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ABSTRACT
The residual entropy is calculated by means of thermodynamic Monte Carlo simulation for several model fluids – hard spheres, one-, two-, and three-centre Lennard-Jones particles, as well as one-centre Mie particles with various repulsion exponents. It turns out that, over wide ranges of density and temperature, the residual molar entropy is an approximately linear function of the reciprocal mean Monte Carlo displacement parameter. The parameters of this function depend on the interaction potential and the acceptance ratio only. Therefore, this function can be used to estimate entropies or free energies with a negligible computational effort.
Acknowledgments
The authors thank the computer center of the University of Cologne (RRZK) for providing CPU time on the CHEOPS supercomputer (funded by Deutsche Forschungsgemeinschaft) as well as for technical support. Simin Yazdi Nezhad gratefully acknowledges a scholarship of the German Academic Exchange Service (DAAD).
Disclosure statement
No potential conflict of interest was reported by the authors.
| Symbols | ||
| A | = | Helmholtz energy |
| a | = | MC displacement acceptance ratio |
| B2 | = | second virial coefficient |
| b | = | bond length |
| G | = | Gibbs energy |
| H | = | enthalpy |
| kB | = | Boltzmann's constant |
| l | = | characteristic length of a radial free-space distribution function |
| m | = | attraction exponent of the Mie potential |
| N | = | number of molecules, simulation ensemble size |
| NA | = | Avogadro's constant |
| n | = | repulsion exponent of the Mie potential |
| P+, P− | = | probability of acceptance of a particle addition or removal during an MC simulation |
| p | = | pressure |
| R | = | universal gas constant |
| r | = | distance |
| = | location vector of a molecule | |
| S | = | entropy |
| T | = | temperature |
| U | = | internal energy |
| u(r) | = | pair potential |
| = | unit vector with random orientation | |
| V | = | volume |
| y(λ) | = | radial free-space distribution function |
| Z | = | compression factor, Z = pVm/(RT) |
| ϵ | = | energy parameter of the Lennard-Jones and Mie pair potentials |
| λ | = | MC displacement parameter |
| λ a | = | mean displacement parameter at constant acceptance ratio a |
| μ | = | chemical potential |
| ξ | = | random number, 0 ≤ ξ ≤ 1 |
| ρ | = | number density, ρ = N/V |
| σ | = | size parameter of the hard-sphere, Lennard-Jones, and Mie potentials |
| Superscripts | ||
| r | = | residual property |
| Subscripts | ||
| i | = | belonging to molecule i |
| m | = | molar property |
| ∞ | = | value at infinite distance |
Notes
1. In some MC applications, a unit cube is used, but this does not affect the arguments presented here.
2. The authors use the term ‘excess entropy’ (SEX), but an excess property is defined as difference between a real mixture and an ideal mixture at the same temperature and pressure. The difference between a real fluid and an ideal gas at the same density should be called a residual property.
3. Strictly speaking, the ln a term should be replaced by a more complicated expression, but this is of no consequence here.