Pippard Relations Research Articles

Calculation of the thermodynamic functions from the Raman frequency shifts close to the 𝜀–δloc–δ transitions and Pippard relations in nitrogen

Thermodynamic functions of the thermal expansion [Formula: see text], isothermal compressibility [Formula: see text] and the difference in the heat capacity [Formula: see text] are calculated as a function of temperature ([Formula: see text] GPa) close to the transitions of [Formula: see text][Formula: see text]–[Formula: see text][Formula: see text] and [Formula: see text][Formula: see text]–[Formula: see text][Formula: see text] in the solid nitrogen. This calculation is performed by using the observed Raman frequency shifts of vibrons [Formula: see text] and [Formula: see text]. Also, by using the observed [Formula: see text]–[Formula: see text] data, those thermodynamic functions are predicted at various pressures for the fluid–solid transition in nitrogen. For both calculations, observed data are used from the literature. From the temperature and pressure dependences of the thermodynamic functions studied, the Pippard relations are examined close to the [Formula: see text][Formula: see text]–[Formula: see text][Formula: see text][Formula: see text]–[Formula: see text][Formula: see text] transitions and also fluid–solid transition in nitrogen.We find that the thermodynamic functions can be predicted from the Raman frequency shifts and that the Pippard relations can be established for both the [Formula: see text][Formula: see text]–[Formula: see text][Formula: see text][Formula: see text]–[Formula: see text][Formula: see text] and fluid–solid transitions in nitrogen. This method of predicting the thermodynamic functions can also be applied to some other molecular solids.

Resonant frequency shifts related to the elastic constants near the α-β transition in quartz

The resonant frequency shifts of vibrational modes are related to the elastic constants at various temperatures for the α-β transition in quartz (Tc = 573.0 °C for α-quartz and 574.3 °C for β-quartz). This correlation is constructed on the basis of the spectroscopic modifications of the Pippard relations near the α-β transition in quartz by using the observed data from the literature.Our results indicate that the elastic properties can be described from the spectroscopic parameters (frequency shifts) as we have studied here for the α-β transition in quartz. Using those relations, elastic constants can also be obtained from the frequency shifts near the phase transitions of various crystalline systems.

Thermodynamic Parameters of Cholesteric/Smectic A Transition in Cholesteric Myristate and Its Binary Mixture CM/PCPB

Thermodynamic properties of the cholesteryl myristate (CM) and its binary mixture CM/PCPB (p-pentylphenyl-2-chloro-4(p-pentylbenzoyl)-benzoate) are studied at the concentrations of xPCPB = 0.052 and 0.219 as a function of temperature near the cholosteric/smectic A transition. By analyzing the observed molar volume from the literature, the temperature dependences of the thermal expansion, isothermal compressibility and the difference in the specific heat are calculated and, the Pippard relations are established for those compounds close to the cholesteric/smectic A transition. Predictions of the thermodynamic quantities and the Pippard relations can be examined by the experimental measurements of the CM and its binary mixture of CM/PCPB close to the cholesteric/smectic A transition.

Calculation of the thermodynamic quantities from the Raman frequency shifts and the Pippard relations for the [formula omitted], [formula omitted] and [formula omitted] phases of oxygen

Thermodynamic quantities such as the isothermal compressibility, thermal expansion and the difference in the specific heat are calculated as a function of pressure (T = 10 K) for the α, α- and α' phases of oxygen using the observed Raman frequencies of the two librons and one vibron from the literature through the mode Grüneisen parameter for this molecular solid. Calculated thermodynamic quantities decrease with increasing pressure with some discrepancies due to the different pressure – induced Raman frequency shifts of these modes studied for the α, α- and α' phases of O2. A linear variation of the difference in the specific heat with the thermal expansion and also a linear variation of the thermal expansion with the isothermal compressibility (Pippard relations) are obtained for the α, α- and α' phases of oxygen as studied here.Calculated isothermal compressibility, thermal expansion and the difference in the specific heat with the Pippard relations which we obtained here at various pressures (T = 10 K), can be examined with the experimental measurements for the α, α- and α' phases of oxygen molecule.

Pippard relations for cubic gauche nitrogen

The Pippard relations (C P vs. a P and a P vs. k T ) are examined at various temperatures up to 1200 K at zero pressure (P = 0) for the cubic gauche nitrogen. The specific heat (C P ) is related to the thermal expansion ( a P ) and the a P is also related to the isothermal compressibility ( k T ) as the Pippard relations for this solid structure. For those relations, experimental data are used from the literature. It is found that the variation of the C P with the a P and a P with the k T is linear. Using the linear relations dP/dT is predicted, which can be compared with the experimental measurements in the P – T phase diagram of cubic gauche nitrogen.

Pressure effect on the cholesteric – smectic A transition in cholestryl myristate

ABSTRACTCholesteryl myristate exhibits tricritical behavior under high pressures close to its cholesteric-smectic A (Ch/SA) transition, as observed experimentally. We study here the thermodynamic properties of this liquid crystal by calculating the pressure dependence of the isothermal compressibility, thermal expansion and the specific heat close to the Ch/SA transition. This calculation is performed at constant temperatures (80.8 and 105.0 °C) using the observed volume-pressure (V-P) data from the literature.We find that the thermal expansion and the isothermal compressibility increase linearly, whereas the specific heat exhibits anomalous behaviour with increasing pressure close to the Ch/SA transition in cholesteryl myristate. From the pressure dependence of the thermodynamic quantities studied, the Pippard relations are examined and their linear variations are obtained at various pressures at constant temperatures of 80.8 and 105.0°C for this liquid crystal close to the Ch/SA transition. Our results also indicate that the tricritical point (TCP) is located at higher pressure (about 1 kbar) for the Ch/SA transition in cholesteryl myristate, as suggested previously.

Molar volume calculated at various pressures and the Pippard relations close to the melting point in benzene

ABSTRACTThe molar volume of solid and liquid benzene was calculated at various pressures (at constant temperatures), and the Pippard relations were examined close to the melting point in this organic molecule.The molar volume calculated is in good agreement with the observed data, which decreases as the pressure increases up to about 150 MPa. The Pippard relations are also valid within this pressure range at constant temperatures studied here for the solid and liquid phases of benzene.

α-β Transition in Quartz: Temperature and Pressure Dependence of the Thermodynamic Quantities for β-Quartz and β-Cristobalite as Piezoelectric Materials

Temperature and pressure dependencies of the thermal expansivity (αp), isothermal compressibility (κT) and the specific heat (Cp − Cv) are studied for piezoelectric materials, in particular, for β-quartz. By analyzing the temperature (at 1 atm) and pressure (at 848 K) dependence of the observed volume V from the literature, the thermodynamic functions (αp, κT and Cp − Cv) are obtained and the Pippard relations (Cp − Cv vs. Vαp and αp vs. κT) close to the transition from the β-quartz to the β-cristobalite are examined.

Thermodynamic quantities and Pippard relations close to the solid I–II transition in benzene

Thermodynamic quantities and Pippard relations close to the solid I–II transition in benzene

Pressure Dependence of the Thermodynamic Quantities in Phase II of Solid Benzene

AbstractThe thermodynamic quantities such as the thermal expansion (αp), isothermal compressibility (KT) and the specific heat (CP − CV), are predicted at various pressures up to 26 GPa (T = 540 K) in the solid phase II of benzene using volume data from the literature. The Pippard relations are examined using the pressure dependence of αp, KT and CP − CV and the value of the slope dP/dT is deduced for the solid phase II of benzene. The thermodynamic quantities studied here decrease with increasing pressure, as expected, which can be compared with the experimental measurements for the solid phase II of benzene. We find that the Pippard relations are validated within the pressure range considered, in particular, at high pressures for the solid phase II of this molecular crystal.

Analysis of the Raman Frequency Shifts for the Lattice Modes and Vibrons Related to the Thermodynamic Quantities in the η Phase of Solid Nitrogen

AbstractThe thermodynamic quantities of the isothermal compressibility, thermal expansion and the specific heat are calculated here as a function of pressure by using the observed Raman frequencies of the lattice modes and vibrons in the η phase of solid nitrogen. The Pippard relations and their spectroscopic modifications are constructed, and the slope dP/dT is deduced from the Raman frequency shifts in this phase of N2. It is shown that the thermodynamic quantities can be predicted from the Raman frequency shifts, in particular, in the η phase of solid nitrogen.

Application of the Pippard Relations to Cyclohexane near the Melting Point

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.

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

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 expansivity obtained for various pressures at constant temperatures of 302 and 325 K in this system. By calculating the isothermal compressibility and the specific heat in the same pressure region of the premelting region of hexadecane, we obtain that varies linearly with and also that varies linearly with for this system. This indicat es that some molecular organic compounds, such as solid hexadecane studied here, can exhibit -phase transitions prior to melting which are expected to verify the Pippard relations.

PIPPARD RELATIONS MODIFIED FOR THE RAMAN FREQUENCY SHIFTS CLOSE TO I–II PHASE TRANSITION IN S-TRIAZINE

The pressure dependence of the frequency shifts 1/ω(∂ω/∂T)Pis calculated from the 1/ω(∂ω/∂P)Tfor the Raman modes I and IV according to the spectroscopic modifications of the Pippard relations. The observed Raman frequencies of those modes are used for this calculation close to the I–II phase transition in s-triazine. From the frequency shifts, the isothermal compressibility κTand the specific heat CPare also calculated as a function of pressure close to the I–II phase transition in s-triazine. Our calculated values of 1/ω(∂ω/∂T)PκTand CPcan be compared with the experimental measurements at various pressures in the phases I and II of s-triazine.

Application of the Pippard relations to solid ammonia at high pressures

Application of the Pippard relations to solid ammonia at high pressures

Pressure dependence of the Raman frequency shifts related to the thermodynamic quantities in phase II of s-triazine

The temperature-induced frequency shifts (1/ω)(∂ω/∂T) p are calculated from the pressure induced frequency shifts (1/ω)(∂ω/∂P) T using the observed frequencies at various pressures for the Raman modes of II and III in phase II (P > P C ) of s-triazine. The values of the mode Gruneisen parameters γ T (isobaric) and γ T (isothermal) of those Raman modes studied here, are determined through the Pippard relations in the spectroscopically modified forms for this crystal. The temperature-induced and the pressure-induced frequency shifts are then used to predict the pressure dependence of the isothermal compressibility, thermal expansion and the specific heat in phase II (P > P C ) of s-triazine.

Calculation of thermodynamic quantities for carbon tetrachloride (CCl4) close to the III–IV phase transition

We calculate the thermal expansion αp, isothermal compressibility κT and the specific heat, Cp-Cv, as a function of pressure using the observed V-P data at room temperature close to the III–IV phase transition in CCl4. Calculated κT, αp and Cp-Cv decrease as the pressure increases from phase III to phase IV in this molecular crystal system, as expected. On the basis of our calculations of the κT, αp and Cp-Cv, the Pippard relations are established close to the III–IV phase transition in CCl4, which can be verified experimentally.

Correlations of the thermodynamic quantities with the frequency shifts of a lattice mode close to the I–II transition in NH 4Br

We correlate here the specific heat C p with the frequency shifts (1/ V) (∂ V/∂ T) p and the thermal expansivity α p with the (1/ ν) (∂ V/∂ P) T close to the I–II transition in NH 4Br. This correlation is performed for the Raman mode of ν s (140 cm −1) using the molar volume data for NH 4Br. It is shown here that spectroscopic modifications of the Pippard relations are applied satisfactorily to the I–II phase transition by using a lattice mode of NH 4Br.

Investigation of thermal expansion, phase diagrams, and barocaloric effect in the (NH4)2WO2F4 and (NH4)2MoO2F4 oxyfluorides

The thermal expansion along the principal crystallographic axes of the (NH4)2WO2F4 and (NH4)2MoO2F4 oxyfluorides has been studied. The anomalous behavior of αi(T) due to the phase transitions has been revealed at T1 = 271.4 K and T2 ≈ 180 K for the molybdate and at T1 = 201.5 K and T2 ≈ 161 K for the tungstate. The quantities dT/dp and dT/dσi, which characterize the dependence of the phase transition temperatures on the hydrostatic and uniaxial pressures, have been determined from analyzing the results of studies of the thermal expansion and heat capacity with the use of the Pippard relations. The p-T and αi-T phase diagrams reflect different characters of the influence of the pressure on the stability of the initial and distorted phases of the oxyfluorides. The magnitudes of the extensive and intensive barocaloric effects determined in the vicinity of the structural phase transitions are as follows: ΔSBCE varies from approximately −10 to −17 J/mol K and ΔTAD ≈ 8−17 K for the molybdate and ΔSBCE varies from approximately −10 to −17 J/mol K and ΔTAD ≈ 8−13 K for the tungstate.

Pippard relations modified by the Raman frequency shifts for the phases I and II in NH 4I

Pippard relations in terms of the frequency shifts of the ν 7 (40 cm −1) lattice mode, are established here for the phases I and II in NH 4I. The temperature-induced frequency shift, (1/ ν)(∂ ν/∂ T) P is calculated through the isobaric mode Grüneisen parameter γ P using the lattice parameter data at zero pressure for the phases I and II in NH 4I. From the frequency shifts, the specific heat C P is calculated for the phases studied at zero pressure. Also, the pressure-induced frequency shift, (1/ ν)(∂ ν/∂ P) T is calculated through the isothermal mode Grüneisen parameter γ T at 300 K and it is related to the thermal expansivity α P for the phases I and II in NH 4I. Linear variation of C P with the (1/ ν)(∂ ν/∂ T) P and also a linear variation of α P with the (1/ ν)(∂ ν/∂ P) T , indicate that the frequency shifts obtained for the ν 7 (40 cm −1) lattice mode exhibit similar critical behaviour as C P and α P close to the I–II phase transition in NH 4I.