The van der Waals equation: analytical and approximate solutions

The thermodynamic properties, enthalpy of vaporization, entropy, Helmholtz function, Gibbs function, but especially the heat capacity at constant volume of a van der Waals gas (and liquid) at the phase transition are examined in two different limit approximations. The first limit approximation is at the near-critical temperatures, i.e., for T/T _c → 1, where T _c is the critical temperature, the other limit approximation is at the near-zero temperatures, T→ 0. In these limits, the analytical equations for liquid and gas concentrations at saturated conditions were obtained. Although the heat capacities at constant volume of a van der Waals gas and liquid do not depend on the volume, they have different values and their change during the phase transition was calculated. It should be noticed that for real substances the equations obtained at the near-zero temperature are only valid for T > T _{triple point} and T ≪ T _c , which means that found equations can be used only for substances with T _{triple point} ≪ T _c .     

Thermodynamic characteristics of triphenylantimony bis(acetophenoneoximate)

The temperature dependence of heat capacity C _p ^° = f(T) of triphenylantimony bis(acetophenoneoximate) Ph₃Sb(ONCPhMe)₂ was measured for the first time in an adiabatic vacuum calorimeter in the range of 6.5–370 K and a differential scanning calorimeter in the range of 350–463 K. The temperature, enthalpy, and entropy of fusion were determined. Treatment of low-temperature (20 K ≤ T ≤ 50 K) heat capacity was performed on the basis of Debye’s theory of the heat capacity of solids and its multifractal model and, as a consequence, a conclusion was drawn on the type of structure topology. Standard thermodynamic functions C _p ^°(T), H°(T) — H°(0), S°(T), and G°(T) — H°(0) were calculated according to the experimental data obtained for the compound mentioned in the crystalline and liquid states for the range of T → 0–460 K. The standard entropy of the formation of crystalline Ph₃Sb(ONCPhMe)₂ was determined at T = 298.15 K.     

Heat capacity and heat content of BiNb<Subscript>5</Subscript>O<Subscript>14</Subscript>

The heat capacity and the heat content of bismuth niobate BiNb₅O₁₄ were measured by the relaxation time method, DSC and drop method, respectively. The temperature dependence of heat capacity in the form C _pm=455.84+0.06016T–7.7342·10⁶/T ² (J K^–1 mol^–1) was derived by the least squares method from the experimental data. Furthermore, the standard molar entropy at 298.15 K S _m=397.17 J K^–1 mol^–1 was derived from the low temperature heat capacity measurement.     

The thermodynamic functions of fullerene derivative (<Emphasis Type="Italic">t</Emphasis>-Bu)<Subscript>12</Subscript>C<Subscript>60</Subscript> over the temperature range from <Emphasis Type="Italic">T</Emphasis> → 0 to 420 K

The temperature dependence of heat capacity C _p ^o = f(T) of fullerene derivative (t-Bu)₁₂C₆₀ has been measured by a adiabatic vacuum calorimeter over the temperature range T = 6–350 K and by a differential scanning calorimeter over the temperature range T = 330–420 K for the first time. The low-temperature (T ≤ 50 K) dependence of the heat capacity was analyzed based on Debye’s the heat capacity theory of solids and its fractal variant. As a consequence, the conclusion about structure heterodynamicity is given. The experimental results have been used to calculate the standard thermodynamic functions C _p ^o (T), H ^o(T)−H ^o(0), S ^o(T) and G ^o(T) − H ^o(0) over the range from T → 0 to 420 K. The standard entropy of formation at 298.15 K of fullerene derivative under study was calculated. The temperature of decomposition onset of derivative was determined by differential scanning calorimetery and thermogravimetric analysis. The standard thermodynamic characteristics of (t-Bu)₁₂C₆₀ and C₆₀ fullerite were compared.     

Determining the Maximum Excess Heat of Solution Differentials Using the Carnot Cycle Temperature Efficiency

The (T ₁−T ₂)/T ₁ efficiency of a very small temperature differential thermodynamic cycle operating near a solvent’s critical locus, where T ₁ is the higher temperature and T ₂ the lower, is shown to be a simple tool for determining the maximum difference in the excess heat of solution between the cycle’s low-temperature liquid side and its high-temperature supercritical fluid side.     

Heat capacity of crystalline GaN

The heat capacity of gallium nitride has been measured by DSC method using DuPont Thermal Analyst 2100, DSC 951 unit in the temperature range (300–850 K). The temperature dependence of the heat capacity can be presented in the following form: C _p=32.960+0.162·10⁻¹ T+2360170T ⁻²-775370000T ⁻³.     

Thermodynamic properties of strontium metaniobate SrNb<Subscript>2</Subscript>O<Subscript>6</Subscript>

The heat capacity and the enthalpy increments of strontium metaniobate SrNb₂O₆ were measured by the relaxation method (2-276 K), micro DSC calorimetry (260-320 K) and drop calorimetry (723-1472 K). Temperature dependence of the molar heat capacity in the form C _pm=(200.47±5.51)+(0.02937±0.0760)T-(3.4728±0.3115)·10⁶/T ² J K⁻¹ mol⁻¹ (298-1500 K) was derived by the least-squares method from the experimental data. Furthermore, the standard molar entropy at 298.15 K S _m⁰ (298.15 K)=173.88±0.39 J K⁻¹ mol⁻¹ was evaluated from the low temperature heat capacity measurements. The standard enthalpy of formation Δ_f H ⁰ (298.15 K)=-2826.78 kJ mol⁻¹ was derived from total energies obtained by full potential LAPW electronic structure calculations within density functional theory.     

Low-temperature heat capacities and thermodynamic properties of 2,2-dimethyl-1,3-propanediol

The molar heat capacities C _p,m of 2,2-dimethyl-1,3-propanediol were measured in the temperature range from 78 to 410 K by means of a small sample automated adiabatic calorimeter. A solid-solid and a solid-liquid phase transitions were found at T-314.304 and 402.402 K, respectively, from the experimental C _p-T curve. The molar enthalpies and entropies of these transitions were determined to be 14.78 kJ mol⁻¹, 47.01 J K⁻¹ mol⁻ for the solid-solid transition and 7.518 kJ mol⁻¹, 18.68 J K⁻¹ mol⁻¹ for the solid-liquid transition, respectively. The dependence of heat capacity on the temperature was fitted to the following polynomial equations with least square method. In the temperature range of 80 to 310 K, C _p,m/(J K⁻¹ mol⁻¹)=117.72+58.8022x+3.0964x ²+6.87363x ³−13.922x ⁴+9.8889x ⁵+16.195x ⁶; x=[(T/K)−195]/115. In the temperature range of 325 to 395 K, C _p,m/(J K⁻¹ mol⁻¹)=290.74+22.767x−0.6247x ²−0.8716x ³−4.0159x ⁴−0.2878x ⁵+1.7244x ⁶; x=[(T/K)−360]/35. The thermodynamic functions H _T−H _298.15 and S _T−S _298.15, were derived from the heat capacity data in the temperature range of 80 to 410 K with an interval of 5 K. The thermostability of the compound was further tested by DSC and TG measurements. The results were in agreement with those obtained by adiabatic calorimetry.     

Thermodynamic investigation of several natural polyols (II)

The low-temperature heat capacity C _p,m of sorbitol was precisely measured in the temperature range from 80 to 390 K by means of a small sample automated adiabatic calorimeter. A solid-liquid phase transition was found at T=369.157 K from the experimental C _p-T curve. The dependence of heat capacity on the temperature was fitted to the following polynomial equations with least square method. In the temperature range of 80 to 355 K, C _p,m/J K⁻¹ mol⁻¹=170.17+157.75x+128.03x ²-146.44x ³-335.66x ⁴+177.71x ⁵+306.15x ⁶, x= [(T/K)−217.5]/137.5. In the temperature range of 375 to 390 K, C _p,m/J K⁻¹ mol⁻¹=518.13+3.2819x, x=[(T/K)-382.5]/7.5. The molar enthalpy and entropy of this transition were determined to be 30.35±0.15 kJ mol⁻¹ and 82.22±0.41 J K⁻¹ mol⁻¹ respectively. The thermodynamic functions [H _T-H _298.15] and [S _T-S 298.15], were derived from the heat capacity data in the temperature range of 80 to 390 K with an interval of 5 K. DSC and TG measurements were performed to study the thermostability of the compound. The results were in agreement with those obtained from heat capacity measurements.     

Low-temperature thermodynamic properties of <Emphasis Type="Italic">L</Emphasis>-cysteine

Heat capacity C _p(T) of the orthorhombic polymorph of L-cysteine was measured in the temperature range 6–300 K by adiabatic calorimetry; thermodynamic functions were calculated based on these measurements. At 298.15 K the values of heat capacity, C _p; entropy, S _m⁰(T)-S _m⁰(0); difference in the enthalpy, H _m⁰(T)-H _m⁰(0), are equal, respectively, to 144.6±0.3 J K⁻¹ mol⁻¹, 169.0±0.4 J K⁻¹ mol⁻¹ and 24960±50 J mol⁻¹. An anomaly of heat capacity near 70 K was registered as a small, 3–5% height, diffuse ‘jump’ accompanied by the substantial increase in the thermal relaxation time. The shape of the anomaly is sensitive to thermal pre-history of the sample.     

Thermal behaviour of transition-and tetravalentmetal oxides and phosphorous oxide composites

The heat capacity of the solid indium nitride was measured, using the Calvet TG-DSC 111 differential scanning microcalorimeter (Setaram, France), in the temperature between (314–978 K). The temperature dependence of the heat capacity can be presented in the following form: C _p=41.400+0.499·10⁻³ T−135502T ⁻²−26169900 T ⁻³.     

Heat capacity,enthalpy and entropy of calcium niobates

Heat capacity and enthalpy increments of calcium niobates CaNb₂O₆ and Ca₂Nb₂O₇ were measured by the relaxation time method (2–300 K), DSC (260–360 K) and drop calorimetry (669–1421 K). Temperature dependencies of the molar heat capacity in the form C _pm=200.4+0.03432T−3.450·10⁶/T ² J K⁻¹ mol⁻¹ for CaNb₂O₆ and C _pm=257.2+0.03621T−4.435·10⁶/T ² J K⁻¹ mol⁻¹ for Ca₂Nb₂O₇ were derived by the least-squares method from the experimental data. The molar entropies at 298.15 K, S _m⁰(CaNb₂O₆, 298.15 K)=167.3±0.9 J K⁻¹ mol⁻¹ and S _m⁰(Ca₂Nb₂O₇, 298.15 K)=212.4±1.2 J K⁻¹ mol⁻¹, were evaluated from the low temperature heat capacity measurements. Standard enthalpies of formation at 298.15 K were derived using published values of Gibbs energy of formation and presented heat capacity and entropy data: Δ_f H ⁰(CaNb₂O₆, 298.15 K)= −2664.52 kJ mol^t-1 and Δ_f H ⁰(Ca₂Nb₂O₇, 298.15 K)= −3346.91 kJ mol⁻¹.     

Heat capacity of GdBaSr(Cu<Subscript>3−x</Subscript><Emphasis Type="Italic">M</Emphasis><Subscript>x</Subscript>)O<Subscript>7−δ</Subscript> superconductor (<Emphasis Type="Italic">M</Emphasis>=Zn and Ni)

In this study, GdBaSr(Cu_3−x M _x)O_7−δ bulk samples (M=Zn and Ni; 0≤x≤0.1) were prepared via solid-state reaction. Specific heat measurement (measured with thermal relaxation technique using PPMS) shows an obvious specific heat jump around the T _c for GdBaSrCu₃O_7−δ sample as observed in most of the high temperature superconductors. It shifts towards lower temperature with increasing of both Zn and Ni doping contents, whose tendency is similar to the decreasing of T _c. Debye temperature, Θ_D (derived from specific heat measurements) calculated at around 10 K is found to be directly proportional to the T _c.     

Thermodynamic properties of poly[<Emphasis Type="Italic">bis</Emphasis>(trifluoroethoxy) phosphazene] in the range from <Emphasis Type="Italic">T</Emphasis>→0 TO 620 K

By adiabatic vacuum and dynamic calorimetry, heat capacity for poly[bis(trifluoroethoxy)phosphazene] has been determined over the 6–620 K range. Physical transformations of the polymer on its heating and cooling have been detected and characterized. Smoothed heat capacity C _p⁰(T) and standard thermodynamic functions (H ⁰(T)-H ⁰(0), S ⁰(T) and G ⁰(T)-H ⁰(0)) of poly[bis(trifluoroethoxy)phosphazene] have been evaluated for the temperature range from T→0 to 560 K. The standard entropy of formation Δ_f S ⁰ at T=298.15 K has been also determined. Fractal dimensions D in the heat capacity function of the multifractal variant of Debye’s theory of heat capacity of solids characterizing the heterodynamics of the tested polymer have been determined.     

The thermodynamic properties of (2,2′-dipyridyl)bis(4-chloro-3,6-di-tert-butyl-<Emphasis Type="Italic">o</Emphasis>-benzosemiquinone)cobalt

The heat capacity of paramagnetic (2,2′-dipyridyl)bis(4-chloro-3,6-di-tert-butyl-o-benzosemi-quinone)cobalt was studied over the temperature range 8–390 K by precision adiabatic vacuum and high-accuracy dynamic calorimetry. The physical transformation observed at 309–388 K was caused by the transition of the semiquinone-catecholate to bis-semiquinone form of the complex. Above 388 K, thermal destruction was superimposed on the physical transition. The experimental data were used to calculate the standard thermodynamic functions C _p ^o (T), H ^o(T)−H ^o(0), S ^o(T), and G ^o(T)−H ^o(0) at temperatures from T → 0 to 300 K. An analysis of the low-temperature heat capacity of the complex in terms of the Debye theory of the heat capacity of solids and its multifractal generalization led us to conclude that the complex had a predominantly chain structure.     

A short review and an empirical method for estimating the absorbed enthalpy of formation and the absolute enthalpy of dried microbial biomass for use in studies on the thermodynamics of microbial growth

Calculations are made using the equations Δ_r G = Δ_r H − TΔ_r S and Δ_r X = Δ_r H − Δ_r Q where Δ_r X represents the free energy change when the exchange of absorbed thermal energy with the environment is represented by Δ_r Q. The symbol Q has traditionally represented absorbed heat. However, here it is used specifically to represent the enthalpy listed in tabulations of thermodynamic properties as (H _T  − H ₀) at T = 298.15 K, the reason being that for a given substance TS equals 2.0 Q for solid substances, with the difference being greater for liquids, and especially gases. Since Δ_r H can be measured, and is tangibly the same no matter what thermodynamics are used to describe a reaction equation, a change in the absorbed heat of a biochemical growth process system as represented by either Δ_r Q or TΔ_r S would be expected to result in a different calculated value for the free energy change. Calculations of changes in thermodynamic properties are made which accompany anabolism; the formation of anabolic, organic by-products; catabolism; metabolism; and their respective non-conservative reactions; for the growth of Saccharomyces cerevisiae using four growth process systems. The result is that there is only about a 1% difference in the average quantity of free energy conserved during growth using either Eq. 1 or 2. This is because although values of TΔ_r S and Δ_r Q can be markedly different when compared to one another, these differences are small when compared to the value for Δ_r G or Δ_r X.     

Heat Capacities and Thermodynamic Properties of a H2O + Li2B4O7 Solution in the Temperature Range from 80 to 356 K

The molar heat capacities of an aqueous Li₂B₄O₇ solution were measured with a precision automated adiabatic calorimeter in the temperature range from 80 to 356 K at a concentration of 0.3492 mol⋅kg⁻¹. The occurrence of a phase transition was determined based on the changes in the curve of the heat capacity with temperature. A phase transition was observed at 271.72 K corresponding to the solid-liquid phase transition; the enthalpy and entropy of the phase transition were evaluated to be Δ H _m = 4.110 kJ⋅mol⁻¹ and Δ S _m = 15.13 J⋅K⁻¹⋅mol⁻¹, respectively. Using polynomial equations and thermodynamic relationship, the thermodynamic functions [H _T −H _298.15] and [S _T −S _298.15] of the aqueous Li₂B₄O₇ solution relative to 298.15 K were calculated in temperature range 80 to 355 K at intervals of 5 K. Values of the relative apparent molar heat capacities of the aqueous Li₂B₄O₇ solution, C _p,φ, were calculated at every 5 K in temperature range from 80 to 355 K from the experimental heat capacities of the solution and the heat capacities of pure water.     

Melting behavior of poly(tetrahydrofuran)-S and their blends

Melting behavior of poly(tetrahydrofuran)-s (PTHF) and their blend with different molecular masses has been studied by TM-DSC. PTHF and their blend show two endothermic peaks on their curve. The melting peak temperatures T _m1 and T _m2, entropy of fusion ΔS _f1 and ΔS _f2, and mean relaxation time for melting τ_f1 and τ_f2 have been estimated, and their dependence on the molecular mass has been examined. Plots of Tm1 to the reciprocal of their molecular mass fit a simple equation (T _m=a-b/M _n). Plots of T _m2 to their molecular mass also fit the equation with different factors. There seems to be a boundary around molecular mass 1200 in the molecular mass dependence of ΔS _fand τ_f. Effect of blending appeared on the τ_f and the non-reversing heat flow. This revised version was published online in July 2006 with corrections to the Cover Date.     

Molar heat capacity and thermodynamic properties of 1,2-cyclohexane dicarboxylic anhydride [C8H10O3]

The molar heat capacity C _p,m of 1,2-cyclohexane dicarboxylic anhydride was measured in the temperature range from T=80 to 390 K with a small sample automated adiabatic calorimeter. The melting point T _m, the molar enthalpy Δ_fus H _m and the entropy Δ_fus S _m of fusion for the compound were determined to be 303.80 K, 14.71 kJ mol⁻¹ and 48.43 J K⁻¹ mol⁻¹, respectively. The thermodynamic functions [H _T-H _273.15] and [S _T-S _273.15] were derived in the temperature range from T=80 to 385 K with temperature interval of 5 K. The thermal stability of the compound was investigated by differential scanning calorimeter (DSC) and thermogravimetry (TG), when the process of the mass-loss was due to the evaporation, instead of its thermal decomposition.