dU ¼ TdS À pdV þ ldN
ð154AÞ
for a system interacting with a single component.
Equation (154A) can be generalized either for open systems interacting with
n components each with its corresponding membrane partition (with no bulk mass
flow in or out other than diffusion through membranes) or for closed systems (with
no mass exchange but with possible Q or/and W exchange) of multiple components
with chemical reaction (see below)
dU ¼ TdS À pdV þ
X n
i¼1
l i dN i
ð154BÞ
This is the fundamental differential for multicomponent systems, which is also
known under the name of Gibbs equation. Note that
l i ¼ h i À Ts i
and the corresponding fundamental function of state
U ¼ U S; V; N 1 ; . . .N n
ð
Þ
ð 155Þ
Equations (154B) and (155) form the starting point of Gibbsian thermodynamics.
Callen noted “The postulatory formulation of thermodynamics features states,
rather than processes, as fundamental constructs. Statements about Carnot cycles
and about the impossibility of perpetual motion of various kinds do not appear in
the postulates, but state functions, energy, and entropy become the fundamental
concepts. An enormous simplification in the mathematics is obtained, for processes
then enter simply as differentials of the state functions” [1:viii (1960)]. While I
disagree with this statement in the general formulation of thermodynamics concerning the central issues of heat and work, Callen’s comment does apply to
Gibbsian thermodynamics, which addresses physical/chemical properties and
chemical and phase equilibrium problems and in which the questions of heat and
work (especially the useful work) and the concept of reversibility are absent. It is
useful to note that, in this regard, Eqs. (154B) and (155) are universally valid, since
in the equations the nature of processes is immaterial.
1
1
It is useful to remind the readers that among the following set of equations:
dU ¼ dQ À dW
dU ¼ TdS À dW
dU ¼ dQ À pdV
dU ¼ TdS À pdV or dU ¼ TdS À pdV þ
P n
i¼1 l i dN i ;
only the first and the fourth are always valid for all processes, while the second and third are valid
for processes that meet internal reversibility condition (see Sect. 6.5).
242
9 Applications to Special States of Thermodynamic Equilibrium …
ð154AÞ
for a system interacting with a single component.
Equation (154A) can be generalized either for open systems interacting with
n components each with its corresponding membrane partition (with no bulk mass
flow in or out other than diffusion through membranes) or for closed systems (with
no mass exchange but with possible Q or/and W exchange) of multiple components
with chemical reaction (see below)
dU ¼ TdS À pdV þ
X n
i¼1
l i dN i
ð154BÞ
This is the fundamental differential for multicomponent systems, which is also
known under the name of Gibbs equation. Note that
l i ¼ h i À Ts i
and the corresponding fundamental function of state
U ¼ U S; V; N 1 ; . . .N n
ð
Þ
ð 155Þ
Equations (154B) and (155) form the starting point of Gibbsian thermodynamics.
Callen noted “The postulatory formulation of thermodynamics features states,
rather than processes, as fundamental constructs. Statements about Carnot cycles
and about the impossibility of perpetual motion of various kinds do not appear in
the postulates, but state functions, energy, and entropy become the fundamental
concepts. An enormous simplification in the mathematics is obtained, for processes
then enter simply as differentials of the state functions” [1:viii (1960)]. While I
disagree with this statement in the general formulation of thermodynamics concerning the central issues of heat and work, Callen’s comment does apply to
Gibbsian thermodynamics, which addresses physical/chemical properties and
chemical and phase equilibrium problems and in which the questions of heat and
work (especially the useful work) and the concept of reversibility are absent. It is
useful to note that, in this regard, Eqs. (154B) and (155) are universally valid, since
in the equations the nature of processes is immaterial.
1
1
It is useful to remind the readers that among the following set of equations:
dU ¼ dQ À dW
dU ¼ TdS À dW
dU ¼ dQ À pdV
dU ¼ TdS À pdV or dU ¼ TdS À pdV þ
P n
i¼1 l i dN i ;
only the first and the fourth are always valid for all processes, while the second and third are valid
for processes that meet internal reversibility condition (see Sect. 6.5).
242
9 Applications to Special States of Thermodynamic Equilibrium …