Biomedical Engineering Reference
In-Depth Information
8.4 Entropy S
Finally, since
U
−
A
S
=
T
we have
k
B
T
∂ ln
Q
∂
T
S
=
V
,
N
+
k
B
ln
Q
(8.10)
8.5 Equation of State and Pressure
The pressure is related to the Helmholtz energy by
∂
A
∂
V
=−
p
T
,
N
and so we find
k
B
T
∂ ln
Q
∂
V
=
p
(8.11)
T
,
N
This equation is sometimes called the
equation of state
. The enthalpy and the Gibbs energy
can be derived using similar arguments. They turn out to be
k
B
T
2
∂ ln
Q
∂
T
k
B
TV
∂ ln
Q
∂
V
V
,
N
+
H
=
T
,
N
k
B
TV
∂ ln
Q
∂
V
G
=−
k
B
T
ln
Q
+
(8.12)
T
,
N
8.6 Phase Space
Sophisticated methods such as those due to Hamilton and to Lagrange exist for the system-
atic treatment of problems in particle dynamics. Such techniques make use of
generalized
coordinates
(written
q
1
,
q
2
, ...,
q
n
) and
generalized momenta
(written
p
1
,
p
2
, ...,
p
n
);in
Hamilton's method we write the total energy as the
Hamiltonian H
.
H
is the sum of the
kinetic energy and the potential energy, and it is a constant provided that the potentials are
time independent.
H
has to be written in terms of the
p
's and the
q
's in a certain way, and
systematic application of Hamilton's equations gives a set of differential equations for the
system.
To fix our ideas, consider the particle of mass
m
undergoing simple harmonic motion as
discussed in Chapter 4. In this one-dimensional problem I wrote the potential as
1
2
k
s
(
R
=
−
R
e
)
2
U
