Simultaneous solidification of two catalyst wastes and their effect on the early stages of cement hydration

Two catalyst wastes (RNi and RAI) from polyol production were considered as hazardous, due to their respective high concentration of nickel and aluminum contents. This article presents the study, done to avoid environmental impacts, of the simultaneous solidification/stabilization of both catalyst wastes with type II Portland cement (CP) by non-conventional differential thermal analysis (NCDTA). This technique allows one to monitor the initial stages of cement hydration to evaluate the accelerating and/or retarding effects on the process due to the presence of the wastes and to identify the steps where the changes occur. Pastes with water/cement ratio equal to 0.5 were prepared, into which different amounts of each waste were added. NCDTA has the same basic principle of Differential Thermal Analysis (DTA), but differs in the fact that there is no external heating or cooling system as in the case of DTA. The thermal effects of the cement paste hydration with and without waste presence were evaluated from the energy released during the process in real time by acquiring the temperature data of the sample and reference using thermistors with 0.03 °C resolution, coupled to an analog–digital interface. In the early stages of cement hydration retarding and accelerating effects occur, respectively due to RNi and RAl presence, with significant thermal effects. During the simultaneous use of the two waste catalysts for their stabilization process by solidification in cement, there is a synergic resulting effect, which allows better hydration operating conditions than when each waste is solidified separately. Thermogravimetric (TG) and derivative thermogravimetric analysis (DTG) of 4 and 24 h pastes allow a quantitative information about the main cement hydrated phases and confirm the same accelerating or retarding effects due to the presence of wastes indicated from respective NCDTA curves.     

Study of dehydration and rehydration processes of portlandite in mature and young cement pastes

The hydration process of the cements induces the formation of different kinds of hydration products. The main products of hydration are C?CS?CH gel and portlandite [Ca(OH)₂]. The C?CS?CH gel is an amorphous compound that is discomposed progressivity with the temperature until approximately 1,000?°C, while the portlandite is discomposed between 450 and 550?°C. Also, calcium carbonate can be formed as a consequence of the portlandite carbonation. All of these processes can be analysed and quantified by simultaneous differential thermal analysis and thermogravimetric analysis. And by X-ray diffraction it is possible to identify the crystalline phases. Some authors have corroborated that the portlandite can be rehydrated, after dehydration processes due to thermal exposition of the cement paste. But all of these experiments have been made with young cement pastes or at temperatures lower than 650?°C. In this work the behaviour of young and mature cement pastes have been studied in relation with the portlandite decomposition and the possibility of the rehydration of it in water presence. We found that young pastes and old pastes, stored at laboratory conditions, and later burned, show a certain grade of rehydration, specially the pastes burned at 650?°C (with ??80% of reformation of portlandite) with respect to the pastes burned at 1,000?°C (between 20 and 40%). It is corroborate that the rehydration process is directly related to the formation of CaO during the burning. Also, a formation of unstable portlandite is detected in young pastes burned at 650?°C, which can be rehydrated easily. Although, the mature pastes that have been burned initially and stored under laboratory conditions cannot be rehydrated, due to the formation of stable products during the storage.     

Study of the MS response by TG–MS in an acid mine drainage efflorescence

The aim of this study is to employ a thermogravimetric analyzer coupled to a mass spectrometer to research into the influence of heating rate and sample mass on the response of the detector. That response is examined by means of a particular efflorescence taken from an acid mine drainage environment. This mixture of weathered products is mainly composed by secondary sulfate minerals, which are formed in evaporation conditions, appearing as efflorescence salts. Thermogravimetry coupled to mass spectrometry has been used to analyze the three main loss steps that happen when this combination of minerals is heated from 30 to 1,100 °C. This inorganic material is based on a mixture of hexahydrite, zinc sulfate hexahydrate, apjonite, gypsum, plumbojarosite, calcite, quartz, and magnetite. While heating, three main effluent gases evolved from this efflorescence. At a standard heating rate of 10 °C/min, loss of water (dehydration) occurred over 30–500 °C in four major steps, loss of carbon dioxide (decarbonisation) occurred over 200–800 °C in three steps, and loss of sulfur trioxide (desulfation) occurred over 400–1,100 °C in three steps. According to the results, thermal analysis is an excellent technique for the study of decomposition in these systems.     

Thermal characteristics of bitumen pyrolysis

The pyrolysis behavior of bitumen was investigated using a thermogravimetric analyzer–mass spectrometer system (TG–MS) and a differential scanning calorimeter (DSC) as well as a pyrolysis-gas chromatograph/mass spectrometer system (Py-GC/MS). TG results showed that there were three stages of weight loss during pyrolysis—less than 110, 110–380, and 380–600 °C. Using distributed activation energy model, the average activation energy of the thermal decomposition of bitumen was calculated at 79 kJ mol⁻¹. The evolved gas from the pyrolysis showed that organic species, such as alkane and alkene fragments had a peak maximum temperature of 130 and 480 °C, respectively. Benzene, toluene, and styrene released at 100 and 420 °C. Most of the inorganic compounds, such as H₂, H₂S, COS, and SO₂, released at about 380 °C while the CO₂ had the maximum temperature peaks at 400 and 540 °C, respectively. FTIR spectra were taken of the residues of the different stages, and the results showed that the C–H bond intensity decreased dramatically at 380 °C. Py-GC/MS confirmed the composition of the evolved gas. The DSC revealed the endothermic nature of the bitumen pyrolysis.     

Synthesis of high surface area transitional alumina from Al(OPh)<Subscript>3</Subscript>

Al(OPh)₃ involving sterically hindered phenyl groups on ultrasonic assisted micro hydrolysis yielded a mixture of boehmite and bayerite as deduced from the FTIR and powder X-ray diffraction pattern. In the thermogravimetric trace, the complete removal of decomposable moieties of the hydrolyzed gel occurred around 530 °C. Calcining the gel at temperatures 600, 700, 800 and 900 °C showed crystalline tetragonal δ-Al₂O₃ to be the product at 900 °C as deduced from FTIR, ²⁷Al NMR and PXRD techniques. δ-Al₂O₃ showed a surface area of 135 m²/g with rectangular bar like morphology with the sizes below 50 nm in the TEM images.     

Sensitive measurement of ultratrace phenols in natural water by purge-and-trap with in situ acetylation coupled with gas chromatography-mass spectrometry

A novel purge-and-trap method coupled with gas chromatography-mass spectrometry (GC-MS) is developed for the analysis of trace and ultratrace phenols based on their derivatization with acetic anhydride. Parameters affecting the extraction efficiency, such as purge temperature, concentration of sodium chloride, purge time, and volume of derivatization reagent, were investigated. The optimized conditions were addition of 150 μL acetic anhydride, purge time of 25 min at the purge temperature of 60 °C with 30% NaCl. The linear range was 0.2–100 μg L⁻¹ for phenols. The limits of detection (LODs) ranged from 0.08 to 0.15 μg L⁻¹ and the relative standard deviations (RSDs) for most of the phenols at the 10 μg L⁻¹level were below 10%. Natural water samples collected from a pool were successfully analyzed using the proposed method. The recovery of spiked water samples was 72.9–84.2%.     

Pyrolysis study of sewage sludge by TG-MS and TG-GC-MS coupled analyses

The pyrolysis of an urban plant sewage sludge carried out under He atmosphere was studied by thermogravimetric-mass spectrometric (TG-MS) and thermogravimetric-gas chromatographic-mass spectrometric (TG-GC-MS) analyses. The sludge was thermally degraded with a heating rate of 10°C min^–1 in the 20–1000°C interval; its mass loss is 51.8% up to 600°C and 61.4% up to 1000°C. Gas chromatographic analyses of the gas released during major thermogravimetric events allow the identification of various chemical species. Water, carbon mono- and di-oxide, several hydrocarbons (up to C5, both saturated and unsaturated) were the major detected species. Minor amount of pollutant species, such as cyano-compounds, were also detected. Among the released species hydrocarbons constituted the major fraction (53%), thus suggesting their immediate exploitation as fuel gas. The pyrolysis behaviour of this sample was then compared with other sewage sludge arising from wastewater treatment plants subjected to anaerobic digestion as found in literature data.     

Purge and trap with monolithic sorbent for gas chromatographic analysis of pesticides in honey

Polydivinylbenzene (P-DVB) and silica monolithic materials were synthesised in capillaries and then used to adsorb nine organonitrogen pesticides extracted from honey. After adsorption, each monolith-capillary was connected to a gas chromatograph (GC) with nitrogen-phosphorous detector (NPD). The silica monolith gave better peak shape and intensity than the P-DVB monolith. Formulation of silica monolith with tetramethoxysilane (TMOS) as a sole initiator provided better separation and higher peak intensity, compared to those with methyltrimethoxysilane (MTMS) and (3:1) TMOS/MTMS initiators. The optimum internal diameter and length of silica monolith-capillary were 0.25 mm and 5 cm, respectively. The purge and trap conditions were optimised by purging with nitrogen and heating the honey sample solution at 100 °C for 60 min. Recovery of the method was in a range of 84.95–99.71% and the detection limits of the pesticides determined by using GC-NPD and GC-mass spectrometric detector (MSD) ranged between 0.36–1.75 and 0.13–0.25 ng g⁻¹, respectively.     

Effect of the AL-MCM-41 catalyst on the catalytic pyrolysis of atmospheric petroleum residue (ATR)

Thermogravimetry (TG) was used in this study to evaluate thermal and catalytic pyrolysis of Atmospheric Petroleum Residue (ATR) which can be found in the state of Rio Grande do Norte/Brazil, after a process of atmospheric distillation of petroleum. The utilized sample in the process of catalytic pyrolysis was Al-MCM-41, a mesoporous material. The procedures for obtaining the thermogravimetric curves were performed in a thermobalance with heating rates of 5, 10, and 20 °C min⁻¹. From TG, the activation energy was determined using the Flynn–Wall kinetic method, which decreased from 161 kJ mol⁻¹, for the pure ATR, to 71 kJ mol⁻¹, in the presence of the Al-MCM-41, showing the efficiency of the catalyst in the pyrolysis of Atmospheric Petroleum Residue.     

Thermal,structural and magnetic studies on chromite spinel synthesized using citrate precursor method and annealed at 450 and 650 °C

Chromite Spinel materials were synthesized in this study by the citrate precursor method using four divalent cations (Ni²⁺, Co²⁺, Zn²⁺, and Cu²⁺). Citrate precursors consisting of mixed chromium citrates were first subjected to a thermogravimetric (TG) analysis for determining optimum temperatures for annealing. TG of coprecipitated chromium(III) citrate–zinc citrate gel has been carried out separately in N₂ and O₂ atmospheres. In both the cases, dehydration is followed by a four-step decomposition. The TG data were subjected to kinetic/mechanistic analysis, and the values of activation energy and Arrhenius factor were approximated. TG curves of various powders which were obtained on annealing at the two temperatures did exhibit thermal instability when carried out in N₂ atmosphere. A large coercivity of 2701.01 Oe was observed for NiCr₂O₄ at 650 °C. On the basis of the results, 450 °C has been chosen for annealing treatment of the four gels. The samples were accordingly annealed at two different temperatures (450 and 650 °C) in a muffle furnace for 1 h in each case. The annealed powders were characterized using X-ray diffraction (XRD), SEM, and vibrating sample magnetometer (VSM). The XRD patterns show that annealing of CuCr₂O₄, NiCr₂O₄, and CoCr₂O₄ at 450 °C yields very small crystallites with poor Bragg reflections, although ZnCr₂O₄ samples show better peaks in XRD data. Annealing at 650 °C resulted in particle size range of 8–89 nm in the four cases. In the case of ZnCr₂O₄, the particle size was 8 nm.     

Pyrolysis and TG Analysis of Shivee Ovoo Coal from Mongolia

The coal sample of the Shivee Ovoo deposits has been non-isothermally pyrolysed in a thermogravimetric analyser to determine the influence of temperature, heating rate and purge gas employed on the thermal degradation of the sample. The heating rates investigated in the TG were 10–50 K min^–1 to final temperature of 1000°C. N₂or CO₂ were employed as well as type of purge gas on the process of thermal degradation of the coal sample. The coal was also investigated in a fixed bed reactor to determine the influence of temperature and heating rate of the pyrolysis on the yield of products and composition of the gases evolved. The main gases produced were H₂, CH₄, C₂H₂, C₂H₄, C₂H₆, C₃H₆ and C₃H₈ and also minor concentrations of other gases. This revised version was published online in August 2006 with corrections to the Cover Date.     

Determination of Aroma Compounds from Alcoholic Beverages in Spiked Blood Samples by Means of Dynamic Headspace GC–MS

Some aroma compounds found in alcoholic beverages are characteristic of a certain beverage (i.e. 2,4-decadienoic acid ethyl ester is characteristic of pear spirit and 5-butyltetrahydro-4-methylfuran-2-on “whiskey lactone” is characteristic of aged spirits like whiskey). These substances were detectable in beverages but not in blood samples. The aim of this investigation was to find a sensitive sampling technique for aroma compounds in whole blood samples. This technique may be used in forensic toxicology for examination of drinking claims. The method comprises dynamic headspace sampling using a purge and trap concentrator, followed by quantitative gas chromatography–mass spectrometry (dynamic HS–GC–MS). The influence of sample preparation, trap adsorbents and sample temperature as well as desorption time and purge time on the quality of the analytical results were investigated. The following optimal parameters were determined: stirred and diluted whole blood sample without salt addition, use of Carbotrap C as trap material, sample temperature at 80 °C, desorption time 20 min and purge time 30 min. These optimal parameters were used for the determination of detection limits (LOD). The LOD of aroma compounds by means of dynamic headspace sampling were compared with the results of conventional sampling: the static headspace technique. Limits of detection for the aroma compounds with conventional static headspace GC are in the range 400–10,000 μg L⁻¹. Dynamic headspace–GC was found to be a more sensitive sampling technique for most of the aroma compounds investigated (e.g. C₄–C₈ ethyl esters, benzoic acid ethyl ester, linalool oxide and 4-ethylguaiacol) with detection limits between 1 and 50 μg L⁻¹, but there were also limits to the sampling of substances with lower volatility like decanoic acid ethyl ester, 2,4-decadienoic acid ethyl ester, eugenol and whiskey lactone with detection limits of about 1,000 μg L⁻¹.

     

Determination of Aroma Compounds from Alcoholic Beverages in Spiked Blood Samples by Means of Dynamic Headspace GC–MS

Some aroma compounds found in alcoholic beverages are characteristic of a certain beverage (i.e. 2,4-decadienoic acid ethyl ester is characteristic of pear spirit and 5-butyltetrahydro-4-methylfuran-2-on “whiskey lactone” is characteristic of aged spirits like whiskey). These substances were detectable in beverages but not in blood samples. The aim of this investigation was to find a sensitive sampling technique for aroma compounds in whole blood samples. This technique may be used in forensic toxicology for examination of drinking claims. The method comprises dynamic headspace sampling using a purge and trap concentrator, followed by quantitative gas chromatography–mass spectrometry (dynamic HS–GC–MS). The influence of sample preparation, trap adsorbents and sample temperature as well as desorption time and purge time on the quality of the analytical results were investigated. The following optimal parameters were determined: stirred and diluted whole blood sample without salt addition, use of Carbotrap C as trap material, sample temperature at 80 °C, desorption time 20 min and purge time 30 min. These optimal parameters were used for the determination of detection limits (LOD). The LOD of aroma compounds by means of dynamic headspace sampling were compared with the results of conventional sampling: the static headspace technique. Limits of detection for the aroma compounds with conventional static headspace GC are in the range 400–10,000 μg L^?1. Dynamic headspace–GC was found to be a more sensitive sampling technique for most of the aroma compounds investigated (e.g. C₄–C₈ ethyl esters, benzoic acid ethyl ester, linalool oxide and 4-ethylguaiacol) with detection limits between 1 and 50 μg L^?1, but there were also limits to the sampling of substances with lower volatility like decanoic acid ethyl ester, 2,4-decadienoic acid ethyl ester, eugenol and whiskey lactone with detection limits of about 1,000 μg L^?1.     

Artificial pozzolanic cement pastes containing burnt clay with and without silica fume

Pozzolanic cement blends were prepared by the partial substitution of ordinary Portland cement (OPC) with different percentages of burnt clay (BC), Libyan clay fired at 700 °C, of 10, 20, and 30%. The pastes were made using an initial water/solid ratio of 0.30 by mass of each cement blend and hydrated for 1, 3, 7, 28, and 90 days. The pozzolanic OPC–BC blend containing 30% BC was also admixed with 2.5 and 5% silica fume (SF) to improve the physicomechanical characteristics. The hardened pozzolanic cement pastes were subjected to compressive strength and hydration kinetics tests. The results of compressive strength indicated slightly higher values for the paste made of OPC–BC blend containing 10% BC The results of DSC and XRD studies indicated the formation and later the stabilization of calcium silicates hydrates (CSH) and calcium aluminosilicate hydrates (C₃ASH₄ and C₂ASH₈) as the main hydration products in addition to free calcium hydroxide (CH). Scanning electron microscopic (SEM) examination revealed that the pozzolanic cement pastes made of OPC–BC mixes possesses a denser structure than that of the neat OPC paste. Furthermore, the addition of SF resulted in a further densification of the microstructure of the hardened OPC–BC–SF pastes; this was reflected on the observed improvement in the compressive strength values at all ages of hydration.     

Non-isothermal crystallization kinetics of some aventurine decorative glaze

Thermal and structural properties of three clays (sepiolite and two kaolinites) from Turkey were studied by thermal analysis (TG–DTA), X-ray diffraction (XRD), X-ray fluorescence (XRF), Fourier transform infrared (FT-IR), and surface area measurement techniques The adsorption of sulfur dioxide (SO₂) gas by these clays was also investigated. SO₂ adsorption values of K1, K2, and S clay samples were measured at 20 °C and pressures up to 106 kPa. Sepiolite sample (S) primarily consists of pure sepiolite, only dolomite present as accompanying mineral. Both kaolinite samples, K1 and K2, mainly contain kaolinite as the major clay mineral and quartz as impurity. In K2 sample, muscovite phase is also present. Simultaneous TG–DTA curves of all clay samples were obtained at three different heating rates 10, 15, and 20 °C min⁻¹ over the temperature range 30–1200 °C. It was found that the retention value of SO₂ by S clay (2.744 mmol/g) was higher than those of K1 (0.144 mmol/g) and K2 (0.164 mmol/g) samples.     

Examination of the oxidation resistance of Cr–Mo–V tool steel by thermal analysis

In the present study, the oxidation behavior of Cr–Mo–V tool steel was examined at different temperatures in air. The examination was conducted by means of thermogravimetric analysis, scanning electron microscopy, and X-ray diffraction (XRD). After non-isothermal oxidation from ambient temperature to 1000 °C, it was revealed that the specimen begins to oxidize over 700 °C, while over 800 °C the oxidation rate increases significantly. Finally over 900 °C, this rate has a considerable value, and the specimen's oxidation resistance is inadequate. From these results, four different oxidation temperatures (805, 835, 865, and 895 °C) were selected for the isothermal test, as referred above, which correspond to different oxidation rates, to determine the oxidation activation energy of the Cr–Mo–V specimens. Energy dispersive X-ray spectroscopy (EDX) and XRD phase identification of the as-formed scales showed that in every case, it contains two distinguishable regions. The inner layer is a mixture of chrome and iron oxides and the outer layer contains iron oxides and is also characterized by high porosity. This phenomenon was explained by the different diffusion coefficients of every element in the steel matrix.     

Characterization of dental composites by thermal analysis, infrared spectroscopy and scanning electron microscopy

Commercial light-cured dental composites were used in this study. Two laboratorial composites, Resilab (Wilcos/Brazil), Epricord (Kuraray/Japan) were compared under cured and uncured conditions. Thermal analysis, infrared spectroscopy and scanning electron microscopy were used to evaluate the dental composites. The mass change and heat flow signals (TG–DSC) were recorded simultaneously by using STA 409 PC Luxx (NETZSCH), in the 25–800 °C temperature range at a heating rate of 10 °C/min under nitrogen atmosphere (70 mL/min). Employing thermo-microbalance TG 209 C F1 Iris (NETZSCH) coupled to the BRUKER Optics FTIR TENSOR, the samples were analyzed by combined thermogravimetric and spectroscopic methods (TG–FTIR). The initial sample mass was about ~12 mg, the data collection have been done in the 35–800 °C temperature range at a heating rate of 20 K/min in nitrogen atmosphere (flow rate: 40 mL/min). Finally, superficial topographic was analyzed by scanning electron microscopy (SEM). Dental composite evaluation suggests a high thermal stability and inorganic content in RES D sample. Degrees of conversion (DC) values were almost the same and there was no direct relationship between DC and amount of particles and size. Similar compositions were found in all samples.     

Evaluation of CO2 adsorption capacity of solid sorbents

The CO₂ adsorption capacity of the low-cost solid sorbents of waste tire char (TC) and chicken waste char (CW) was compared with commercial active carbon (AC) and 5 ? zeolite (ZA) using thermogravimetric analysis (TG), pressurized TG, and differential scanning calorimetry (DSC). The sorbents were degassed in a TG up to 150 °C to release all gases on the surface of the sample, then cooled down to the designed temperature for adsorption. TG results indicated that the CO₂ adsorption capacity of TC was higher than that of CW, but lower than those of AC and ZA. The maximum adsorption rate of TC at 50 °C was 0.61% min⁻¹, lower than that of AC, but higher than that of CW, 0.44% min⁻¹. The maximum adsorption rate of ZA at 50 °C was 3.1% min⁻¹. When the pressure was over 4 bar, the adsorption rate of ZA was lower than that of TC and AC. At 30 bar, the total CO₂ uptake of TC was 20 wt%, higher than that of CW and ZA but lower than that of AC. The temperature, nitrogen concentration, and water content also influenced the CO₂ adsorption capacity of sorbents to some extent. DSC results showed that adsorption was an exothermic process. The heat of CO₂ adsorption per mole of CO₂ of TC at 50 °C was 24 kJ mol⁻¹ while the ZA had the largest heat of adsorption at 38 kJ mol⁻¹. Comparing the characteristics of TC and CW, TC may be a promising sorbent for removal of CO₂.     

Release characteristics of semi-volatile heavy metals during co-combustion of sewage sludge and coal under the O<Subscript>2</Subscript>/CO<Subscript>2</Subscript> atmosphere

Oil well cementing is a vital operation to assure casing stability and zonal isolation for oil and gas exploration. However, some scenarios demand the cemented region to withstand high thermal gradients and imposed deformations, as occurs in the case of oil wells subjected to cyclic steam injection at temperatures up to 250 °C, to reduce oil viscosity and to increase well pressure to facilitate heavy oil recovery. In this paper, the hydration of ductile special cement systems using styrene-butadiene latex (SBR) and carboxylated styrene-butadiene latex (XSBR) addition was studied by conduction calorimetry. The resulting heat flow curves, presented in log–log plots, were used to analyze the influence of those copolymers on the hydration stages of three families of cement pastes of different complexity. The simpler cement systems (SCCS) contained water, oil well Portland cement class G and SBR or XSBR in its composition. In medium complexity systems silica fume was added and in the higher complexity ones (HCCS), superplasticizer as well. The primary objective of adding those copolymers into the Portland cement paste is to obtain higher ductility properties after setting, silica fume to have good thermal stability up to 300 °C, while superplasticizer was added to guarantee good workability. Rheological tests were carried out to evaluate the effect of the copolymers on the composite viscosity. Thermogravimetric analysis of selected SCCS and HCCS samples was performed to quantify the main formed phases up to 24 h of cement hydration. From the obtained results, it was noticed that SBR and XSBR addition substantially affects hydration kinetics at all early age stages. Starting from pre-induction and induction periods, the main observed effect during these stages, was related to the increased viscosity of the pastes, which was higher in XSBR containing pastes, retarding the hydration reactions of respective following stages, when compared to pastes with the same cementitious matrix without copolymer addition.     

Branched-chain organic compounds. Part 11: Validation of a new method for determination of the potential energy of cohesion of an organic crystal and deduction of the heat capacity of the vapour

Abstract  

The cohesion energy of ethyl 3-cyano-3-(3,4-dimethyloxyphenyl)-2,2,4-trimethylpentanoate, as obtained from the change of kinetic and potential energies in the heat of sublimation of the crystal, E _p,coh = −46.7 kJ mol⁻¹ (78.6 °C), has been validated. A safe physicomathematic test based on the balance of entropy for the sublimation and Planck’s equation for changes of state, extended to entropy, was devised to ascertain the kinetic energies of the crystal and the gas molecule. Entropic equations were developed for the phase equilibrium to find precisely and with simplicity the vibrational energy of the crystal by using the vapour pressure exclusively and independently from the internal rotational and vibrational motion of the gas molecule. The heat capacity of the vapour was determined in this way, which in this case releases the solid allowing vibrational movement in the gas phase to meet the pressure of sublimation, C _p (T)/J K⁻¹ mol⁻¹ = 1.268 T/K + 58.62 (71.1–86.1 °C). An independent variational method of deducing the vibrational entropy, energy, or heat capacity of the gas molecule from each other was compared with the equations and was shown to yield the quantities with high accuracy. Values of the Nernst–Lindemann functions are tabulated.