Closed systems with compositional change

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The internal energy in eq. dE = TdSpdV 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:

E = E(S,V,{n_1},{n_2},...{n_N})\qquad \qquad(1)

Expanding eq. (1) 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(2)

Where ji. The first two terms on the right side of eq. (2) refer to conditions of constant composition, as represented by eq. dE = TdSpdV from Maxwell Relations. Comparing eqs. (2) and dE = TdSpdV, the coefficients of the first two terms in eq. (2) are

{\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = T\qquad \qquad(3)

{\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} =  - p\qquad \qquad(4)

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

{\mu _i} = {\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)_{S,V,{n_{j \ne i}}}}\qquad \qquad(5)

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

dE = TdS - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(6)

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 = 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(7)

dF =  - SdT - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(8)

dG = Vdp - SdT + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(9)

It is therefore readily determined from eqs. (7) – (9) 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(10)

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(11)

 - p = {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,{n_i}}}\qquad \qquad(12)

V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial G}}{{\partial p}}} \right)_{T,{n_i}}}\qquad \qquad(13)

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


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