# Closed systems with compositional change

### From Thermal-FluidsPedia

The internal energy in eq. *d**E* = *T**d**S* − *p**d**V* 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:

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Expanding eq. (1) in terms of each independent variable, while holding all other properties constant, produces the following:

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Where *j**i*. The first two terms on the right side of eq. (2) refer to conditions of constant composition, as represented by eq. *d**E* = *T**d**S* − *p**d**V* from Maxwell Relations. Comparing eqs. (2) and *d**E* = *T**d**S* − *p**d**V*, the coefficients of the first two terms in eq. (2) are

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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

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Therefore, the above expanded fundamental equation for a multicomponent system, as seen in eq. (2), can be rewritten as

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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* + *p**V*), Helmholtz free energy (*F* = *E* − *T**S*) and Gibbs free energy (*G* = *E* − *T**S* + *p**V*), i.e.,

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It is therefore readily determined from eqs. (7) – (9) that other expressions of chemical equilibrium exist; these are

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In addition, the following expressions for the fundamental thermodynamic properties are valid:

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which will be very useful in stability analysis in the next subsection.