88
2 Macroscopic Thermodynamics
carried out at constant T and P , we shall choose them to be the appropriate physical
variables to describe a chemically reacting system. Entropy production is thus
uniquely associated with the chemical reaction, so that δQ will be independent
of any changes in T and P that may occur concurrently.
2.7 Extension to Multicomponent Systems
We shall now consider extending our thermodynamic system from a pure substance to a mixture of substances, with corresponding numbers, n α , of moles
of chemical species α. The generalization of the internal energy then becomes
U ≡ U(S, V , {n α }), rather than simply U ≡ U(S, V , n). Whether we are allowed
to describe multicomponent systems in terms of a closed thermodynamic system
rather than requiring an open system depends upon whether or not we have mass
sinks or sources. If the processes that take place within the system satisfy the
conservation of mass principle, then they may be treated within the context of a
closed thermodynamic system, just as processes that satisfy the conservation of
energy principle maybe treated in the context of an isolated system.
If we return briefly to the differential form for the combined first and second
laws for a pure substance, dU = T dS − P dV , we note that it is structured as
a sum involving the products of intensive thermodynamic variables multiplying
differentials of extensive thermodynamic variables. Extension of the differential
form for the combined first and second laws of thermodynamics thus logically
requires that we add terms with the same structure, i.e., intensive thermodynamic
variables multiplying differentials of extensive thermodynamic variables. The
obvious extensive thermodynamic variables are the numbers of moles of substances
constituting the thermodynamic system. We shall see that the corresponding
intensive thermodynamic variables are quantities called chemical potentials, which
are designated by the symbol μ α . The combined first and second law expression for
a multicomponent system thus becomes
dU = T dS − P dV +
M
α=1
μ α dn α ,
(2.7.1)
with P the total pressure for the mixture and M the number of distinct chemical
components in it. This expression reduces to Eq. (2.2.12), which applies to a singlecomponent closed system, as dn = 0 for a closed system.
If we recall that U is a homogeneous function, then we may apply Euler’s
theorem (see Sect. B.1.2) to U ≡ U(S, V , {n α }) to give
U = S
∂U
∂S
V ,{n α }
+ V
∂U
∂V
S,{n α }
+
M
α=1
n α
∂U
∂n α
S,V ,n β =n α
.
2 Macroscopic Thermodynamics
carried out at constant T and P , we shall choose them to be the appropriate physical
variables to describe a chemically reacting system. Entropy production is thus
uniquely associated with the chemical reaction, so that δQ will be independent
of any changes in T and P that may occur concurrently.
2.7 Extension to Multicomponent Systems
We shall now consider extending our thermodynamic system from a pure substance to a mixture of substances, with corresponding numbers, n α , of moles
of chemical species α. The generalization of the internal energy then becomes
U ≡ U(S, V , {n α }), rather than simply U ≡ U(S, V , n). Whether we are allowed
to describe multicomponent systems in terms of a closed thermodynamic system
rather than requiring an open system depends upon whether or not we have mass
sinks or sources. If the processes that take place within the system satisfy the
conservation of mass principle, then they may be treated within the context of a
closed thermodynamic system, just as processes that satisfy the conservation of
energy principle maybe treated in the context of an isolated system.
If we return briefly to the differential form for the combined first and second
laws for a pure substance, dU = T dS − P dV , we note that it is structured as
a sum involving the products of intensive thermodynamic variables multiplying
differentials of extensive thermodynamic variables. Extension of the differential
form for the combined first and second laws of thermodynamics thus logically
requires that we add terms with the same structure, i.e., intensive thermodynamic
variables multiplying differentials of extensive thermodynamic variables. The
obvious extensive thermodynamic variables are the numbers of moles of substances
constituting the thermodynamic system. We shall see that the corresponding
intensive thermodynamic variables are quantities called chemical potentials, which
are designated by the symbol μ α . The combined first and second law expression for
a multicomponent system thus becomes
dU = T dS − P dV +
M
α=1
μ α dn α ,
(2.7.1)
with P the total pressure for the mixture and M the number of distinct chemical
components in it. This expression reduces to Eq. (2.2.12), which applies to a singlecomponent closed system, as dn = 0 for a closed system.
If we recall that U is a homogeneous function, then we may apply Euler’s
theorem (see Sect. B.1.2) to U ≡ U(S, V , {n α }) to give
U = S
∂U
∂S
V ,{n α }
+ V
∂U
∂V
S,{n α }
+
M
α=1
n α
∂U
∂n α
S,V ,n β =n α
.