scholarly journals Thermodynamic analysis based on the second-order variations of thermodynamic potentials

2008 ◽  
Vol 35 (1-3) ◽  
pp. 215-234 ◽  
Author(s):  
Vlado Lubarda
Keyword(s):  
Thermodynamic Analysis ◽  
Thermodynamic Equilibrium ◽  
Second Order ◽  
Constrained Minimization ◽  
Specific Heats ◽  
Thermodynamic Relationships ◽  
Thermodynamic Potentials ◽  
Convexity Properties ◽  
Classical Derivation

An analysis of the Gibbs conditions of stable thermodynamic equilibrium based on the constrained minimization of the four fundamental thermodynamic potentials, is presented with a particular attention given to the previously unexplored connections between the second-order variations of thermodynamic potentials. These connections are used to establish the convexity properties of all potentials in relation to each other, which systematically deliver thermodynamic relationships between the specific heats, and the isentropic and isothermal bulk moduli and compressibilities. The comparison with the classical derivation is then given.

SPECIFIC HEATS OF CERTAIN SALTS OF IRON GROUP ELEMENTS FROM 65° TO 300°K.

1950 ◽  
Vol 28a (4) ◽  
pp. 367-376 ◽  
Author(s):  
H. D. Vasileff ◽  
H. Grayson-Smith
Keyword(s):  
Low Temperature ◽  
Specific Heat ◽  
Cobalt Nitrate ◽  
Second Order ◽  
Nickel Nitrate ◽  
Chromium Nitrate ◽  
Iron Group ◽  
Anhydrous Salt ◽  
Specific Heats ◽  
Entropy Changes

Using a new low temperature calorimeter, which is briefly described in the paper, the specific heats have been measured from 65° to 300°K. for the following salts: chromium sulphate (hydrated and anhydrous), chromium nitrate, cobalt nitrate, and nickel nitrate (hydrated). Hydrated chromium sulphate was found to have a transition of the second order at 195°K., while the specific heat of the anhydrous salt was quite regular. The hydrated nitrates all showed second order transitions in the neighborhood of 150°K. The entropy changes associated with these transitions have been estimated approximately, and vary from about 0.4 R for cobalt nitrate to 1.65 R for chromium nitrate, where R is the gas constant. Pending further evidence, it is tentatively suggested that the transitions are due to the onset of partial rotation of the H2O groups in the crystals.

scholarly journals Efficient Solutions of Interval Programming Problems with Inexact Parameters and Second Order Cone Constraints

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 270
Author(s):  
Ali Sadeghi ◽  
Mansour Saraj ◽  
Nezam Amiri
Keyword(s):  
Objective Function ◽  
Main Idea ◽  
Efficient Solutions ◽  
Second Order ◽  
Interval Programming ◽  
Properly Efficient Solutions ◽  
Cone Constraints ◽  
Second Order Cone ◽  
Convexity Properties ◽  
Interval Valued

In this article, a methodology is developed to solve an interval and a fractional interval programming problem by converting into a non-interval form for second order cone constraints, with the objective function and constraints being interval valued functions. We investigate the parametric and non-parametric forms of the interval valued functions along with their convexity properties. Two approaches are developed to obtain efficient and properly efficient solutions. Furthermore, the efficient solutions or Pareto optimal solutions of fractional and non-fractional programming problems over R + n ⋃ { 0 } are also discussed. The main idea of the present article is to introduce a new concept for efficiency, called efficient space, caused by the lower and upper bounds of the respective intervals of the objective function which are shown in different figures. Finally, some numerical examples are worked through to illustrate the methodology and affirm the validity of the obtained results.

Thermodynamic Analysis on the Minimum of Oxygen Content in the Deoxidation Equilibrium Curve in Liquid Iron

2016 ◽  
Vol 35 (4) ◽  
pp. 347-351 ◽  
Author(s):  
Pei-Wei Han ◽  
Pei-Xian Chen ◽  
Shao-Jun Chu
Keyword(s):  
Oxygen Content ◽  
Thermodynamic Analysis ◽  
Liquid Iron ◽  
Second Order ◽  
Interaction Parameters ◽  
Equilibrium Curve ◽  
Interaction Coefficients ◽  
First Order ◽  
Deoxidation Equilibrium ◽  
Special Case

AbstractThe minimum of oxygen content in the deoxidation equilibrium in liquid iron was thermodynamically analyzed in the present paper. Two criteria were developed to determine the existence of the minimum. The first criterion was $$0 \le x\gamma _{\rm{M}}^{\rm{M}} + y\gamma _{\rm{O}}^{\rm{M}} \le \min ({x \mathord{\left/{\vphantom {x {4.606[\% {\rm{M}}]_{{\rm{ex}}}^2}}} \right.\kern-\nulldelimiterspace} {4.606[\% {\rm{M}}]_{{\rm{ex}}}^2}},{{{{(xe_{\rm{M}}^{\rm{M}} + ye_{\rm{O}}^{\rm{M}})}^2}} \mathord{\left/{\vphantom {{{{(xe_{\rm{M}}^{\rm{M}} + ye_{\rm{O}}^{\rm{M}})}^2}} {3.474x}}} \right.\kern-\nulldelimiterspace} {3.474x}})$$ with $$xe_{\rm{M}}^{\rm{M}} + ye_{\rm{O}}^{\rm{M}} \lt 0$$, or $$x\gamma _{\rm{M}}^{\rm{M}} + y\gamma _{\rm{O}}^{\rm{M}}{\rm{\lt 0}}$$. And the second criterion was $$(xe_{\rm{M}}^{\rm{O}} + ye_{\rm{O}}^{\rm{O}}) + {y \mathord{\left/{\vphantom {y {2.303{{[\% {\rm{O}}]}_{{\rm{ex}}}}}}} \right.\kern-\nulldelimiterspace} {2.303{{[\% {\rm{O}}]}_{{\rm{ex}}}}}} \gt 0$$. The criteria in terms of first-order activity interaction parameters were the special case of present thermodynamic analysis with neglecting the second-order activity interaction parameters. They were not fit for the case of $$xe_{\rm{M}}^{\rm{M}} + ye_{\rm{O}}^{\rm{M}} \gt 0$$, in which case the criteria in terms of second-order activity interaction parameters should be taken into account to determine the existence of the minimum. The value 0.11 of $$e_{{\rm{Si}}}^{{\rm{Si}}}$$ was smaller based on the existence of the minimum for the Fe-O-Si system. It was guaranteed that the minimum value of oxygen content on the deoxidation equilibrium curve existed at silicon content 20 mass%, when the value 0.32 of $$e_{{\rm{Si}}}^{{\rm{Si}}}$$ was chosen, and the second-order activity interaction coefficients $$\gamma _{{\rm{Si}}}^{{\rm{Si}}}$$ and $$\gamma _{\rm{O}}^{{\rm{Si}}}$$ satisfied the condition $$\gamma _{{\rm{Si}}}^{{\rm{Si}}} + 2\gamma _{\rm{O}}^{{\rm{Si}}} = - 1.54 \times {10^{- 3}}$$.

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