Statistical Mechanics of Rubber

1953 ◽  
Vol 26 (2) ◽  
pp. 302-310
Author(s):  
F. W. Boggs
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Thermodynamic Properties ◽  
Statistical Mechanics ◽  
Partial Pressure ◽  
Solvent Mixtures ◽  
Perfect Gas ◽  
Second Order Transition ◽  
Statistical Studies ◽  
Solvent Systems

Abstract It is shown in this paper that, above the second-order transition, the partition function of a rubber can be represented to a reasonably good approximation by the product of the partition function of a liquid formed of molecules similar to the chain elements and the partition function of the noninteracting chains divided by the partition function of a perfect gas. In terms of the free energy this means that the free energy of rubber is equal to the sum of the free energy of the chain network without interactions and that of a liquid formed of the chain elements alone minus the free energy of a perfect gas having the same number of molecules as the rubber has chain elements. This same procedure has made possible the calculation of the free energy and partial pressure of a rubber solution. This is, however, by no means the only possible application. All the thermodynamic properties of elastomer solvent systems can be obtained. This is a generalization of the methods previously used in the statistical studies of rubber and rubber solvent mixtures.

scholarly journals Twisted Conformal Algebra and Quantum Statistics of Harmonic Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
J. Naji ◽  
S. Heydari ◽  
R. Darabi
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Quantum Statistics ◽  
Conformal Algebra ◽  
Harmonic Oscillators ◽  
Two Dimensional ◽  
Raising And Lowering Operators ◽  
Lowering Operators

We consider noncommutative two-dimensional quantum harmonic oscillators and extend them to the case of twisted algebra. We obtained modified raising and lowering operators. Also we study statistical mechanics and thermodynamics and calculated partition function which yields the free energy of the system.

THERMODYNAMIC PROPERTIES OF ISOTOPIC COMPOUNDS OF SULPHUR

1950 ◽  
Vol 28b (9) ◽  
pp. 567-578 ◽  
Author(s):  
A. P. Tudge ◽  
H. G. Thode
Keyword(s):  
Partition Function ◽  
Thermodynamic Properties ◽  
Statistical Mechanics ◽  
Equilibrium Constants ◽  
Sulphur Isotopes ◽  
Exchange Reactions ◽  
Sulphur Compounds ◽  
Exchange Constants

Using the well known methods of statistical mechanics, partition function ratios for many isotopic sulphur compounds have been calculated. These partition function ratios are used to determine equilibrium constants for many possible exchange reactions involving the isotopes of sulphur. The results indicate that considerable fractionation of the sulphur isotopes can be expected in laboratory and natural occurring processes. The predicted exchange constants are discussed in the light of recent results on the distribution of the sulphur isotopes in nature.

Ensembles in Classical Statistical Mechanics

2019 ◽  
pp. 231-257
Author(s):  
Robert H. Swendsen
Keyword(s):  
Molecular Dynamics ◽  
Monte Carlo ◽  
Free Energy ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Harmonic Oscillators ◽  
Canonical Partition Function ◽  
Classical Statistical Mechanics ◽  
Classical Models ◽  
The Relationship

This chapter explores more powerful methods of calculation than were seen previously. Among them are Molecular Dynamics (MD) and Monte Carlo (MC) computer simulations. Another is the canonical partition function, which is related to the Helmholtz free energy. The derivation of thermodynamic identities within statistical mechanics is illustrated by the relationship between the specific heat and the fluctuations of the energy. It is shown how the canonical ensemble allows us to integrate out the momentum variables for many classical models. The factorization of the partition function is presented as the best trick in statistical mechanics, because of its central role in solving problems. Finally, the problem of many simple harmonic oscillators is solved, both for its importance and as an illustration of the best trick.

A canonical ensemble derivation of the McMillan-Mayer solution theory

1983 ◽  
Vol 48 (10) ◽  
pp. 2888-2892 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Energy Function ◽  
Special Function ◽  
Helmholtz Free Energy ◽  
Canonical Ensemble ◽  
Free Energy Function ◽  
Solution Theory ◽  
Pure Solvent ◽  
The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

scholarly journals First-Principles Study of Mechanical and Thermodynamic Properties of Binary and Ternary CoX (X = W and Mo) Intermetallic Compounds

Materials ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 1404
Author(s):  
Yunfei Yang ◽  
Changhao Wang ◽  
Junhao Sun ◽  
Shilei Li ◽  
Wei Liu ◽  
...  
Keyword(s):  
Free Energy ◽  
Thermodynamic Properties ◽  
Gibbs Free Energy ◽  
Density Functional ◽  
Vibrational Entropy ◽  
Functional Theory ◽  
Lattice Constants ◽  
The Density Functional Theory ◽  
Ductile Behavior ◽  
Pseudo Potential

In this study, the structural, elastic, and thermodynamic properties of DO19 and L12 structured Co3X (X = W, Mo or both W and Mo) and μ structured Co7X6 were investigated using the density functional theory implemented in the pseudo-potential plane wave. The obtained lattice constants were observed to be in good agreement with the available experimental data. With respect to the calculated mechanical properties and Poisson’s ratio, the DO19-Co3X, L12-Co3X, and μ-Co7X6 compounds were noted to be mechanically stable and possessed an optimal ductile behavior; however, L12-Co3X exhibited higher strength and brittleness than DO19-Co3X. Moreover, the quasi-harmonic Debye–Grüneisen approach was confirmed to be valid in describing the temperature-dependent thermodynamic properties of the Co3X and Co7X6 compounds, including heat capacity, vibrational entropy, and Gibbs free energy. Based on the calculated Gibbs free energy of DO19-Co3X and L12-Co7X6, the phase transformation temperatures for DO19-Co3X to L12-Co7X6 were determined and obtained values were noted to match well with the experiment results.

scholarly journals The bulk, surface and corner free energies of the anisotropic triangular Ising model

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190713
Author(s):  
Rodney J. Baxter
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Partition Function ◽  
Temperature Series ◽  
Isotropic Case ◽  
Series Expansions ◽  
Exact Results ◽  
Free Energies ◽  
Bulk Surface ◽  
Bulk Free Energy

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

scholarly journals Kinetics and thermodynamic properties related to the drying of 'Cabacinha' pepper fruits

2016 ◽  
Vol 20 (2) ◽  
pp. 174-180 ◽  
Author(s):  
Hellismar W. da Silva ◽  
Renato S. Rodovalho ◽  
Marya F. Velasco ◽  
Camila F. Silva ◽  
Luís S. R. Vale
Keyword(s):  
Experimental Data ◽  
Activation Energy ◽  
Free Energy ◽  
Thermodynamic Properties ◽  
Gibbs Free Energy ◽  
Removal Rate ◽  
Effective Diffusion ◽  
Drying Time ◽  
Three Samples ◽  
Different Temperatures

ABSTRACT The objective of this study was to determine and model the drying kinetics of 'Cabacinha' pepper fruits at different temperatures of the drying air, as well as obtain the thermodynamic properties involved in the drying process of the product. Drying was carried out under controlled conductions of temperature (60, 70, 80, 90 and 100 °C) using three samples of 130 g of fruit, which were weighed periodically until constant mass. The experimental data were adjusted to different mathematical models often used in the representation of fruit drying. Effective diffusion coefficients, calculated from the mathematical model of liquid diffusion, were used to obtain activation energy, enthalpy, entropy and Gibbs free energy. The Midilli model showed the best fit to the experimental data of drying of 'Cabacinha' pepper fruits. The increase in drying temperature promoted an increase in water removal rate, effective diffusion coefficient and Gibbs free energy, besides a reduction in fruit drying time and in the values of entropy and enthalpy. The activation energy for the drying of pepper fruits was 36.09 kJ mol-1.

Thermodynamic properties of n-alkoxyethanols + organic solvent mixtures

2003 ◽  
Vol 403 (2) ◽  
pp. 223-229 ◽  
Author(s):  
F.J. Carmona ◽  
J.Antonio González ◽  
Isaı́as Garcı́a de la Fuente ◽  
J.Carlos Cobos
Keyword(s):  
Thermodynamic Properties ◽  
Organic Solvent ◽  
Solvent Mixtures

Phase equilibria in solutions of rod-like particles

1956 ◽  
Vol 234 (1196) ◽  
pp. 73-89 ◽  
Keyword(s):  
Free Energy ◽  
Phase Separation ◽  
Partition Function ◽  
Colloidal Particles ◽  
Double Layers ◽  
Positive Energy ◽  
Isotropic Phase ◽  
Electric Double Layers ◽  
Rigid Rod ◽  
Anisotropic Phase

A partition function for a system of rigid rod-like particles with partial orientation about an axis is derived through the use of a modified lattice model. In the limit of perfect orientation the partition function reduces to the ideal mixing law ; for complete disorientation it corresponds to the polymer mixing law for rigid chains. A general expression is given for the free energy of mixing as a function of the mole numbers, the axis ratio of the solute particles, and a disorientation parameter. This function passes through a minimum followed by a maximum with increase in the disorientation parameter, provided the latter exceeds a critical value which is 2e for the pure solute and which increases with dilution. Assigning this parameter the value which minimizes the free energy, the chemical potentials display discontinuities a t the concentration a t which the minimum first appears. Separation into an isotropic phase and a some what more concentrated anisotropic phase arises because of the discontinuity, in confirmation of the theories of Onsager and Isihara, which treat only the second virial coefficient. Phase separation thus arises as a consequence of particle asymmetry, unassisted by an energy term . Whereas for a large-particle asymmetry both phases in equilibrium are predicted to be fairly dilute when mixing is athermal, a comparatively small positive energy of interaction causes the concentration in the anisotropic phase to increase sharply, while the concentration in the isotropic phase becomes vanishingly small. The theory offers a statistical mechanical basis for interpreting precipitation of rod-like colloidal particles with the formation of fibrillar structures such as are prominent in the fibrous proteins. The asymmetry of tobacco mosaic virus particles (with or without inclusion of their electric double layers) is insufficient alone to explain the well-known phase separation which occurs from their dilute solutions at very low ionic strengths. Higher-order interaction between electric double layers appears to be a major factor in bringing about dilute phase separation for these and other asymmetric colloidal particles bearing large charges, as was pointed out previously by Oster.

ChemInform Abstract: Electron Transfer Reactions in Solvent Mixtures: The Excess Component of Solvent Reorganization Free Energy

ChemInform ◽  
2000 ◽  
Vol 31 (37) ◽  
pp. no-no
Author(s):  
M. Morillo ◽  
C. Denk ◽  
P. Perez ◽  
M. Lopez ◽  
A. Sanchez ◽  
...  
Keyword(s):  
Electron Transfer ◽  
Free Energy ◽  
Transfer Reactions ◽  
Solvent Mixtures ◽  
Electron Transfer Reactions ◽  
Excess Component
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