thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...

, the excess chemical potential is defined as the difference between the

chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...

of a given

species In biology, a species is the basic unit of classification and a taxonomic rank of an organism, as well as a unit of biodiversity. A species is often defined as the largest group of organisms in which any two individuals of the appropriate s ...

and that of an

ideal gas 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 a ...

under the same conditions (in particular, at the same

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...

, and

composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...

). The chemical potential of a particle species

i

is therefore given by an ideal part and an excess part. :

\mu_i=\mu_i^\text+\mu_i^\text

Chemical potential of a pure fluid can be estimated by the Widom insertion method.

Derivation and Measurement

For a system of diameter and volume , at constant temperature , the classical canonical partition function :

Q(N,V,T)=\frac\int_^\ldots\int_^ds^\exp \beta U(s^;L) /math>
with  a scaled coordinate, the free energy is given by:
: F(N,V,T)= -k_T\ln Q=-k_T\ln\left(\frac\right)-k_T \ln= : =F_(N,V,T)+F_(N,V,T) Combining the above equation with the definition of chemical potential,

: \mu_= \left(\frac\right)_=\left(\frac\right)_, we get the chemical potential of a sufficiently large system from (and the fact that the smallest allowed change in the particle number is \Delta N=1)

: \mu= \frac\overset-k_T\ln\left(\frac\right) - k_T \ln=\mu_(\rho) + \mu_wherein the chemical potential of an ideal gas can be evaluated analytically. 
Now let's focus on  since the potential energy of an  system can be separated into the potential energy of an  system and the potential of the excess particle interacting with the  system, that is,

: \Delta U\equiv U(s^) - U(s^) and

: \mu_= -k_T \ln \int ds_ \langle \exp(-\beta\Delta U)\rangle_. Thus far we converted the excess chemical potential into an ensemble average, and the integral in the above equation can be sampled by the brute force

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...

. The calculating of excess chemical potential is not limited to homogeneous systems, but has also been extended to inhomogeneous systems by the Widom insertion method, or other ensembles such as NPT and NVE.

References

* Note: the equations and presentation in this article are drawn fro
Excess Chemical Potential via the Widom Method
!-- not a copyright violation--see talk page and Wikipedia talk:Copyright problems#Copyright violations in Excess chemical potential --> {{DEFAULTSORT:Excess Chemical Potential Potentials Thermodynamics Chemical thermodynamics ro:Mărimi molare de exces

Derivation and Measurement

See also

References