Internal Energy - Internal Energy of Multi-component Systems

Internal Energy of Multi-component Systems

In addition to including the entropy S and volume V terms in the internal energy, a system is often described also in terms of the number of particles or chemical species it contains:

where the terms Nj are the numbers of constituents of type j in the system. The internal energy is an extensive function of the extensive variables variables S, V, and the set of components, the internal energy may be written as a linear homogeneous function of first degree:

U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots ) = \alpha U(S,V,N_{1},N_{2},\ldots)\,

where α is a factor describing the growth of the system. The differential internal energy may be written as

where the coefficients are the chemical potentials for the components of type i in the system. The chemical potentials are defined as the partial derivatives of the energy with respect to the variations in composition:

As conjugate variables to the composition, the chemical potentials are intensive properties, intrinsically characteristic of the system, and not dependent on its extent. Because of the extensive nature of U and its variables, the differential dU may be integrated and yields an expression for the internal energy:

.

The sum over the composition of the system is the Gibbs energy:

that arises from changing the composition of the system at constant temperature and pressure. For a single component system, the chemical potential equals the Gibbs energy per amount of substance, i.e. particles or moles according to the original definition of the unit for .

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