Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup
a system in global equilibrium and those that belong to a system out of global equilibrium. The answer is yes; they differ in the nature of their fluctuations, as shown convincingly in the book by Ortiz and Sengers [112]. Equal time correlations around stationary states become long-range. In equilibrium, these correlations are short-range. Zielinskaet al. [7, 116] have discussed the description of fluctuations for an interface.
3.AAppendix: Partial molar properties
We give the fundamental thermodynamic relations that define partial molar quantities, among them the chemical potential. The various contributions to the chemical potential are given and standard states are defined. Symbols are defined in the symbol list.
3.A.1Homogeneous phases
The starting point for our derivations is the Gibbs equation
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(3.43) |
By integrating the Gibbs equation with constant composition, temperature, pressure and displacement field, we obtain
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(3.44) |
The Gibbs energy can then be defined:
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(3.45) |
By using again the Gibbs equation, we have
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(3.46) |
Two equivalent definitions are obtained from these equations for the chemical potential, the partial energy change that follows when we add a particular component to a system:
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(3.47) |
We note that μj = Gj. By using Maxwell relations for Eq. (3.46), we find the following expressions for the partial molar volume, entropy and polarization:
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(3.48) |
This results in the following expression for a change in the chemical potential
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(3.49) |
A frequently used combination of terms is
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(3.50) |
where we used the partial molar entropy. In order to find dμj,T, we differentiate dμj at constant temperature. Equivalent expressions for unpolarized systems are
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(3.51) |
The partial molar volume, the partial molar entropy and the partial molar polarization for the i-phase are, respectively:
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(3.52) |
Furthermore, we have
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(3.53) |
With these partial molar quantities, we can define the partial molar internal energy and enthalpy
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(3.54) |
The Uj and Hj are functions of p, T, ck and Eeq.
3.A.2The surface
The Gibbs equation for the surface is
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(3.55) |
In this expression, Us, Ss, Nis, Ps are extensive quantities, which are proportional to the surface area. These extensive quantities were obtained as excesses of the corresponding three-dimensional densities. The variables Ts, γ, μis, Dseq are intensive variables of the surface, indicated by superscript s. The temperature, surface tension, chemical potentials and the displacement field are intensive variables and have no excesses.
By integration, with constant surface tension, temperature, composition and displacement field, we obtain
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(3.56) |
The Gibbs energy can now be defined by
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(3.57) |
By using Gibbs equation again, we have
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(3.58) |
The chemical potential obtains two equivalent definitions from these relations:
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(3.59) |
where μjs = Gjs. The superscript indicating the surface has been dropped in the subscripts of the differentiation. By using Maxwell relations for the definition of Gs, we find the following expressions for the partial molar quantities:
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(3.60) |
This results in the following expression for a change in the chemical potential