Biomedical Engineering Reference
In-Depth Information
r
D s
D t þr
h
y
r
y
r
0
:
(10.54)
Integral equations such as (3.4) and ( 10.53 ) are global statements because they
apply to an entire object. However the results (3.5) and ( 10.54 ) are local, point wise
conditions valid at the typical point (place) in the object. Thus the transitions
(3.4)
( 10.54 ) are from the global to the local or from the
object to the point (or particle) in the object.
On the molecular level, each macroscopic substate { E ,
!
(3.5) and ( 10.53 )
!
r ( a ) } corresponds to a
large set of molecular states. In other words, the relationship between the molecular
states and the macroscopic states is injective, they are in one-to-one correspon-
dence. If we attribute an equal probability to each molecular state, the probability
that a thermodynamic state { E ,
r ( a ) } is different from another thermodynamic state
{ E* ,
r ( a ) * } can be calculated from the number of molecular states to which it
corresponds. In statistical physics, the concept of entropy is defined as the logarithm
of the number of molecular states that correspond to that particular molecular state
(multiplied by Boltzman constant). The entropy thereby provides a measure of the
relative probability of different macrostates. The second law, stating that entropy
moves towards increasing entropy, simply states that the system has a natural
tendency to evolve from less probable states towards more probable states. In this
sense one can interpret the second law as being almost a tautology.
10.7 The Entropy Inequality for a Mixture
Returning to the continuum model note that, in terms of the internal energy
, E ,
r ( a ) , X ), the temperature, stress and electrochemical (or chemical) potential may be
defined as the derivatives of
e
(
e
(
, E ,
r ( a ) , X ) with respect to entropy, strain, and
volume fraction, respectively:
!
;r ðaÞ ; m ðaÞ ¼
@e
@
@e
@
@e
@r ðaÞ
y ¼
;r ðaÞ ;
T
¼
:
(10.55)
E
E
E
;
e
The time derivative of the internal energy
may then be expressed as follows:
X
N
1 m ðaÞ
D s
D s
D s
r ðaÞ
D t
e
D t ¼ y
D t þ
T
:
D
þ
:
(10.56)
The Helmholtz free energy is defined by
Cðy;
E
; r ðaÞ ;
X
Þ¼eð;
E
; r ðaÞ ;
X
Þy;
(10.57)
Search WWH ::




Custom Search