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
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 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