Biomedical Engineering Reference
In-Depth Information
The thermodynamic functions can be calculated using the equations given in Chapter 8.
We have for example
k
B
T
2
∂ ln
Q
∂
T
U
=
V
and so
3
2
Nk
B
T
For a monatomic ideal gas, the translational kinetic energy is the only energy that an
atom will possess. The internal energy is therefore just what we would expect from the
equipartition of energy principle. In a similar way we find that the entropy is
U
=
5
2
+
ln
2π
mk
B
T
h
2
3/2
V
N
S
=
Nk
B
(17.15)
and this is called the
Sackur-Tetrode equation
.
17.6.2
Ideal Diatomic Gas
Monatomic gas particles can only store translational kinetic energy, whilst, for a polyatomic
molecule, we have to consider:
•
rotational energy;
•
vibrational energy.
We also need to consider electronic energies. In order to progress, we make the assumption
that the different energy modes are independent, and that each set has its own Boltzmann
distribution. We therefore write for a typical molecule
=
ε
trans
+
ε
rot
+
ε
vib
+
ε
elec
(17.16)
ε
and it is easily seen that the molecular partition function is the product of a translational,
rotational, vibrational and electronic molecular partition function. These are related to the
canonical partition function by
1
N
=
(
q
trans
q
rot
q
vib
q
elec
)
N
Q
(17.17)
!
17.6.3
q
rot
The rotational energy levels for a rigid diatomic rotator are given by
h
2
8π
2
I
ε
J
=
J
(
J
+
1)
=
Bhc
0
J
(
J
+
1)
(17.18)
where the rotational quantum number
J
takes integral values 0, 1, 2,...,
I
is the moment of
inertia about an axis through the centre of mass of the molecule and
c
0
is the speed of light
