9.11 The Thermodynamic Entropy of a System
353
dE = T dS − pdV +
k
μ k dN k ,
(9.69)
where the system is allowed to exchange heat, work, and material with the
environment. The N k can be taken as the number of particles of species labeled by
‘k’ added to the system, which carries an energy μ k per unit particle into the system.
The intensive quantities μ k are called the ‘chemical potentials’. Differences of
chemical potentials across space are associated with particle flow, just as differences
in temperature across space are associated with heat flow.
The energy relation Eq. (9.69) is appropriate for chemical reactions among a
subset of molecules in the system, wherein some of these molecules are converted
into others. If the system is closed, then conservation of atoms (N =
N k fixed)
will constrain the changes dN k with
dN k = 0. (To see how this is applied, see
the discussion of biochemical reactions in Sect. 9.21.)
From our microscopic form for heat, Eq. (9.6), and now the connection δQ =
T dS, we can see that entropy production involves the redistribution of the particles
within the system. This observation will be made more explicit later by following
Boltzmann’s statistical interpretation of entropy.
Note that the expression (9.69) for dE tells us that E itself ‘naturally’ depends
on the extensive variables S, V , and all the N k . We say that the ‘natural’ variables 35
of the thermodynamic energy are entropy, volume, and particle numbers. Each of
the coefficients of the differentials on the right-hand-side of the above expression
for dE are intensive quantities, and must also depend only on S, V , and the N k . The
relation for dE can be used to define these measurable quantities:
T =
∂E
∂S
V ,N
(9.70)
p = −
∂E
∂V
S,N
(9.71)
μ k =
∂E
∂N k
S,V
.
(9.72)
These intensive quantities are ‘slopes’ of the extensive energy function E, plotted
in the planes defined by holding all but one of its extensive variables (S, V , {N k })
fixed.
For example, the temperature of a system can be determined by observing how
its internal energy changes by varying its entropy, while holding its volume and
constituents fixed. This is not a very convenient way to measure the temperature of
a system, nor would you use the phrase to give others what the notion of temperature
generalized displacement, making dE =
i f i dX i . The pair {f i , X i } are called thermodynamic
conjugate variables.
35 Natural variables are also called by some authors ‘characteristic variables’.
353
dE = T dS − pdV +
k
μ k dN k ,
(9.69)
where the system is allowed to exchange heat, work, and material with the
environment. The N k can be taken as the number of particles of species labeled by
‘k’ added to the system, which carries an energy μ k per unit particle into the system.
The intensive quantities μ k are called the ‘chemical potentials’. Differences of
chemical potentials across space are associated with particle flow, just as differences
in temperature across space are associated with heat flow.
The energy relation Eq. (9.69) is appropriate for chemical reactions among a
subset of molecules in the system, wherein some of these molecules are converted
into others. If the system is closed, then conservation of atoms (N =
N k fixed)
will constrain the changes dN k with
dN k = 0. (To see how this is applied, see
the discussion of biochemical reactions in Sect. 9.21.)
From our microscopic form for heat, Eq. (9.6), and now the connection δQ =
T dS, we can see that entropy production involves the redistribution of the particles
within the system. This observation will be made more explicit later by following
Boltzmann’s statistical interpretation of entropy.
Note that the expression (9.69) for dE tells us that E itself ‘naturally’ depends
on the extensive variables S, V , and all the N k . We say that the ‘natural’ variables 35
of the thermodynamic energy are entropy, volume, and particle numbers. Each of
the coefficients of the differentials on the right-hand-side of the above expression
for dE are intensive quantities, and must also depend only on S, V , and the N k . The
relation for dE can be used to define these measurable quantities:
T =
∂E
∂S
V ,N
(9.70)
p = −
∂E
∂V
S,N
(9.71)
μ k =
∂E
∂N k
S,V
.
(9.72)
These intensive quantities are ‘slopes’ of the extensive energy function E, plotted
in the planes defined by holding all but one of its extensive variables (S, V , {N k })
fixed.
For example, the temperature of a system can be determined by observing how
its internal energy changes by varying its entropy, while holding its volume and
constituents fixed. This is not a very convenient way to measure the temperature of
a system, nor would you use the phrase to give others what the notion of temperature
generalized displacement, making dE =
i f i dX i . The pair {f i , X i } are called thermodynamic
conjugate variables.
35 Natural variables are also called by some authors ‘characteristic variables’.