9.16 Maxwell Relations
365
The chemical potential for a species of particles determines the ‘pressure’ to cause a
change in the number of those particles, just as ordinary pressure causes changes in
volume. More specifically, the chemical potential is the energy gained by a system
per unit particle number (or moles) when those particles increase in the system, just
as pressure is the energy gained in a system per unit volume loss due to pressure.
The balancing of the chemical potentials we will see also applies when a
chemical reaction generates a transfer of material from one type to another. When
the chemical potentials do not match across regions in a cell, material will flow until
they do. If they do not match in a mixture of compounds, a chemical reaction may be
available to equalize the chemical potentials. Whether the reaction occurs in times
short compared to other activities is outside the purview of thermodynamics, but
rather requires more detailed molecular dynamics, becoming the subject of chemical
kinematics, a topic of Sect. 10.11.
After introducing Maxwell’s thermodynamic relations, we will use the equality
of chemical potentials in different phases to derive the vapor pressure of water and
of a gas dissolved in a liquid, and the equality across a semi-permeable membrane to
derive the osmotic pressure difference across the membrane. Such topics are clearly
important for plants and animals.
9.16 Maxwell Relations
For quasi-static changes that cause a ‘smooth’ change in the energy of a system, we
will have
∂ 2 E
∂S∂V
=
∂ 2 E
∂V ∂S
,
(9.106)
which can be written
∂p
∂S
V
= −
∂T
∂V
S
.
(9.107)
This is the first of such ‘slope’ connections called ‘Maxwell relations’ based on the
‘smoothness’ of the thermodynamic functions in their variables, apart from regions
of phase change. This one is useful in relating an isochoric process (fixed volume) 47
to an adiabatic one (no heat transfer).
Three more Maxwell relations come from the smoothness of the enthalpy, the
Helmholtz free energy and the Gibbs free energy defined as follows. Consider the
enthalpy,
47 Isochoric is from the Greek: isos choros, ‘equal space’.
365
The chemical potential for a species of particles determines the ‘pressure’ to cause a
change in the number of those particles, just as ordinary pressure causes changes in
volume. More specifically, the chemical potential is the energy gained by a system
per unit particle number (or moles) when those particles increase in the system, just
as pressure is the energy gained in a system per unit volume loss due to pressure.
The balancing of the chemical potentials we will see also applies when a
chemical reaction generates a transfer of material from one type to another. When
the chemical potentials do not match across regions in a cell, material will flow until
they do. If they do not match in a mixture of compounds, a chemical reaction may be
available to equalize the chemical potentials. Whether the reaction occurs in times
short compared to other activities is outside the purview of thermodynamics, but
rather requires more detailed molecular dynamics, becoming the subject of chemical
kinematics, a topic of Sect. 10.11.
After introducing Maxwell’s thermodynamic relations, we will use the equality
of chemical potentials in different phases to derive the vapor pressure of water and
of a gas dissolved in a liquid, and the equality across a semi-permeable membrane to
derive the osmotic pressure difference across the membrane. Such topics are clearly
important for plants and animals.
9.16 Maxwell Relations
For quasi-static changes that cause a ‘smooth’ change in the energy of a system, we
will have
∂ 2 E
∂S∂V
=
∂ 2 E
∂V ∂S
,
(9.106)
which can be written
∂p
∂S
V
= −
∂T
∂V
S
.
(9.107)
This is the first of such ‘slope’ connections called ‘Maxwell relations’ based on the
‘smoothness’ of the thermodynamic functions in their variables, apart from regions
of phase change. This one is useful in relating an isochoric process (fixed volume) 47
to an adiabatic one (no heat transfer).
Three more Maxwell relations come from the smoothness of the enthalpy, the
Helmholtz free energy and the Gibbs free energy defined as follows. Consider the
enthalpy,
47 Isochoric is from the Greek: isos choros, ‘equal space’.