Statistical Mechanics Y R K Pathria Solution 3.25
Statistical Mechanics - Homework Assignment 3
Alejandro G´ omez Espinosa
∗
March 4, 2013
Pathria 3.41
A system of N spins at a negative temperature
(
E >
0)
is brought into contact with an ideal-gas thermometer consisting of N' molecules. What will the nature of their state of mutual equilibrium be? Will their commom temperature be negative or positive, and in what manner will it be affected by the ratio
N
/N
?
If the system of N spins has negative temperature its energy must be bounded. Then, if it is brought into contact with a sistem of N' molecules, the equilibrium will be reach when both temperatures are the same. But since the thermometer does not have bounded energy, the temperature in equilibrium should be greater than zero. Then, by conservation of energy we can write:
−
Nε
tanh
ε kT
+ 3 2
N
kT
=
E
T
(1) where we can see that the ratio
N
/N
will not be affected by this result.
Pathria 4.8
Determine the grand partition function of a gaseous system of magnetic atoms (with
J
=
1 2
and
g
= 2
) that can have, in addition to the kinetic energy, a magnetic potential energy equal to
µ
B
H
or
−
µ
B
H
, depending on their orientation with respect to an applied magnetic field H. Derive an expression for the magnetization of the system, and calculate how much heat will be given off by the system when the magnetic field is reduced from H to zero ar constant volume and constant temperature.
In this case, the Hamiltonian of the system is given by:
H
i
=
p
2
i
2
m
i
±
µ
B
H
(2) Then, using the partition function for one particle and taking only the positive magnetic potential energy:
Q
+ 1
=
e
−
βE
d
3
p d
3
q h
3
=
e
−
β
p
2 2
m
+
µ
B
H
d
3
p d
3
q h
3
=
e
−
βµ
B
H
h
3
e
−
β
p
2 2
m
d
3
p d
3
q
=
Ve
−
βµ
B
H
h
3
∞ −∞
e
−
β
p
2 2
m
d
3
p
=
Ve
−
βµ
B
H
h
3
2
m β
√
π
3
=
2
mπ βh
2
3
/
2
Ve
βµ
B
H
∗
gomez@physics.rutgers.edu
1
The total partition function for one particle must be:
Q
1
=
Q
+ 1
+
Q
−
1
=
2
mπ βh
2
3
/
2
Ve
−
βµ
B
H
+
2
mπ βh
2
3
/
2
Ve
−
βµ
B
H
=
2
mπ βh
2
3
/
2
V
e
−
βµ
B
H
+
e
−
βµ
B
H
=
2
mπ βh
2
3
/
2
2
V
cosh(
−
βµ
B
H
) Therefore, the partition function for the system of N particles:
Q
N
=
Q
N
1
N
! = 1
N
!
2
mπ βh
2
3
N/
2
(2
V
)
N
cosh
N
(
−
βµ
B
H
) Thus, the grand partition function, given by (4.3.15) is:
Q
=
∞
N
=0
z
N
Q
N
=
∞
N
=0
z
N
N
!
2
mπ βh
2
3
/
2
2
V
cosh(
−
βµ
B
H
)
N
=
∞
N
=0
1
N
!
2
Vz
2
mπ βh
2
3
/
2
cosh(
−
βµ
B
H
)
N
= exp
2
Vz
2
mπ βh
2
3
/
2
cosh(
−
βµ
B
H
)
For the magnetization:
M
=
−
∂F ∂H
T
=
∂ ∂H kT
ln
Q
N
=
kT ∂ ∂H
ln 1
N
!
2
mπ βh
2
3
/
2
2
V
cosh(
−
βµ
B
H
)
N
=
NkT ∂ ∂H
ln
2
mπ βh
2
3
/
2
2
V
cosh(
−
βµ
B
H
)
−
1
N
ln
N
!
=
NkT ∂ ∂H
ln
2
V
2
mπ βh
2
3
/
2
+ lncosh(
−
βµ
B
H
)
=
NkT ∂ ∂H
(lncosh(
−
βµ
B
H
)) =
−
NkTβµ
B
sinh(
−
βµ
B
H
) cosh(
−
βµ
B
H
) =
−
Nµ
B
tanh(
−
βµ
B
H
) 2
Finally, the heat in the system at constant volume and temperature is given by
Q
=
T
∆
S
. Therefore, let us begin by calculating the entropy of the system:
S
=
−
∂F ∂T
V,N
=
k ∂ ∂T
T
ln 1
N
!
2
mπ βh
2
3
/
2
2
V
cosh(
−
βµ
B
H
)
N
=
k ∂ ∂T
T
ln 1
N
!
2
mπkT h
2
3
/
2
2
V
cosh
−
µ
B
H kT
N
=
k ∂ ∂T
TN
ln
2
mπkT h
2
3
/
2
2
V
cosh
−
µ
B
H kT
−
T
ln
N
!
=
k ∂ ∂T
3 2
TN
ln
2
mπkT h
2
+
T
lncosh
−
µ
B
H kT
−
T
ln 2
V N
!
=
k
3 2
N
ln
2
mπkT h
2
+ 3
N
2 + lncosh
−
µ
B
H kT
+
µ
B
H
sinh
−
µ
B
H kT
kT
cosh
−
µ
B
H kT
−
ln 2
V N
!
= 3 2
Nk
ln
2
mπkT h
2
+ 3 2
Nk
+
k
ln
N
! 2
V
cosh
−
µ
B
H kT
+
µ
B
H T
tanh
−
µ
B
H kT
(3) Then, for the final state,
H
= 0:
S
f
= 3 2
Nk
ln
2
mπkT h
2
+ 3
Nk
2 (4) The difference in entropy (∆
S
) is given by the subtraction of ( 3 ) and ( 4 ), and finally the heat relation is find as:
Q
=
T
(
S
f
−
S
i
) =
−
T
k
ln
N
! 2
V
cosh
−
µ
B
H kT
+
µ
B
H T
tanh
−
µ
B
H kT
=
−
β
ln
N
! 2
V
cosh(
−
βµ
B
H
)
−
µ
B
H
tanh(
−
βµ
B
H
)
Pathria 4.10
A surface with
N
0
adsorption center has
N
(
≤
N
0
)
gas molecules adsorbed on it. Show that the chemical potential of the adsorbed molecules is given by
µ
=
kT
ln
N
(
N
0
−
N
)
a
(
T
) (5)
where
a
(
T
)
is the partition function of a single adsorbed molecule. Solve the problem by constructing the grand partition function as well as the partition function of the system. (Neglect the intermolec- ular interaction among the adsorbed molecules.)
Since
a
(
T
) is the partition function of a single molecule, then the total partition function must be:
Q
N
=
Q
N
1
N
! =
a
N
(
T
)
N
! (6) 3
Statistical Mechanics Y R K Pathria Solution 3.25
Source: https://pt.scribd.com/doc/150767118/Statistical-Mechanics-Pathria-Homework-3
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