Biomedical Engineering Reference
In-Depth Information
8.2
Internal Energy,
U
th
The IUPAC recommended symbol for thermodynamic internal energy is
U
, but I have added
a subscript 'th' for 'thermodynamic' so that there is no confusion with the total potential
energy of a system
U
(often written Φ). Internal energy is obviously related to the ensemble
energy average but we have to exercise caution. Chemical measurements only give changes
in the internal energy, not absolute values. I will therefore write the internal energy as
U
th
−
U
0
, where
U
0
is an arbitrary constant. For most purposes we can take
U
0
to be zero.
We have, for the ensemble of
N
∗
members
i
N
i
E
i
N
∗
U
th
−
U
0
=
(8.4)
and according to the ergodic theorem, this is equal to the time average of
U
th
−
U
0
for any
one cell. Using the Boltzmann expression we have
i
E
i
exp
E
i
k
B
T
−
U
th
−
U
0
=
(8.5)
Q
I can tidy up Equation (8.5) by noting
∂
Q
∂
T
E
i
exp
k
B
T
2
i
1
E
i
k
B
T
V
,
N
=
−
(8.6)
and so on substitution
∂
Q
∂
T
k
B
T
2
Q
U
th
−
U
0
=
(8.7)
V
8.3 Helmholtz Energy,
A
From the definition of
A
we have the following:
A
=
U
−
TS
∂
A
∂
T
S
=−
V
,
N
T
∂
A
∂
T
A
=
U
+
(8.8)
V
,
N
A little manipulation gives
∂
∂
T
A
T
k
B
∂ ln
Q
∂
T
V
,
N
=−
V
,
N
A
−
A
0
=−
k
B
T
ln
Q
(8.9)
Again, the arbitrary constant
A
0
can be taken as zero, since only changes in
A
are ever
measured.
