Closed systems with compositional change

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<center><math>E = E(S,V,{n_1},{n_2},...{n_N})\qquad \qquad(1) </math></center>
<center><math>E = E(S,V,{n_1},{n_2},...{n_N})\qquad \qquad(1) </math></center>
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Expanding eq. (1) in terms of each independent variable, while holding all other properties constant, produces the following:
Expanding eq. (1) in terms of each independent variable, while holding all other properties constant, produces the following:
<center><math>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) </math></center>
<center><math>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) </math></center>
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Where <math>j ≠ i</math>.  The first two terms on the right side of eq. (2) refer to conditions of constant composition, as represented by eq. <math>dE = TdS - pdV</math> from [[Maxwell Relations]]. Comparing eqs. (2) and <math>dE = TdS - pdV</math>, the coefficients of the first two terms in eq. (2) are  
Where <math>j ≠ i</math>.  The first two terms on the right side of eq. (2) refer to conditions of constant composition, as represented by eq. <math>dE = TdS - pdV</math> from [[Maxwell Relations]]. Comparing eqs. (2) and <math>dE = TdS - pdV</math>, the coefficients of the first two terms in eq. (2) are  
<center><math>{\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = T\qquad \qquad(3) </math></center>
<center><math>{\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = T\qquad \qquad(3) </math></center>
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<center><math>{\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} =  - p\qquad \qquad(4) </math></center>
<center><math>{\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} =  - p\qquad \qquad(4) </math></center>
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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
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
<center><math>{\mu _i} = {\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)_{S,V,{n_{j \ne i}}}}\qquad \qquad(5) </math></center>
<center><math>{\mu _i} = {\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)_{S,V,{n_{j \ne i}}}}\qquad \qquad(5) </math></center>
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Therefore, the above expanded fundamental equation for a multicomponent system, as seen in eq. (2), can be rewritten as
Therefore, the above expanded fundamental equation for a multicomponent system, as seen in eq. (2), can be rewritten as
<center><math>dE = TdS - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(6) </math></center>
<center><math>dE = TdS - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(6) </math></center>
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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 <math>(H = E + pV)</math>, Helmholtz free energy <math>(F = E - TS)</math> and Gibbs free energy <math>(G = E - TS + pV)</math>, i.e.,  
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 <math>(H = E + pV)</math>, Helmholtz free energy <math>(F = E - TS)</math> and Gibbs free energy <math>(G = E - TS + pV)</math>, i.e.,  
<center><math>dH = Vdp + TdS + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(7) </math></center>
<center><math>dH = Vdp + TdS + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(7) </math></center>
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<center><math>dF =  - SdT - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(8) </math></center>
<center><math>dF =  - SdT - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(8) </math></center>
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<center><math>dG = Vdp - SdT + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(9) </math></center>
<center><math>dG = Vdp - SdT + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(9) </math></center>
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It is therefore readily determined from eqs. (7) – (9) that other expressions of chemical equilibrium exist; these are
It is therefore readily determined from eqs. (7) – (9) that other expressions of chemical equilibrium exist; these are
<center><math>{\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) </math></center>
<center><math>{\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) </math></center>
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In addition, the following expressions for the fundamental thermodynamic properties are valid:
In addition, the following expressions for the fundamental thermodynamic properties are valid:
<center><math>T = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = {\left( {\frac{{\partial H}}{{\partial S}}} \right)_{p,{n_i}}}\qquad \qquad(11) </math></center>
<center><math>T = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = {\left( {\frac{{\partial H}}{{\partial S}}} \right)_{p,{n_i}}}\qquad \qquad(11) </math></center>
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<center><math> - p = {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,{n_i}}}\qquad \qquad(12) </math></center>
<center><math> - p = {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,{n_i}}}\qquad \qquad(12) </math></center>
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<center><math>V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial G}}{{\partial p}}} \right)_{T,{n_i}}}\qquad \qquad(13) </math></center>
<center><math>V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial G}}{{\partial p}}} \right)_{T,{n_i}}}\qquad \qquad(13) </math></center>
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which will be very useful in stability analysis in the next subsection.  
which will be very useful in stability analysis in the next subsection.  

Revision as of 00:10, 5 February 2010

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.

References

Further Reading

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