Noro–Frenkel law of corresponding states

Last updated • 2 min readFrom Wikipedia, The Free Encyclopedia

The Noro–Frenkel law of corresponding states is an equation in thermodynamics that describes the critical temperature of the liquid-gas transition T as a function of the range of the attractive potential R. It states that all short-ranged spherically symmetric pair-wise additive attractive potentials are characterised by the same thermodynamics properties, if compared at the same reduced density and second virial coefficient [1]

Contents

Description

Johannes Diderik van der Waals's law of corresponding states expresses the fact that there are basic similarities in the thermodynamic properties of all simple gases. Its essential feature is that if we scale the thermodynamic variables that describe an equation of state (temperature, pressure, and volume) with respect to their values at the liquid-gas critical point, all simple fluids obey the same reduced equation of state.

Massimo G. Noro and Daan Frenkel formulated an extended law of corresponding states that predicts the phase behaviour of short-ranged potentials on the basis of the effective pair potential alone – extending the validity of the van der Waals law to systems interacting through pair potentials with different functional forms.

The Noro–Frenkel law suggests to condensate the three quantities which are expected to play a role in the thermodynamics behavior of a system (hard-core size, interaction energy and range) into a combination of only two quantities: an effective hard core diameter and the reduced second virial coefficient. Noro and Frenkel suggested to determine the effective hard core diameter following the expression suggested by Barker [2] based on the separation of the potential into attractive Vatt and repulsive Vrep parts used in the Weeks–Chandler– Andersen method. [3] The reduced second virial coefficient, i.e., the second virial coefficient B2 divided by the second virial coefficient of hard spheres with the effective diameter can be calculated (or experimentally measured) once the potential is known. B2 is defined as

Applications

The Noro–Frenkel law is particularly useful for the description of colloidal and globular protein solutions, [4] for which the range of the potential is indeed significantly smaller than the particle size. For these systems the thermodynamic properties can be re-written as a function of only two parameters, the reduced density (using the effective diameter as length scale) and the reduced second-virial coefficient B*
2
. The gas-liquid critical point of all systems satisfying the extended law of corresponding states are characterized by same values of B*
2
at the critical point.

The Noro-Frenkel law can be generalized to particles with limited valency (i.e. to non spherical interactions). [5] Particles interacting with different potential ranges but identical valence behave again according to the generalized law, but with a different value for each valence of B*
2
at the critical point.

See also

Related Research Articles

<span class="mw-page-title-main">Equation of state</span> An equation describing the state of matter under a given set of physical conditions

In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most modern equations of state are formulated in the Helmholtz free energy. Equations of state are useful in describing the properties of pure substances and mixtures in liquids, gases, and solid states as well as the state of matter in the interior of stars.

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in the fields of physics, biology, chemistry, neuroscience, computer science, information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.

Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. In consequence, the relative lowering of vapor pressure of a dilute solution of nonvolatile solute is equal to the mole fraction of solute in the solution.

<span class="mw-page-title-main">Timeline of thermodynamics</span>

A timeline of events in the history of thermodynamics.

<span class="mw-page-title-main">Ideal gas</span> Mathematical model which approximates the behavior of real gases

An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.

<span class="mw-page-title-main">Van der Waals equation</span> Gas equation of state which accounts for non-ideal gas behavior

The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them. As a result the equation is able to model the phase change, liquid vapor. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is: where is pressure, is temperature, and is molar volume, is the Avogadro constant, is the volume, and is the number of molecules. In addition is the universal gas constant, is the Boltzmann constant, and and are experimentally determinable, substance-specific constants.

Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

<span class="mw-page-title-main">Lennard-Jones potential</span> Model of intermolecular interactions

In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential is an intermolecular pair potential. Out of all the intermolecular potentials, the Lennard-Jones potential is probably the one that has been the most extensively studied. It is considered an archetype model for simple yet realistic intermolecular interactions. The Lennard-Jones potential is often used as a building block in molecular models for more complex substances. Many studies of the idealized "Lennard-Jones substance" use the potential to understand the physical nature of matter.

The virial expansion is a model of thermodynamic equations of state. It expresses the pressure P of a gas in local equilibrium as a power series of the density. This equation may be represented in terms of the compressibility factor, Z, as This equation was first proposed by Kamerlingh Onnes. The terms A, B, and C represent the virial coefficients. The leading coefficient A is defined as the constant value of 1, which ensures that the equation reduces to the ideal gas expression as the gas density approaches zero.

Virial coefficients appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potential between the particles and in general depend on the temperature. The second virial coefficient depends only on the pair interaction between the particles, the third depends on 2- and non-additive 3-body interactions, and so on.

In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of chemical equilibrium. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas.

<span class="mw-page-title-main">Thermodynamic equations</span> Equations in thermodynamics

Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process. Thermodynamics is based on a fundamental set of postulates, that became the laws of thermodynamics.

<span class="mw-page-title-main">Critical point (thermodynamics)</span> Temperature and pressure point where phase boundaries disappear

In thermodynamics, a critical point is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas comes into a supercritical phase, and so cannot be liquefied by pressure alone. At the critical point, defined by a critical temperatureTc and a critical pressurepc, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures, and the ferromagnet–paramagnet transition in the absence of an external magnetic field.

<span class="mw-page-title-main">Compressibility factor</span> Correction factor which describes the deviation of a real gas from ideal gas behavior

In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure. It is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behaviour. In general, deviation from ideal behaviour becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure. Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound-specific empirical constants as input. For a gas that is a mixture of two or more pure gases, the gas composition must be known before compressibility can be calculated.
Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot as a function of pressure at constant temperature.

<span class="mw-page-title-main">Theorem of corresponding states</span>

According to van der Waals, the theorem of corresponding states indicates that all fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same degree.

<span class="mw-page-title-main">Table of thermodynamic equations</span> Thermodynamics

Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows:

<span class="mw-page-title-main">Diffusion</span> Transport of dissolved species from the highest to the lowest concentration region

Diffusion is the net movement of anything generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing.

Pitzer equations are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water. They were first described by physical chemist Kenneth Pitzer. The parameters of the Pitzer equations are linear combinations of parameters, of a virial expansion of the excess Gibbs free energy, which characterise interactions amongst ions and solvent. The derivation is thermodynamically rigorous at a given level of expansion. The parameters may be derived from various experimental data such as the osmotic coefficient, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion activity coefficients and water activities in solutions of high ionic strength for which the Debye–Hückel theory is no longer adequate. They are more rigorous than the equations of specific ion interaction theory, but Pitzer parameters are more difficult to determine experimentally than SIT parameters.

Supercritical liquid–gas boundaries are lines in the pressure-temperature (pT) diagram that delimit more liquid-like and more gas-like states of a supercritical fluid. They comprise the Fisher–Widom line, the Widom line, and the Frenkel line.

In thermodynamics, the Frenkel line is a proposed boundary on the phase diagram of a supercritical fluid, separating regions of qualitatively different behavior. Fluids on opposite sides of the line have been described as "liquidlike" or "gaslike", and exhibit different behaviors in terms of oscillation, excitation modes, and diffusion.

References

  1. M.G. Noro & D. Frenkel (2000). "Extended corresponding-states behavior for particles with variable range attractions". Journal of Chemical Physics . 113 (8): 2941. arXiv: cond-mat/0004033 . Bibcode:2000JChPh.113.2941N. doi:10.1063/1.1288684. hdl:1874/10793. S2CID   98751659.
  2. J.A. Barker & D. Henderson (1976). "What is "liquid"? Understanding the states of matter". Reviews of Modern Physics . 48 (4): 587. Bibcode:1976RvMP...48..587B. doi:10.1103/RevModPhys.48.587.
  3. H.C. Andersen; J.D. Weeks & D. Chandler (1971). "Relationship between the Hard-Sphere Fluid and Fluids with Realistic Repulsive Forces". Physical Review A . 4 (4): 1579. Bibcode:1971PhRvA...4.1597A. doi:10.1103/PhysRevA.4.1597. Archived from the original on September 22, 2017.
  4. R.P. Sear (1999). "Phase behavior of a simple model of globular proteins". Journal of Chemical Physics . 111 (10): 4800–4806. arXiv: cond-mat/9904426 . Bibcode:1999JChPh.111.4800S. doi:10.1063/1.479243. S2CID   15005765.
  5. G. Foffi & F. Sciortino (2007). "On the possibility of extending the Noro-Frenkel generalized law of correspondent states to nonisotropic patchy interactions". Journal of Physical Chemistry B . 111 (33): 9702–5. arXiv: 0707.3114 . doi:10.1021/jp074253r. PMID   17672500. S2CID   6757031.