# Closed systems with compositional change

(Difference between revisions)
 Revision as of 06:56, 3 February 2010 (view source)← Older edit Revision as of 00:10, 5 February 2010 (view source)Newer edit → Line 2: Line 2:
$E = E(S,V,{n_1},{n_2},...{n_N})\qquad \qquad(1)$
$E = E(S,V,{n_1},{n_2},...{n_N})\qquad \qquad(1)$
- (110) + 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:
$dE = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}}dS + {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}}dV + \sum\limits_{i = 1}^N {{{\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)}_{S,V,{n_{j \ne i}}}}d{n_i}} \qquad \qquad(2)$
$dE = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}}dS + {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}}dV + \sum\limits_{i = 1}^N {{{\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)}_{S,V,{n_{j \ne i}}}}d{n_i}} \qquad \qquad(2)$
- (111) + Where $j ≠ i$.  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 Where $j ≠ i$.  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
${\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = T\qquad \qquad(3)$
${\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = T\qquad \qquad(3)$
- (112) +
${\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = - p\qquad \qquad(4)$
${\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = - p\qquad \qquad(4)$
- (113) + 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
${\mu _i} = {\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)_{S,V,{n_{j \ne i}}}}\qquad \qquad(5)$
${\mu _i} = {\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)_{S,V,{n_{j \ne i}}}}\qquad \qquad(5)$
- (114) + 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
$dE = TdS - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(6)$
$dE = TdS - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(6)$
- (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. (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., 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.,
$dH = Vdp + TdS + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(7)$
$dH = Vdp + TdS + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(7)$
- (116) +
$dF = - SdT - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(8)$
$dF = - SdT - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(8)$
- (117) +
$dG = Vdp - SdT + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(9)$
$dG = Vdp - SdT + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}} \qquad \qquad(9)$
- (118) + 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
${\mu _i} = {\left( {\frac{{\partial H}}{{\partial {n_i}}}} \right)_{p,S,{n_{j \ne i}}}} = {\left( {\frac{{\partial F}}{{\partial {n_i}}}} \right)_{T,V,{n_{j \ne i}}}} = {\left( {\frac{{\partial G}}{{\partial {n_i}}}} \right)_{T,p,{n_{j \ne i}}}}\qquad \qquad(10)$
${\mu _i} = {\left( {\frac{{\partial H}}{{\partial {n_i}}}} \right)_{p,S,{n_{j \ne i}}}} = {\left( {\frac{{\partial F}}{{\partial {n_i}}}} \right)_{T,V,{n_{j \ne i}}}} = {\left( {\frac{{\partial G}}{{\partial {n_i}}}} \right)_{T,p,{n_{j \ne i}}}}\qquad \qquad(10)$
- (119) + In addition, the following expressions for the fundamental thermodynamic properties are valid: In addition, the following expressions for the fundamental thermodynamic properties are valid:
$T = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = {\left( {\frac{{\partial H}}{{\partial S}}} \right)_{p,{n_i}}}\qquad \qquad(11)$
$T = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,{n_i}}} = {\left( {\frac{{\partial H}}{{\partial S}}} \right)_{p,{n_i}}}\qquad \qquad(11)$
- (120) +
$- p = {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,{n_i}}}\qquad \qquad(12)$
$- p = {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,{n_i}}}\qquad \qquad(12)$
- (121) +
$V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial G}}{{\partial p}}} \right)_{T,{n_i}}}\qquad \qquad(13)$
$V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_{S,{n_i}}} = {\left( {\frac{{\partial G}}{{\partial p}}} \right)_{T,{n_i}}}\qquad \qquad(13)$
- (122) + 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.