Closed systems with compositional change
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| - | The internal energy in eq. | + | The internal energy in eq. <math>dE = TdS - pdV</math> from [[Maxwell Relations]] is a function of only two independent variables, <math>E = E(S,V),</math> 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 <math>N</math> components: |
<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> | ||
| - | + | ||
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> | ||
| - | + | ||
| - | 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. | + | 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> | ||
| - | + | ||
<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> | ||
| - | + | ||
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> | ||
| - | + | ||
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> | ||
| - | + | ||
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> | ||
| - | + | ||
<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> | ||
| - | + | ||
<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> | ||
| - | + | ||
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> | ||
| - | + | ||
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> | ||
| - | + | ||
<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> | ||
| - | + | ||
<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> | ||
| - | + | ||
which will be very useful in stability analysis in the next subsection. | which will be very useful in stability analysis in the next subsection. | ||
==References== | ==References== | ||
| + | Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA. | ||
==Further Reading== | ==Further Reading== | ||
==External Links== | ==External Links== | ||
Current revision as of 15:09, 20 July 2010
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.