Simple harmonic oscillator
The SHO has non-degenerate levels with
n
1
= (n + 2 ) for n = 0, 1, 2, . . .
h
a. The internal partition function is
(T )
e
=
n
n=0
h
e(n+1/2)
=
n=0
h
= e /2
h
e n ,
n=0
but
1
= 1 + x + x2 +
1x
so
for
0 x < 1,
1
h
1 e
h
(T )
Debye frequency
In general
G() d.
3N =
0
So in the Debye model
3N =
D
3V
2 2 c3
s
2 d =
0
3
3V D
2 c3 3
2 s
whence
D = cs
3
6 2 N
.
V
In summary,
9N 2
3
D
G() =
0
for < D
for > D
Debye model energy and heat capacity
The total energy is
E(T, V, N ) =
E(
Cool mountain air
a. Consider a slab of air with area A and mass M :
p(z+dz)A
dz
Mg
p(z)A
The mass of this slab is
N
V
M =m
A dz = m
p
kB T
A dz
where, on the right hand side, weve invoked the ideal gas equation of state.
The forces on the slab are in equ
Isothermal vs. adiabatic compressibility
a. We start with the relation
dS =
S
T
S
p
dT +
p
dp
T
and nd conventional names for the partial derivatives. First, by the denition of heat capacity, we have
S
T
=
p
Cp
.
T
Then, using the Maxwell relation associa
Isothermal compressibility
a. The isothermal compressibility is
T =
1 V
V p
,
(1)
T,N
but = N/V , so V = N/, so
T =
1 [N/]
[N/] p
=
T,N
[1/]
N
N
p
.
(2)
T
In the last step, we changed from a derivative at constant T and N to a derivative at constant T a
Denaturization of DNA
In this solution I record every physically important point but I sometimes skip mathematical steps.
a. The eective Hamiltonian H gives eH as a product, with a factor of
e
q
r
for each segment in the helical state [hi = 1],
for each s