Excess chemical potential
Encyclopedia
The excess chemical potential is defined as the difference between the chemical potential
of a given species and that of an ideal gas
under the same conditions (in particular, at the same pressure
, temperature
, and composition
).
Chemical potential of pure fluid can be estimated by the Widom insertion method
.
For a system of diameter
and volume 
, at constant temperature :
, the classical partition function
s is a scaled coordinate.
combining the above equation with the definition of chemical potential:
we get the chemical potential of a sufficient large system is :
wherein the chemical potential of ideal gas can be evaluated analytically.
Now let's focus on
, since the potential energy of an N+1 particle system can be separated into the potential energy of an N particle system
and the potential of the excess particle interacts with the N particle system, there is
and
thus far we converted the excess chemical potential into an ensemble average, and the integral in the above equation can be sampled by brute force Monte Carlo method
.
The calculating of excess chemical potential is not only limited to homogeneous system, it has also been extended to inhomogeneous system by Widom insertion method
, or other ensembles
such as NPT, NVE
.
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
of a given species and that of an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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.At normal conditions such as...
under the same conditions (in particular, at the same pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
, temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, and composition
Composition
Composition may refer to:* Composition , in which one assumes that a whole has a property solely because its various parts have that property* Compounding is also known as composition in linguistic literature* in computer science...
).
Chemical potential of pure fluid can be estimated by the Widom insertion method
Widom insertion method
The Widom Insertion Method is a statistical thermodynamic approach to the calculation of material and mixture properties. It is named for Benjamin Widom, who derived it in 1963. In general, there are two theoretical approaches to determining the statistical mechanical properties of materials...
.
For a system of diameter
and volume , at constant temperature :, the classical partition functionPartition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
s is a scaled coordinate.
combining the above equation with the definition of chemical potential:
we get the chemical potential of a sufficient large system is :
wherein the chemical potential of ideal gas can be evaluated analytically.
Now let's focus on
, since the potential energy of an N+1 particle system can be separated into the potential energy of an N particle systemParticle system
The term particle system refers to a computer graphics technique to simulate certain fuzzy phenomena, which are otherwise very hard to reproduce with conventional rendering techniques...
and the potential of the excess particle interacts with the N particle system, there is
and
thus far we converted the excess chemical potential into an ensemble average, and the integral in the above equation can be sampled by brute force Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
.
The calculating of excess chemical potential is not only limited to homogeneous system, it has also been extended to inhomogeneous system by Widom insertion method
Widom insertion method
The Widom Insertion Method is a statistical thermodynamic approach to the calculation of material and mixture properties. It is named for Benjamin Widom, who derived it in 1963. In general, there are two theoretical approaches to determining the statistical mechanical properties of materials...
, or other ensembles
Statistical ensemble (mathematical physics)
In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1878, an ensemble is an idealization consisting of a large number of mental copies of a system, considered all at once, each of which represents a possible state that the real...
such as NPT, NVE
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
.