# 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> | ||

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(111) | (111) | ||

- | 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> |

## Revision as of 06:56, 3 February 2010

The internal energy in eq. *d**E* = *T**d**S* − *p**d**V* 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:

(110)

Expanding eq. (1) in terms of each independent variable, while holding all other properties constant, produces the following:

(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. *d**E* = *T**d**S* − *p**d**V* from Maxwell Relations. Comparing eqs. (2) and *d**E* = *T**d**S* − *p**d**V*, the coefficients of the first two terms in eq. (2) are

(112)

(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

(114)

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

(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* + *p**V*), Helmholtz free energy (*F* = *E* − *T**S*) and Gibbs free energy (*G* = *E* − *T**S* + *p**V*), i.e.,

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(117)

(118)

It is therefore readily determined from eqs. (7) – (9) that other expressions of chemical equilibrium exist; these are

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

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(122)

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