Closed systems with compositional change
From Thermal-FluidsPedia
The internal energy in eq. dE = TdS − pdV from Maxwell Relations is a function of only two independent variables, E = E(S,V), when dealing with a single phase, single-component system. When a compositional change is possible, i.e., for multicomponent systems, internal energy must also be a function of the number of moles of each of the N components:
Expanding eq. (1) in terms of each independent variable, while holding all other properties constant, produces the following:
Where ji. The first two terms on the right side of eq. (2) refer to conditions of constant composition, as represented by eq. dE = TdS − pdV from Maxwell Relations. Comparing eqs. (2) and dE = TdS − pdV, the coefficients of the first two terms in eq. (2) are
The third term on the right-hand side of eq. (2) corresponds to the effects of the presence of multiple components. The chemical potential can be defined as
Therefore, the above expanded fundamental equation for a multicomponent system, as seen in eq. (2), can be rewritten as
which is known as the internal energy representation of the fundamental thermodynamic equation of multi-component systems. Other representations can be directly obtained from eq. (6) by using the definitions of enthalpy (H = E + pV), Helmholtz free energy (F = E − TS) and Gibbs free energy (G = E − TS + pV), i.e.,
It is therefore readily determined from eqs. (7) – (9) that other expressions of chemical equilibrium exist; these are
In addition, the following expressions for the fundamental thermodynamic properties are valid:
which will be very useful in stability analysis in the next subsection.
References
Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.