5), the pressure dependence of the specific heat Cp can be written as a power-law formula:
(7)Cp=ATV(dPmdT)(P-Pm)-γ.In the vicinity of the melting point of solid hexadecane, the specific heatCp can be related to the thermal expansivity αp linearly, according to the first Pippard relation:
(8)Cp=TV(dPmdT)αp+T(dSdT)m.
Similarly, the thermal expansivity αp can be related linearly with the isothermal compressibility κT by the relation in the vicinity of the melting point:
(9)αp=(dPmdT)κT+1V(dVdT)m
according to the second Pippard relation. Thus, linear plots of Cp against αp (8) and αp against κT (9) can be obtained near the melting point in solid hexadecane. From those linear plots, (8) and (9), we can extract the values of the slope dPm/dT numerically for solid hexadecane close to the melting point.We reanalyzed here the thermal expansivityαp measured at various pressures in the premelting region close to the melting point for constant temperatures of 302 and 325 K in solid hexadecane due to Pruzan et al. who also analyzed their experimental data [9]. For our analysis, we followed the steps, as given below.(1)
We used the power-law formula (2) in the logarithmic form obtained as
(10)lnαp=lnA-γln(P-Pm)
for the premelting region of hexadecane. We then extracted the values of the critical exponent γ as 2.06±0.05 and 1.97±0.04 with the values of the amplitude A for constant temperatures of 302 and 325 K, respectively, as given in Table 1. Similarly, from their analysis of the thermal expansivity αp, Pruzan et al. [9] have obtained the value of the critical exponent as γm=1.9 in the melting zone of hexadecane. Figure 1 gives the thermal expansion αp plotted as a function of P-PL in a log-log scale for constant temperatures of 302 K and 325 K. Since we used for our analysis the experimental data in the melting zone, which was obtained in the range P-PL<50 MPa where PL is the liquid or melting (Pm) pressure [9], we were able to describe the critical behaviour of the thermal expansivity αp in the premelting region of hexadecane according to (10). On the other hand, P-PL>50 MPa corresponds to the precursor zone (solid phase) for hexadecane [9]. These pressure ranges have also been reported in the work of Pruzan et al. [9] as P1-PL=140 MPa and P2-PL=60 MPa, where P1 represents the beginning of the precursor range and P2 is the pressure for a change of regime, which is closer to the critical pressure PC for the solid hexadecane.(2)
Using those values ofγ and A which we obtained (Table 1), we then calculated the isothermal compressibility κT as a function of pressure according to (4) where we used the experimental value of dPm/dT=10 MPa/K [9].(3)
The next step was to calculate the pressure dependence of the volumeVT(P) in the premelting region of hexadecane near the melting point for constant temperatures of 302 and 325 K using (6). In this equation, the values of the melting volume Vm [16], as given in Table 1, were used with the experimental value of dPm/dT=10 MPa/K.(4)
Finally, the pressure dependence of the specific heatCp was calculated by (7) for the premelting region of hexadecane near the melting point for those constant temperatures considered using VT(P) values (6) with the dPm/dT value [5] given above.Table 1
Values ofγ and A extracted (Figure 1) for αp using the experimental data [9] and the Vm values for the solid hexadecane at temperatures indicated.
T (K)
γ
A
(
MPa
)
γ
/K
V
m
×
10
-
6 (m3/mol)
302
2.06
±
0.05
1.9837
294.93 at 303 K
325
1.97
±
0.04
1.3624
300.13 at 323 KFigure 1
α
p versus P in a log-log scale close to Pm in the premelting region of hexadecane for T=302 and 325 K (10) when Pm=PL [9]. The experimental data [9] with the uncertainties extracted (error bars) are also given here.Since we obtained the pressure dependence of the thermal expansivityαp (2), isothermal compressibility κT (4), and the specific heat Cp (7) in the same pressure interval for constant temperatures of 302 and 325 K in the premelting  region of hexadecane, we were then able to establish the Pippard relations ((8) and (9)) in this crystalline system. We plot the specific heat Cp against the thermal expansivity αp in the premelting region of hexadecane near the melting point for constant temperatures of 302 K and 325 K in Figure 2.Figure 2
C
p versus αp at various P close to Pm for T=302 and 325 K in the premelting region of hexadecane (8). Observed αp values [9] are shown here.We also examined the second Pippard relation (9) by αp as a function of κT in the premelting region of hexadecane close to the melting point. Figure 3 gives our plot for constant temperatures of 302 and 325 K. From our plots of Cp against αp (first Pippard relation) and αp against κT (second Pippard relation), we extracted the value of the slope as dPm/dT=10 MPa/K. Since the experimental value of dPm/dT=10 MPa/K [5] was used as the initial value in (4) (step 2), (6) (step 3), and (7) (step 4) to calculate κT, VT(P), and CP, respectively, as a function of pressure, we should expect the same slope value of 10 MPa/K from (8) as well.Figure 3
α
p versus κT close to the premelting region of hexadecane at various P for Pm, T=302 and 325 K (9). Observed αp values [9] are shown here.


## 4. Discussion

We examined here whether the specific heatCp varies linearly with the thermal expansivity αp (first Pippard relation) and also αp varies linearly with the isothermal compressibility κT (second Pippard relation) in the vicinity of the melting point in the premelting region of hexadecane. As seen from our plots (Figures 2 and 3), we established the Pippard relations ((8) and (9)) with the slope dPm/dT=10 MPa/K which was extracted for this system. This indicates that those thermodynamic quantities, namely, Cp, αp, and κT exhibit similar critical behaviour close to the melting point in the premelting region of hexadecane.From our analysis of the measured thermal expansivity, we extracted the value ofγ=2 as the critical exponent for constant temperatures (302 K and 325 K), in the premelting region of hexadecane (Table 1). Our exponent value is very close to the value with the uncertainties (Table 1) of γ=1.9 as obtained by Pruzan et al. [9]. They analyzed their experimental data according to the power-law relation (2) with the Pm=PC. The critical pressure PC at which αp diverges due to the precursor transformation is always between P2 and PL [9]. For hexadecane, PC is close to P2, as indicated previously. Within the pressure ranges P1-PL=140 MPa and P2-PL=60 MPa, Pruzan et al. [9] obtained the values of γS=0.5 in the solid phase, γP=0.7 in the precursor zone, and γm=1.9 in the melting zone, where the pressure range is greater than the melting pressure Pm. The exponent value of 1.9 or 2 from both analyses in the melting zone can be compared with an effective exponent greater than 1 for the critical behaviour of the thermal expansivity αp due to surface melting model of Lipowsky and Speth [17, 18], as also pointed out previously [9]. It has been interpreted that the second regime (liquid phase) observed in molecular organic compounds might be ascribed to impurity effects for the critical exponents close to 2 [5]. Regarding the exponent value of 1.9 or 2, the melting zone can be shortened as the impurity decreases [19] close to the melting point in the premelting region of hexadecane. Our values of the critical exponent (γm=2) for hexadecane can also be compared with those values for other organic compounds, which we obtained by Pruzan et al. [9]. Close to the melting point, they obtained the values of γm=2 (cyclohexane), 1.8 (carbon tetrachloride), 1.6 (benzene), and 1.5 (butanediol), as given in Table 2 in [5]. Similarly, the γm values of 1.85 for hexamethylbenzene (HMB), 2 for paradichlorobenzene (PDCB), and the value of 1.7 for both paraozoxydianisol (PAA, C–N transition) and heptyloxyazoxybenzene (HOAOB, C–S transition) compounds (Table 2 in [9]) can also be compared with our exponent value of γm=2 for hexadecane or γm=1.9 as obtained by Pruzan et al. [9]. The organic compounds indicated above have similar chemical structure (C–H–⋯ bonds) within the close melting points and pressure ranges [9]. This shows that within the pressure ranges studied all those organic compounds exhibit similar divergence behaviour of the thermal expansion αp near the melting point according to a power-law formula (2). Thus, our exponent value is within expectations in regard to the experimental results for various organic compounds stated above.Ourγ values (Table 1) were used to calculate here the pressure dependence of volume VT(P) (6), isothermal compressibility κT (4), and the specific heat Cp (7), as indicated above. This calculation of Cp and κT then led us to plot Cp against αp (Figure 2) and αp against κT (Figure 3) with the zero values of the intercepts T(dS/dT)m and (dV/dT)m/V, respectively, at the melting point in the premelting region of hexadecane. Those intercepts would have nonzero values according to the Pippard relations ((8) and (9)) if the temperature dependences of the entropy and volume are known experimentally or their functional forms can be derived for solid hexadecane. Values of the critical exponent γ (Table 1) and also the slope dP/dT were extracted from one source of experimental data [9] for hexadecane in the melting region. Regarding the anomalous behavior of αp, κT, and Cp close to the melting point in the premelting region of hexadecane, the other sources of the experimental data rather than the measurements of the thermal expansivity αp due to Pruzan et al. [9] can be used to examine our relations given here. Some new experimental measurements of the αp, κT, and Cp at various pressures for constant temperatures can be performed in the premelting region near the melting point for hexadecane. When this experimental data is available in the literature which can be obtained to high accuracy, our analysis can then be reconsidered to examine the Pippard relations given here.Regarding the uncertainties in the measurements of the thermal expansivityαp for hexadecane, it was pointed out [9] that the uncertainty in the αp data depends on the respective uncertainties in the heat of compression (1 or 2%) and in the pressure step. It was also pointed out [9] that far from the transition lines, pressure steps around 20 MPa caused an uncertainty in αp of about 5%. Close to the melting lines, within the pressure steps of 0.1–10 MPa, the uncertainty in αp increased, whereas for the measurements of the molar volume V the uncertainty was considerably small (~0.3%) [9].Using the experimental data [9], we extracted that the uncertainties in αp are about 5% close to the melting point, which may be higher than this value, as pointed out by Pruzan et al. [9]. We represented the error bars in Figure 1 for T=302 K and 325 K, as stated above. This gave us the uncertainties in the critical exponent γ for the thermal expansivity αp, which changes from 1.9 to 2.1 (Figure 1).On the basis of the volume measured or calculated as a function of temperature in the premelting region of hexadecane, the temperature dependence of the thermal expansivityαp, isothermal compressibility κT, and the specific heat Cp can be obtained, and the Pippard relations ((8) and (9)) can also be established at various temperatures near the melting point for constant pressures in this system.


## 5. Conclusions

We studied here the Pippard relations (Cp versus αp and αp versus κT) close to the melting point in the premelting region of hexadecane. Linear variations of Cp with αp (first Pippard relation) and αp with κT (second Pippard relation) were both obtained at various pressures for constant temperatures of 302 K and 325 K in this system.The Pippard relations can be also be applied to the solid hexadecane in the premelting region using the temperature dependence ofCP, αP, and κT for constant pressures. They can be verified for some other molecular organic compounds near the melting point when the experimental data for the thermodynamics quantities is available.



---

*Source: 892913-2012-11-13.xml*