Download Boltzmann factors and partition functions revisited

Chem 390
Physical Chemistry II
Spring 2007
Boltzmann factors and partition functions revisited
A brief summary of material from McQuarrie & Simon, Chapters 17 and 18, on the
partition function and its use in the calculation of some equilibrium properties. You have
already seen this material in Chem 389. The concepts outlined here will be applied in
Chem 390 to a number of important problems.
Partition function as a normalization factor for probabilities
For a system in equilibrium at (absolute) temperature T , the probability of finding the system in the
quantum state with energy Ej is proportional to the Boltzmann factor
pj ∝ e−Ej /kB T ≡ e−βEj
(1)
where kB is Boltzmann’s constant, and we have defined the useful combination β ≡ 1/kB T . Each pj is
(and has to be ) > 0. As the probability of finding the system in any state j is 1, we must have
X
pj = 1,
(2)
j
A suitable normalization factor is then 1/Q, where Q is the partition function
X
Q≡
e−Ej /kB T ,
(3)
j
so that the properly normalized probabilities pj are [McQ&S, eq. (17.13)]
pj =
e−βEj
e−βEj
= P −βEj .
Q
je
(4)
Calculation of average energy
For a system of N particles (for example, a gas of particles in a container of volume V ), the energies
Ej are the eigenvalues of the Schrödinger equation
ĤΨj = Ej Ψj .
(5)
The system energies Ej will naturally be functions of how many particles there are (N ) and how big
the box is (V ), so Ej = Ej (N, V ), and the full dependence of Q and the pj s is
X
Q(N, V, T ) =
e−Ej (N,V )/kB T
(6a)
j
pj (N, V, T ) =
e−βEj (N,V )
Q(N, V, T )
(6b)
or
Q(N, V, β) =
X
e−βEj (N,V )
(7a)
j
pj (N, V, β) =
e−βEj (N,V )
Q(N, V, β)
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(7b)
Physical Chemistry II
Chem 390
Spring 2007
where it is important to note that we can use either the temperature T or β = 1/kB T as an independent
variable in addition to N and V .
The average energy of the system hEi, which we equate with the observed energy U , is calculated
by evaluating the sum of each energy Ej multiplied by the corresponding probability pj
hEi =
X
1 X
Ej (N, V )e−βEj (N,V )
Q j
Ej p j =
j
(8)
which is (McQ&S, equations (17.20) and (17.21))
∂ln Q
2 ∂ln Q
hEi = −
= kB T
.
∂β N,V
∂T
N,V
(9)
We therefore have the first essential route from the quantum levels Ej to an equilibrium bulk property
{Ej } =⇒ Q(N, V, T ) =⇒ U (N, V, T ).
(10)
Heat capacity (constant volume)
Once we have (in principle) the average energy as a function of T , N and V , we can calculate the rate
at which hEi changes as we change T at constant N and V : this is the (constant volume) heat capacity
CV (eq. (17.25))
∂U
∂hEi
=
.
(11)
CV =
∂T N,V
∂T N,V
Calculation of the pressure
For a macroscopic system in level j, energy Ej , the associated level pressure Pj is directly related to
the rate at which the energy Ej (N, V ) changes as the volume of the system varies:
∂Ej
Pj (N, V ) = −
.
(12)
∂V N
The equilibrium pressure p at temperature T is obtained by averaging the level pressures Pj over the
probabilities pj
X
p ≡ hPi =
pj Pj
(13a)
j
=−
X
pj
j
=−
1 X
Q j
∂Ej
∂V
∂Ej
∂V
(13b)
N
e−βEj .
(13c)
N
We therefore have (17.32)
p=
1
β
∂ln Q
∂V
= kB T
N,T
∂ln Q
∂V
(14)
N,T
and a second essential route from the quantum levels Ej to an equilibrium bulk property
{Ej } =⇒ Q(N, V, T ) =⇒ p(N, V, T ).
In principle we can calculate the equation of state, p = p(N, V, T ) from the {Ej }.
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Physical Chemistry II
Chem 390
Spring 2007
The partition function for independent subsystems: distinguishable versus
indistinguishable
Independent subsystems
Independent means that the interaction energy between the particles is effectively zero. The total energy
for the N particle system, Ej , can then be written as a sum of contributions εα
i from independent
subsystems (molecules) α
X
Ej =
εα
(16)
i .
α
Distinguishable subsystems
Distinguishable means that, although the particles have the same chemical identity, they are localized
in distinct regions of space and are (in principle) experimentally identifiable (“taggable”) according to
their location. (Example: molecules fixed at distinct sites in a crystal lattice.)
From (16), the partition function Q for independent and distinguishable particles factors into the
product of partition functions qα for each identical subsystem, (17.35):
Q(N, T, V ) =
Y
qα (V, T ) = [q(V, T )]N
(17)
X
(18)
α
with
q(V, T ) =
e−εj /kB T .
j
Indistinguishable subsystems
Indistinguishable means that there is no way in principle to identify individual particles according to
their location in space. (Example: identical gas molecules moving around inside a container of volume
V .)
If the criterion: number of quantum states per particle with energy εj . kB T N is satisfied, the
partition function Q for N independent and indistinguishable particles is given by Boltzmann statistics,
(17.38)
Q(N, V, T ) =
[q(V, T )]N
.
N!
(19)
The partition function for a subsystem (molecule) whose energy is the sum
of separable contributions
Quantized molecular energy levels can often be written to very good approximation as the sum of
independent contributions from translational, rotational, vibrational and electronic motions (17.45)
vib
elec
ε = εtrans
+ εrot
i
j + εk + ε `
(20)
so that the single-molecule partition function q(V, T ) has the product form (17.46)
q(V, T ) = qtrans qrot qvib qelec .
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Physical Chemistry II
Chem 390
Spring 2007
Partition functions for molecular motions
• Translation
Consider a particle of mass m in a 1D box of length L. Replacing the sum over quantum states
with an integral we have
1/2
mkB T
L
(22)
q1D (V, T ) =
2π~2
For a particle of mass m in a 3D volume V at temperature T ,
mkB T
qtrans (V, T ) =
2π~2
3/2
V
McQ&S, eq. (18.20)
(23)
• Rotation
Consider a rigid heteronuclear diatomic, with rotational energy levels
EJ = BJ(J + 1), J = 0, 1, . . .
B = ~2 /2I. The rotational partition function is
X
qrot =
(2J + 1)e−Θrot J(J+1)/T
McQ&S, eq. (18.33)
(24)
(25)
J=0,1,...
where Θrot = B/kB .
For T Θrot we can replace the sum over rotational quantum number J by an integral,
⇒ qrot =
2IkB T
T
=
Θrot
~2
McQ&S, eq. (18.34) .
In general
qrot =
T
σΘrot
(26)
(27)
where σ is the symmetry number, σ = 1 for a heteronuclear diatomic, σ = 2 for a homonuclear.
• Vibration
For a single molecular vibrational mode treated as a harmonic oscillator, vibrational frequency ν,
vibrational quantum hν = ~ω,
qvib =
X
1
e−~ω(n+ 2 )/kB T
(28a)
n=0,1,...
e−β~ω/2
1 − e−β~ω
e−Θvib /2T
=
1 − e−Θvib /T
=
McQ&S, eq. (18.23)
with Θvib = ~ω/kB . For T Θvib ,
qvib =
4 of 4
McQ&S, eq. (18.24) .
T
.
Θvib
(28b)
(28c)
(29)