# Application of the Pippard Relations to Cyclohexane near the Melting Point

Authors: Hamit Yurtseven
Journal: ISRN Thermodynamics
 (2012)

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## Abstract

The Pippard relations are verified near the melting point for cyclohexane. The experimental data for the thermal expansivity is analyzed according to a power-law formula using the critical exponent and the thermodynamic quantities are calculated close to the melting point for this molecular organic compound. This applies to those compounds showing a second order transition prior to melting.


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## Body

## 1. Introduction

Various properties of cyclohexane have been studied extensively in the literature. Using different experimental techniques, its thermodynamics [1], dynamic [2–8], and structural [9–11] properties have been investigated.Cyclohexane undergoes the sequence of phase changes at ambient temperature under high pressures. From liquid to phase I, cyclohexane solidifies as the pressure increases. At 0.5 GPa there occurs I to III transition which follows III to IV at 1.3 GPa, as observed experimentally using X-ray diffraction [9], infrared spectroscopy [3–8], Raman spectroscopy [11–13]. Below 186.1 K, an orientationally ordered structure, namely, phase II, occurs [11]. It has also been indicated that between the liquid phase (T>279.8 K) and the monoclinic solid phase (T<186 K), cyclohexane forms an orientationally disordered cubic phase [14]. In a recent study using X-ray diffraction at room temperature for high pressures to 40 GPa [15], a different sequence of phase transitions (i) between 4.14 and 6.91 GPa (phase IV to V), (ii) between 6.91 and 12.3 GPa (phase V to VI), and (iii) above 15.1 GPa (phase VI to VII) have been investigated.Regarding the structures of the phases that occur in cyclohexane, plastic phase I is cubic, phase III is orthorhombic, and phase IV is monoclinic. It has been reported that phase V retains structural similarities with the monoclinic phase IV [16], as also pointed out previously [12].It has been observed experimentally that IV-V transition which takes place at 3.2 GPa is accompanied by a large hysteresis [12]. These hysteresis effects have also been observed in cyclohexane [8, 13]. Near the melting point, the thermal expansivity exhibits anomaly at various pressures for a constant temperature (T=299.7 K), as observed experimentally in cyclohexane [17]. The differential thermal analysis (DTA) measurement has also revealed that cyclohexane confined in metal-organic framework exhibits thermal anomaly at 121, 213, and 306 K [18].On the basis of the anomalous behaviour of the thermal expansivityαp, as observed experimentally [17], we examine in this study the Pippard relations which relate the specific heat Cp to the thermal expansivity αp (Cp versus αp) and αp to the isothermal compressibility κT (αp versus κT) for cyclohexane close to the melting point. As cyclohexane melts from the solid phase to the liquid when the temperature increases or pressure decreases, the Pippard relations are constructed for this molecular organic compound by analyzing the experimental data for thermal expansivity αp at various pressures near the melting point [17] according to a power-law formula. We then calculate the temperature dependence of the thermodynamic quantities, namely, αp, κT, and Cp. By plotting Cp versus αp and αp versus κT for the same temperature interval, we obtain linear relations close to the melting point for cyclohexane.


## 2. Theory

In order to obtain correlations between the specific heatCp and the thermal expansivity αp (first Pippard relation), and also correlations between the αp and isothermal compressibility κT (second Pippard relation), we derive the temperature and pressure dependencies of those thermodynamic quantities. We start with a power-law formula for the pressure dependence of the thermal expansivity αp expressed as [17]
(1)αp=A(P-Pm)-γ,
where γ is the critical exponent for αp, A is the amplitude, and Pm is melting pressure. Close to the melting point, one can approximate the rate of change of pressure with the temperature as
(2)P-Pm(T)Tm(P)-T=dPmdT,
where the melting pressure Pm and the melting temperature Tm depend upon the temperature and pressure, respectively, along the P-T melting curve. Thus, from (2) the temperature dependence of the thermal expansivity αp can be expressed as
(3)αp=A(dPmdT)-γ(Tm-T)-γ
according to (1). Using the thermodynamic relation
(4)αpκT=dPmdT
close to the melting point, the pressure and temperature dependence of the isothermal compressibility κT will be obtained as
(5)κT=A(P-Pm)-γ(dPm/dT),(6)κT=A(dPm/dT)-γ-1(Tm-T)-γ,
respectively. By means of the thermodynamic relation
(7)(∂P∂T)s=Cp(TVαp)
and using the approximate relation
(8)(∂P∂T)s=dPmdT=(∂P∂T)V
near the melting point [17], the pressure and temperature dependence of specific heat Cp can be obtained as
(9)Cp=ATV(dPmdT)(P-Pm)-γ,(10)Cp=ATV(dPmdT)-γ+1(Tm-T)-γ,
respectively.From the pressure and temperature dependence of the specific heatCp (see (9) and (10)), the thermal expansivity αp (see (1) and (3)), and the isothermal compressibility κT (see (5) and (6)), the first and second Pippard relations can be established, respectively, as given below:
(11)Cp=TV(dPmdT)αp+T(dSdT)m,(12)αp=(dPmdT)κT+(1V)(dVdT)m,
where the entropy change with the temperature, (dS/dT)m, and the volume change with the temperature, (dV/dT)m, are taken at the melting point.


## 3. Calculations and Results

The Pippard relations given above (see (11) and (12)) were established in this study by analyzing the experimental data [17] for the thermal expansivity αp of cyclohexane at various pressures for T=299.7 K close to the melting pressure. This analysis of thermal expansivity αp was performed according to the power-law formula (1) in the solid phase (P>Pm) prior to the melting. By taking the logarithm of (1), we get
(13)lnαp=lnA-γln(P-Pm).
From the analysis of the experimental data, the values we obtained are tabulated in Table 1. Using the values of the critical exponent γ, the amplitude A (Table 1), and the experimental value of dPm/dT=5 MPa/K [17], the temperature dependence of the thermal expansivity αp (3), the isothermal compressibility κT (6), and the specific heat Cp (10) were calculated. The temperature dependence of the specific heat per unit volume (Cp/V) is plotted against that dependence of the thermal expansivity αp near the melting point of cyclohexane according to the first Pippard relation (11) in Figure 1. A plot of the temperature dependence of the thermal expansivity αp is given in Figure 2 as a function of the isothermal compressibility κT near the melting point according to the second Pippard relation (12) for cyclohexane.Table 1
Values of the critical exponentγ for thermal expansivity αp and the amplitude A for cyclohexane according to a power-law formula (1). The experimental (dPm/dT) value [17] is also given here.
γ
A
[
K
-
1
(
GPa
)
γ
]
Pressure interval (GPa)
d
P
m
/
d
T (MPa)
1.3
1.764
0.023
<
(
P
-
P
m
)
<
0.143
5Figure 1
The specific heat (Cp/V) as a function of the thermal expansivity αp according to the first Pippard relation (11) for cyclohexane near the melting point.Figure 2
The thermal expansivityαp as a function of the isothermal compressibility κT according to the second Pippard relation (12) for cyclohexane near the melting point.


## 4. Discussion

The Pippard relations were constructed in this study for cyclohexane close to the melting point according to (11) and (12), as shown in Figures 1 and 2, respectively. For these plots (Cp/V versus αp and αp versus κT), the experimental data [17] for the thermal expansivity αp was analyzed using a power-law formula (1) with the critical exponent value of γ=1.3 within the pressure interval 0.023<P-Pm<0.143 (GPa) (Table 1) close to the melting point in cyclohexane. This analysis of the thermal expansivity was also performed previously by Pruzan et al. [17] using a power-law formula (1). They obtained values of the critical exponent γ for thermal expansivity as follows: γs=0.2 (solid phase), γp=0.7 (precursor zone), and γm=2 (melting zone). They described the pressure ranges on the basis of their experimental measurements as follows: P1 beginning of the precursor zone, P2 change of regime, and PL end of melting, with the difference in pressure as P1-PL=35 MPa and P2-PL=15 MPa [17]. Thus, the experimental αp versus P plot gives that a straight line in the solid phase changes to a curve at P1 followed by a reverse λ-type peak at P2 [17]. Below P2 as the pressure decreases down to the liquid pressure PL, this peak disappears and the homogenous liquid phase appears [17]. So, the solid-liquid transformation occurs in the melting zone (P2-PL) by taking PL very close to the melting pressure Pm and from the liquid state, the crystalization takes place at around P2 in cyclohexane [17].From our analysis given here, the exponent value (γ=1.3) lies on the pressure range between precursor and melting zones, which is a small pressure range in comparison with that of the solid phase. In fact, the exponent value of γs=0.2 describes a second order transition in the solid phase as obtained by Pruzan et al. [17], which is followed by the melting process. The exponent values of γp=0.7 (precursor zone) and γm=2 (melting zone) due to Pruzan at al. [17] or our value of γ=1.3 describe this melting process which occurs below the pressure P2 where instability takes place and the solid state is transformed into the homogenous liquid state.Using our value ofγ=1.3 and the experimental value of the slope dPm/dT=5 MPa/K [17], the temperature dependence of the thermal expansivity αp, specific heat Cp, and the isothermal compressibility κT were calculated, which gave us as straight lines when plotted Cp/V versus αp (Figure 1) and αp versus κT (Figure 2) near the melting point in cyclohexane. Since we used the experimental dPm/dT value to calculate αp (3), κT (6), and Cp (10) as functions of temperature, the same slope value was deduced from our plots (Figures 1 and 2), as expected. This linear relationship between the specific heat Cp and the thermal expansivity αp (Figure 1) expresses the validity of the first Pippard relation (11) close to the melting point in cyclohexane. Also, the second Pippard relation (12) was verified by a linear variation of αp with the isothermal compressibility κT, as shown in Figure 2 near the melting point in this compound.


## 5. Conclusions

Linear variations of the specific heatCp with the thermal expansivity αp, and also αp with the isothermal compressibility κT were established close to the melting point in cyclohexane. This was obtained by analyzing the thermal expansivity data from the literature near the melting point in this molecular organic compound. From a power-law analysis, Cp, αp, and κT were calculated as functions of temperature close to the melting point, and the Pippard relations were verified.This shows that the thermodynamic quantities can be calculated and the Pippard relations can be constructed close to the melting point for those molecular organic compounds which exhibit a second order phase transformation prior to melting.



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*Source: 701209-2012-11-19.xml*