410
9 Bioenergetics
If we know how the temperature, pressure, and chemical potentials depend on
the internal energy, volume, and concentration of chemical type, we will know a
fundamental equation for the system expressed as S = S(U, V , {N i }).
In general form, the entropy change in a closed system due to non-dissipative
processes has the general form
dS =
i
∂S
∂Y i
dY i ≡ −k B
i
y i dY i ,
(9.240)
where
y i ≡ −
1
k B
∂S
∂Y i
(9.241)
and Y i is one of the independent variables for the subsystem. For simple systems,
∂S/∂E = 1/T , ∂S/∂V = p/T and ∂S/∂N i = −μ i /T .
Compare with conservative mechanical forces satisfying F = −∇, so the work
done by such a force while a body ‘displaces’ dY is given by dW = −∇ · dY =
−dd. In the present case, we have an expression for the entropy change which
formally appears as dE does, namely, a sum over gradients times the corresponding
changes in the independent variables. For this reason, the y i are called entropic
thermodynamic ‘forces’, and the Y i ‘entropic thermodynamic displacements’. In
addition, each of the terms y i dY i must be integrable, because they are perfect
differentials when one holds all the other Y k fixed.
If we take the Y i to be the extensive variables, We can find the result of this
integration using the fact that S(λY ) = λS(Y ), which makes
S = −k B
i
y i Y i .
(9.242)
The form of this expression has led to the nomenclature that the y i and Y i are
‘conjugate thermodynamic variables’. As S has the units of k B , each y i has the
inverse units of Y i .
Let s be the local entropy in a small open subsystem per unit total particle number
in the subsystem. If there are no dissipative processes, the disorder of a system with
fixed extensive variables will not change. If the system is open, then there could be
a flow of disorder (entropy) across the boundaries of the system. Such a flow will be
denoted by J s giving the entropy per unit area per unit time carried across an area
whose normal point in flow direction. 68
Let ˙
σ be the excess rate of entropy production due to dissipative processes within
a small subsystem. Then the local rate of entropy change is given by
68 A good example of such a flow of disorder is that from a chicken egg as the embryo develops. In
this case, the entropy flow is carried by heat flowing out of the egg.
9 Bioenergetics
If we know how the temperature, pressure, and chemical potentials depend on
the internal energy, volume, and concentration of chemical type, we will know a
fundamental equation for the system expressed as S = S(U, V , {N i }).
In general form, the entropy change in a closed system due to non-dissipative
processes has the general form
dS =
i
∂S
∂Y i
dY i ≡ −k B
i
y i dY i ,
(9.240)
where
y i ≡ −
1
k B
∂S
∂Y i
(9.241)
and Y i is one of the independent variables for the subsystem. For simple systems,
∂S/∂E = 1/T , ∂S/∂V = p/T and ∂S/∂N i = −μ i /T .
Compare with conservative mechanical forces satisfying F = −∇, so the work
done by such a force while a body ‘displaces’ dY is given by dW = −∇ · dY =
−dd. In the present case, we have an expression for the entropy change which
formally appears as dE does, namely, a sum over gradients times the corresponding
changes in the independent variables. For this reason, the y i are called entropic
thermodynamic ‘forces’, and the Y i ‘entropic thermodynamic displacements’. In
addition, each of the terms y i dY i must be integrable, because they are perfect
differentials when one holds all the other Y k fixed.
If we take the Y i to be the extensive variables, We can find the result of this
integration using the fact that S(λY ) = λS(Y ), which makes
S = −k B
i
y i Y i .
(9.242)
The form of this expression has led to the nomenclature that the y i and Y i are
‘conjugate thermodynamic variables’. As S has the units of k B , each y i has the
inverse units of Y i .
Let s be the local entropy in a small open subsystem per unit total particle number
in the subsystem. If there are no dissipative processes, the disorder of a system with
fixed extensive variables will not change. If the system is open, then there could be
a flow of disorder (entropy) across the boundaries of the system. Such a flow will be
denoted by J s giving the entropy per unit area per unit time carried across an area
whose normal point in flow direction. 68
Let ˙
σ be the excess rate of entropy production due to dissipative processes within
a small subsystem. Then the local rate of entropy change is given by
68 A good example of such a flow of disorder is that from a chicken egg as the embryo develops. In
this case, the entropy flow is carried by heat flowing out of the egg.