Extended law of corresponding states: square-well oblates

2021 ◽  
Author(s):  
Miguel Gómez de Santiago ◽  
Peter Gurin ◽  
Szabolcs Varga ◽  
Gerardo Odriozola
Keyword(s):  
Short Range ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Corresponding States ◽  
Pair Potentials ◽  
Single Master Curve ◽  
Law Of Corresponding States ◽  
Square Well ◽  
Reduced Density ◽  
Reduced Temperature

Abstract The vapour-liquid coexistence collapse in the reduced temperature, Tr=T/Tc, reduced density, ρr= ρ/ρc, plane is known as a principle of corresponding states, and Noro and Frenkel have extended it for pair potentials of variable range. Here, we provide a theoretical basis supporting this extension and show that it can also be applied to short-range pair potentials where both repulsive and attractive parts can be anisotropic. We observe that the binodals of oblate hard ellipsoids for a given aspect ratio (κ=1/3) with varying short-range square-well interactions collapse into a single master curve in the Δ B*2--ρr plane, where Δ B*2= (B2(T)-B*2(Tc))/v0, B2 is the second virial coefficient, and v0 is the volume of the hard body. This finding is confirmed by both REMC simulation and second virial perturbation theory for varying square-well shells, mimicking uniform, equator, and pole attractions. Our simulation results reveal that the extended law of corresponding states is not related to the local structure of the fluid.

The analytical evaluation of the first-order density cluster integral in the radial distribution function

1970 ◽  
Vol 320 (1542) ◽  
pp. 323-348 ◽  
Keyword(s):  
Hard Sphere ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Scattering Data ◽  
X Ray ◽  
Lennard Jones ◽  
X Ray Scattering ◽  
Cluster Integrals ◽  
Square Well ◽  
Three Body

It is shown how to evaluate the two-body, and three-body cluster integrals, ɳ 3 , ɳ * 3 , β 3 , β * 3 (equations (1.1) to (1.4)) for the hard-sphere, square-well and Lennard-Jones ( v :½ v ) potentials; the three-body potential used is the dipole-dipole-dipole potential of Axilrod & Teller. Explicit expressions are presented for the integrals ɳ * 3 , β * 3 using the above potentials; in the case of the first integral, its values for both small and large values of the separation distance are also given, for the Lennard-Jones ( v :½ v ) potential. Similar considerations have been carried out for ɳ 3 and β 3 , except that explicit expressions for the hard-sphere, and square-well potentials are not given, since these had been done before by other authors. The intermediate expressions for the four cluster integrals, are in terms of single integrals, and such expressions are valid for any continuous potential. Numerical results based on some of the expressions in this paper are compared with the results of numerical evaluation of the above integrals by other authors, and the agreement is seen to be good. Making use of the Mikolaj-Pings relation, the above results are used to obtain relationships between the second virial coefficient, and X-ray scattering data, as well as a means of deducing the pair potential at large separations, directly from a knowledge of X-ray scattering data, and the second virial coefficient.

Second Virial Coefficient Calculations for Square-Well Chain Molecules

1994 ◽  
Vol 27 (10) ◽  
pp. 2744-2756 ◽  
Author(s):  
John M. Wichert ◽  
Carol K. Hall
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Chain Molecules ◽  
Square Well

The second virial coefficients of mixed organic vapours

1952 ◽  
Vol 210 (1103) ◽  
pp. 557-564 ◽  
Keyword(s):  
Hydrogen Bonding ◽  
Binary Mixtures ◽  
Diethyl Ether ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Virial Coefficients ◽  
Corresponding States ◽  
Linear Variation ◽  
Second Virial Coefficients

The second virial coefficients of some binary mixtures of organic vapours have been measured at temperatures between 50 and 120° C. Mixtures of n -hexane with chloroform and of n -hexane with diethyl ether show a linear variation of second virial coefficient with composition. This is shown to be in accordance with prediction from the principle of corresponding states. Mixtures of chloroform with diethyl ether show a linear variation at 120° C, but pronounced curvature at lower temperatures. This is interpreted quantitatively as being due to association by hydrogen bonding with an energy of 6020 cal/mole.

scholarly journals Determination of Second Virial Coefficient of Gold by a Modified Berthelot Equation of State

2020 ◽  
Vol 1 (4) ◽  
pp. 175-180
Author(s):  
M. Sivakumar ◽  
R. Balasubramanian
Keyword(s):  
Equation Of State ◽  
Critical Point ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Similarity Parameter ◽  
Thermodynamic Similarity ◽  
Van Der Waals Equation ◽  
Wide Range ◽  
Law Of Corresponding States

The present study aims to determine the second virial coefficient of gold over a wide range of temperatures from the boiling point to the critical point. A three - parameter modified Berthelot equation of state has been employed to determine the second virial coefficient of gold. The parameters of the equation of state are determined through the critical - point parameters of gold. The temperature -dependence of the second virial coefficient of gold has been investigated. The obtained results are compared with that of the van der Waals equation of state, Berthelot equation of state, Tsonopoulus correlation, and McGlashan correlation. The results of this work agree well with that of other correlations in the vicinity of the critical point. It is also established that gold obeys the single - parameter law of corresponding states. And, the new parameter introduced in the attractive term of the equation of state is found to be a thermodynamic similarity parameter. Doi: 10.28991/HEF-2020-01-04-02 Full Text: PDF

STICKY SPHERES IN QUANTUM MECHANICS

1994 ◽  
Vol 06 (05a) ◽  
pp. 947-975 ◽  
Author(s):  
M. D. PENROSE ◽  
O. PENROSE ◽  
G. STELL
Keyword(s):  
Quantum Mechanics ◽  
Brownian Motion ◽  
Gaseous Phase ◽  
Virial Coefficient ◽  
Hard Spheres ◽  
Second Virial Coefficient ◽  
Dimensional System ◽  
3 Dimensional ◽  
Fermi Statistics ◽  
Square Well

For a 3-dimensional system of hard spheres of diameter D and mass m with an added attractive square-well two-body interaction of width a and depth ε, let BD, a denote the quantum second virial coefficient. Let BD denote the quantum second virial coefficient for hard spheres of diameter D without the added attractive interaction. We show that in the limit a → 0 at constant α: = ℰma2/(2ħ2) with α < π2/8, [Formula: see text] The result is true equally for Boltzmann, Bose and Fermi statistics. The method of proof uses the mathematics of Brownian motion. For α > π2/8, we argue that the gaseous phase disappears in the limit a → 0, so that the second virial coefficient becomes irrelevant.

The extended law of corresponding states when attractions meet repulsions

Soft Matter ◽  
2018 ◽  
Vol 14 (19) ◽  
pp. 3704-3715 ◽  
Author(s):  
K. van Gruijthuijsen ◽  
M. Obiols-Rabasa ◽  
P. Schurtenberger ◽  
W. G. Bouwman ◽  
A. Stradner
Keyword(s):  
Phase Diagram ◽  
Interaction Potential ◽  
Short Range ◽  
Corresponding States ◽  
Law Of Corresponding States

Rescaling of the phase diagram via B2* (Baxter stickiness) and the interaction potential for short-range attractive colloids with shorter- and longer-ranged repulsions.

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