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= Chemical Potential: Definition and Mathematical Representation =

== Introduction ==
The '''chemical potential''' is a fundamental thermodynamic quantity that plays a crucial role in understanding how particles and energy distribute in physical systems. It is particularly significant in the study of phase equilibria, chemical reactions, and processes involving the transfer of matter.

In essence, the chemical potential represents the change in a system's internal energy (or other thermodynamic potentials) when an additional amount of a substance is introduced, keeping temperature, pressure, and the number of particles of other substances constant.

== Definition ==
In thermodynamics, the chemical potential ''μ'' of a component in a system is defined as the partial derivative of the system's Gibbs free energy ''G'' with respect to the number of moles ''nᵢ'' of the component, at constant temperature ''T'' and pressure ''P'':

 :μᵢ = (∂G/∂nᵢ) at constant T, P, and nⱼ (j ≠ i)

Alternatively, depending on the thermodynamic potential being used, chemical potential can also be defined using internal energy ''U'', enthalpy ''H'', or Helmholtz free energy ''A''. For example, in terms of internal energy:

    μᵢ = (∂U/∂nᵢ) at constant S, V, and nⱼ (j ≠ i)

Here, 
* ''G'' is the Gibbs free energy,
* ''U'' is the internal energy,
* ''S'' is the entropy,
* ''V'' is the volume,
* ''nᵢ'' is the number of moles of component ''i'',
* The subscript ''j ≠ i'' indicates that the number of moles of all other species is held constant.

== Physical Interpretation ==
The chemical potential can be thought of as the "escaping tendency" of a species from a phase or a system. If two phases or systems are in equilibrium, the chemical potential of each component must be the same in both:

    μᵢ^(1) = μᵢ^(2)


This condition ensures no net flow of particles between the phases, indicating chemical equilibrium.

== Importance in Multicomponent Systems ==
In multicomponent thermodynamic systems, the chemical potential governs:
* The direction of chemical reactions.
* The distribution of substances between phases.
* The flow of species in diffusion and osmosis.

It is a central concept in the formulation of equilibrium criteria and in the application of the Gibbs phase rule.

== Related Concepts ==
* [[Gibbs Free Energy]]
* [[Thermodynamic Potential]]
* [[Phase Equilibrium]]
* [[Entropy]]
* [[Chemical Thermodynamics]]

== References ==
* Callen, H. B. (1985). ''Thermodynamics and an Introduction to Thermostatistics''. Wiley.
* Atkins, P., & de Paula, J. (2010). ''Atkins' Physical Chemistry''. Oxford University Press.

== See Also ==
* [[Fugacity]]
* [[Partial Molar Properties]]
* [[Equilibrium Constant]]