The Triple Product Rule is Incorrect

The triple product rule, also known as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, the Mayer's relation and the heat capacity ratio, which describe the difference between isobaric and isochoric heat capacities, are also untrue. Also, the relationship linking thermal expansion and isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of the author.

The Triple Product Rule is Incorrect

The triple product rule, also known as the cyclic chain rule, cyclic relation, cyclical rule or Euler’s chain rule, relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, the Mayer’s relation and the heat capacity ratio, which describe the difference between isobaric and isochoric heat capacities, are also untrue. Also, the relationship linking thermal expansion and isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of the author.

Thermodynamic Mechanism of the Density and Compressibility Anomalies of Water in the Range -30<T(oC) <100

Compared to normal liquids, water exhibits a variety of anomalous thermal behaviors. This fact has been known for centuries. However, the thermodynamic mechanisms behind them have not been elucidated despite the efforts of many researchers. Under such circumstances, the author theoretically reproduced the measured values of the density-temperature curve at 1 atm for water above 0 oC. Then, the mystery of negative thermal expansion was clarified in relation to the shapes of the intermolecular interactions. In this paper, the author develops this line of work further and presents the interactions between water molecules to simultaneously reproduce the measured values of both the density-temperature curve and the isothermal compressibility-temperature curve in the range -30<100 at 1 atm. Then, the thermodynamic mechanism that produces these thermal behaviors is clarified in relation to the shapes of the interactions between molecules. Unraveling the mystery of related phenomena in relation to the shapes of the interaction between molecules has been a traditional and fundamental method in physics since the days of Newton.

Developing an Extreme Learning Machine-Based Model for Estimating the Isothermal Compressibility of Biodiesel

Nowadays, the high consumption of fossil fuels has caused many pollutants and environmental problems. Biodiesel has recently been considered as a clean and renewable alternative to fossil fuels. They are found in some molecular structures including fatty acid ethyl esters (FAEEs) and also fatty methyl esters (FAMEs), having various thermophysical characteristics. Thus, it appears essential to select the suitable methods for a particular diesel engine to estimate the ester characteristics. The current research sets out to develop a new and robust method predicting isothermal compressibility of long-chain fatty acid methyl and ethyl esters directly from several basic efficient parameters (pressure, temperature, normal melting point, and molecular weight). Therefore, as a novel and prevailing mathematical method in this field, an extreme learning machine was implemented for isothermal compressibility on the massive dataset. According to statistical evaluations, this novel established model had high accuracy and applicability (R2 = 1 and RMSE = 0.0018714) which is more accurate than previous models presented by former researchers. Among various factors of the sensitivity analysis, temperature and pressure had the greatest effect on the output values, so that the output parameter has a direct relationship with temperature and an inverse relationship with pressure with relevancy factors of 22.44% and −79.81%.

Enhancement and maximum in the isobaric specific-heat capacity measurements of deeply supercooled water using ultrafast calorimetry

Knowledge of the temperature dependence of the isobaric specific heat (Cp) upon deep supercooling can give insights regarding the anomalous properties of water. If a maximum in Cp exists at a specific temperature, as in the isothermal compressibility, it would further validate the liquid–liquid critical point model that can explain the anomalous increase in thermodynamic response functions. The challenge is that the relevant temperature range falls in the region where ice crystallization becomes rapid, which has previously excluded experiments. Here, we have utilized a methodology of ultrafast calorimetry by determining the temperature jump from femtosecond X-ray pulses after heating with an infrared laser pulse and with a sufficiently long time delay between the pulses to allow measurements at constant pressure. Evaporative cooling of ∼15-µm diameter droplets in vacuum enabled us to reach a temperature down to ∼228 K with a small fraction of the droplets remaining unfrozen. We observed a sharp increase in Cp, from 88 J/mol/K at 244 K to about 218 J/mol/K at 229 K where a maximum is seen. The Cp maximum is at a similar temperature as the maxima of the isothermal compressibility and correlation length. From the Cp measurement, we estimated the excess entropy and self-diffusion coefficient of water and these properties decrease rapidly below 235 K.