Closed systems with compositional change

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6.3 Closed Systems with Compositional Change

The internal energy in eq. (98) 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:

E = E(S,V,{n_1},{n_2},...{n_N})\qquad \qquad( )

(110)

Expanding eq. (110) in terms of each independent variable, while holding all other properties constant, produces the following:

dE = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}}dS + {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}}dV + \sum\limits_{i = 1}^N {{{\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)}_{S,V,{n_{j \ne i}}}}d{n_i}} \qquad \qquad( )

(111)

Where ji. The first two terms on the right side of eq. (111) refer to conditions of constant composition, as represented by eq. (98). Comparing eqs. (111) and (98), the coefficients of the first two terms in eq. (111) are

{\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = T\qquad \qquad( )

(112)

{\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} =  - p\qquad \qquad( )

(113)

The third term on the right-hand side of eq. (111) corresponds to the effects of the presence of multiple components. The chemical potential can be defined as

{\mu _i} = {\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)_{S,V,{n_{j \ne i}}}}\qquad \qquad( )

(114)

Therefore, the above expanded fundamental equation for a multicomponent system, as seen in eq. (111), can be rewritten as

dE = TdS - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad( )

(115)

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. (115) by using the definitions of enthalpy (H = E + pV), Helmholtz free energy (F = ETS) and Gibbs free energy (G = ETS + pV), i.e.,

dH = Vdp + TdS + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad( )

(116)

dF =  - SdT - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad( )

(117)

dG = Vdp - SdT + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad( )

(118)

It is therefore readily determined from eqs. (116) – (118) that other expressions of chemical equilibrium exist; these are

{\mu _i} = {\left( {\frac{{\partial H}}{{\partial {n_i}}}} \right)_{p,S,{n_{j \ne i}}}} = {\left( {\frac{{\partial F}}{{\partial {n_i}}}} \right)_{T,V,{n_{j \ne i}}}} = {\left( {\frac{{\partial G}}{{\partial {n_i}}}} \right)_{T,p,{n_{j \ne i}}}}\qquad \qquad( )

(119)

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

T = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = {\left( {\frac{{\partial H}}{{\partial S}}} \right)_{p,{n_i}}}\qquad \qquad( )

(120)

 - p = {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,{n_i}}}\qquad \qquad( )

(121)

V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial G}}{{\partial p}}} \right)_{T,{n_i}}}\qquad \qquad( )

(122)

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

References

Further Reading

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