182
4 The Thermodynamics of Real Systems
4.5
Multicomponent Systems
In an equilibrium open multicomponent system with one gas or liquid phase the number
of variables required to specify the equilibrium macroscopic state is c + 2, where
c stands for the number of independent substances, called components. The number
of components is equal to the number of substances whose amounts can separately be
varied under the given conditions. It is also equal to the minimum number of substances
from which the system can be prepared under the given conditions. For example, if a
gaseous system at equilibrium contains NO 2 it will also contain N 2 O 4 . The system can
be prepared by adding only NO 2 or by adding only N 2 O 4 and allowing the system to
equilibrate. Two substances are present, but there is only one component.
The Chemical Potential and Partial Molar Quantities
In an equilibrium one-phase simple system containing c components, the Gibbs energy
depends on c + 2 variables, which can be chosen as T , P, and the mole fractions of all
components:
G G(T , P, n 1 , n 2 , …, n c )
(4.5-1)
where n i is the amount of substance number i (measured in moles). The differential
of G is
dG
∂G
∂T
P,n
dT +
∂G
∂P
T ,n
dP +
c
i1
∂G
∂n i
T ,P,n
dn i
(4.5-2)
where the subscript n stands for keeping the amounts of all of the components fixed
and the subscript n stands for keeping the amount of every component fixed except
for component number i.
The first two partial derivatives in Eq. (4.5-2) are no different from the partial
derivatives in Eqs. (4.2-20) and (4.2-21) for a closed system. In those equations, the
amounts of all substances present were held fixed because the system was closed. In
Eq. (4.5-2), the amounts of all substances are held fixed because that is how partial
derivatives are defined. Therefore, we can write
dG −SdT + VdP +
c
i1
µ i dn i
(4.5-3)
where µ i is called the chemical potential and is defined by
µ i
∂G
∂n i
T ,P,n
(definition of the
chemical potential)
(4.5-4)
Equation (4.5-3) is called the Gibbs equation or the fundamental relation of chemical
thermodynamics. We could also choose to write
dG
∂G
∂T
V ,n
dT +
∂G
∂V
T ,n
dV +
c
i1
∂G
∂n i
T ,V ,n
dn i
(4.5-5)
4 The Thermodynamics of Real Systems
4.5
Multicomponent Systems
In an equilibrium open multicomponent system with one gas or liquid phase the number
of variables required to specify the equilibrium macroscopic state is c + 2, where
c stands for the number of independent substances, called components. The number
of components is equal to the number of substances whose amounts can separately be
varied under the given conditions. It is also equal to the minimum number of substances
from which the system can be prepared under the given conditions. For example, if a
gaseous system at equilibrium contains NO 2 it will also contain N 2 O 4 . The system can
be prepared by adding only NO 2 or by adding only N 2 O 4 and allowing the system to
equilibrate. Two substances are present, but there is only one component.
The Chemical Potential and Partial Molar Quantities
In an equilibrium one-phase simple system containing c components, the Gibbs energy
depends on c + 2 variables, which can be chosen as T , P, and the mole fractions of all
components:
G G(T , P, n 1 , n 2 , …, n c )
(4.5-1)
where n i is the amount of substance number i (measured in moles). The differential
of G is
dG
∂G
∂T
P,n
dT +
∂G
∂P
T ,n
dP +
c
i1
∂G
∂n i
T ,P,n
dn i
(4.5-2)
where the subscript n stands for keeping the amounts of all of the components fixed
and the subscript n stands for keeping the amount of every component fixed except
for component number i.
The first two partial derivatives in Eq. (4.5-2) are no different from the partial
derivatives in Eqs. (4.2-20) and (4.2-21) for a closed system. In those equations, the
amounts of all substances present were held fixed because the system was closed. In
Eq. (4.5-2), the amounts of all substances are held fixed because that is how partial
derivatives are defined. Therefore, we can write
dG −SdT + VdP +
c
i1
µ i dn i
(4.5-3)
where µ i is called the chemical potential and is defined by
µ i
∂G
∂n i
T ,P,n
(definition of the
chemical potential)
(4.5-4)
Equation (4.5-3) is called the Gibbs equation or the fundamental relation of chemical
thermodynamics. We could also choose to write
dG
∂G
∂T
V ,n
dT +
∂G
∂V
T ,n
dV +
c
i1
∂G
∂n i
T ,V ,n
dn i
(4.5-5)