Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 14. Thermodynamics formalism and examples 133
quantity in any region does only change by flows out or in to the region. Compare the electrostatic
flow features in Section 5B.
We write the equations for the particle numbers and the energy:
n
( n v
n v
n v
i i i, x) ( i i, y) ( i i,
z)
ni
t x
y
z
t
( ji
) 0
(
nu)
( nu)
( Jux) x ( Juy) y ( Juz)
z
∇Ju 0
t
t
(14.33)
v i,x , v i,y , v i,z are the components of the local velocity of component “i”; is the vector notation
for derivatives as defined in the equations. The above notation is generally used.
j i n i v i is the particle flux, u is the energy density; Ju x , Ju y , Ju z are the components of the
energy flux, Ju, which contains various contributions: convection, un i v i , heat flux, J q ,
energy from external work, e.g. electric current, J e , energy transported by diffusion J diff .
Next, we introduce the entropy changes. It is customary to divide the entropy change in
two parts: one that is produced in the system and another that is transferred from the environment
(as heat is transferred). We write:
dS d e S d i S (14.34)
d e S stands for a transferred entropy through heat from the outside. For this, we use the
reversible expression:
Q
dS e
T
(14.35)
Heat can be positive (when it is transferred to the system) or negative (when taken away).
Thus this transferred entropy can be both positive and negative.
The internal entropy change d i S is what is governed by the second law. That change is
never negative:
d i S 0 (14.36)
The total entropy change is not necessarily positive. The two parts of the entropy change
provide two principally different contributions. The “external entropy change”, d e S can be
expressed as a flux term:
dS
Q
T
(14.37)