9.3 Open Systems with Semi-permeable Membrane
Opening, and Multicomponent Closed Systems
Gibbs introduced the idea of chemical potential in his magnum opus entitled On the
Equilibrium of Heterogeneous Substances [2, (1875) and (1878)]. The heterogeneous substance system may be either an open system with a semi-permeable
membrane opening or a closed system of multicomponents.
Consider the former case of a mixing substance system in C (of uniform state at
U-V-S-p-T-m [or N]), while a homogeneous substance n of mass dm in the cylinder
inlet (or outlet), with a semi-permeable membrane partition separating C and the
cylinder [3:582–583]. This implies that the pressure force of the cylinder piston
equals the partial pressure p n of the n component of the mixing substance. It will be
assumed that the temperature of the homogeneous substance n equals that of the
mixing substance. In this manner, adding or extracting mass dm will be reversible,
and this will enable us to write the second law as well as the first law for the change
in the open system of mixing substance.
We shall consider no useful work W useful here but admit a deforming boundary
C as shown in Fig. 9.1 with work due to deformable C (which is absent in
Eq. (152)),
dW ¼ pdV
Note that p is the total pressure of the mixing substance, not the partial pressure p n
of its n-component, while h n ¼ u n À p n v n . With these stipulations, Eq. (152)
becomes
dU ¼ dQ þ h n dm
ð
ÞÀpdV
ð153Þ
Since the process is reversible, and the system entropy exchange assumes
d E S ¼ s n dm þ
dQ
T , Eq. (84B) in Chapter 6 becomes
dS ¼ d E S ¼ s n dm þ
dQ
T
Substitution of which into Eq. (153) yields
dU ¼ TdS À pdV þ h n À Ts n
ð
Þ dm
ð154Þ
The coefficient of dm is known as the chemical potential, which was first
introduced by Gibbs. We shall opt for the molar representation dN replacing the
mass dm. Denote the mole-based chemical potential by l. Equation (154) is then
written as
9.3 Open Systems with Semi-permeable Membrane Opening …
241
Opening, and Multicomponent Closed Systems
Gibbs introduced the idea of chemical potential in his magnum opus entitled On the
Equilibrium of Heterogeneous Substances [2, (1875) and (1878)]. The heterogeneous substance system may be either an open system with a semi-permeable
membrane opening or a closed system of multicomponents.
Consider the former case of a mixing substance system in C (of uniform state at
U-V-S-p-T-m [or N]), while a homogeneous substance n of mass dm in the cylinder
inlet (or outlet), with a semi-permeable membrane partition separating C and the
cylinder [3:582–583]. This implies that the pressure force of the cylinder piston
equals the partial pressure p n of the n component of the mixing substance. It will be
assumed that the temperature of the homogeneous substance n equals that of the
mixing substance. In this manner, adding or extracting mass dm will be reversible,
and this will enable us to write the second law as well as the first law for the change
in the open system of mixing substance.
We shall consider no useful work W useful here but admit a deforming boundary
C as shown in Fig. 9.1 with work due to deformable C (which is absent in
Eq. (152)),
dW ¼ pdV
Note that p is the total pressure of the mixing substance, not the partial pressure p n
of its n-component, while h n ¼ u n À p n v n . With these stipulations, Eq. (152)
becomes
dU ¼ dQ þ h n dm
ð
ÞÀpdV
ð153Þ
Since the process is reversible, and the system entropy exchange assumes
d E S ¼ s n dm þ
dQ
T , Eq. (84B) in Chapter 6 becomes
dS ¼ d E S ¼ s n dm þ
dQ
T
Substitution of which into Eq. (153) yields
dU ¼ TdS À pdV þ h n À Ts n
ð
Þ dm
ð154Þ
The coefficient of dm is known as the chemical potential, which was first
introduced by Gibbs. We shall opt for the molar representation dN replacing the
mass dm. Denote the mole-based chemical potential by l. Equation (154) is then
written as
9.3 Open Systems with Semi-permeable Membrane Opening …
241