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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^ /kBT^ ≡ e−βEj^ (1)
where kB is Boltzmann’s constant, and we have defined the useful combination β ≡ 1 /kBT. Each pj is (and has to be ) > 0. As the probability of finding the system in any state j is 1, we must have ∑
j
pj = 1, (2)
A suitable normalization factor is then 1/Q, where Q is the partition function
Q ≡
j
e−Ej^ /kBT^ , (3)
so that the properly normalized probabilities pj are [McQ&S, eq. (17.13)]
pj = e−βEj Q
e−βEj ∑ j e −βEj.^ (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 Schr¨odinger equation
HˆΨj = 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
Q(N, V, T ) =
j
e−Ej^ (N,V^ )/kBT^ (6a)
pj (N, V, T ) =
e−βEj^ (N,V^ ) Q(N, V, T )
(6b)
or
Q(N, V, β) =
j
e−βEj^ (N,V^ )^ (7a)
pj (N, V, β) =
e−βEj^ (N,V^ ) Q(N, V, β)
(7b)
where it is important to note that we can use either the temperature T or β = 1/kBT as an independent variable in addition to N and V. The average energy of the system 〈E〉, which we equate with the observed energy U , is calculated by evaluating the sum of each energy Ej multiplied by the corresponding probability pj
〈E〉 =
j
Ej pj =
Q
j
Ej (N, V )e−βEj^ (N,V^ )^ (8)
which is (McQ&S, equations (17.20) and (17.21))
〈E〉 = −
∂ln Q ∂β
N,V
= kBT 2
∂ln Q ∂T
N,V
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 〈E〉 changes as we change T at constant N and V : this is the (constant volume) heat capacity CV (eq. (17.25))
CV =
∂〈E〉
∂T
N,V
∂U
∂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:
Pj (N, V ) = −
∂Ej ∂V
N
The equilibrium pressure p at temperature T is obtained by averaging the level pressures Pj over the probabilities pj
p ≡ 〈P〉 =
j
pj Pj (13a)
j
pj
∂Ej ∂V
N
(13b)
Q
j
∂Ej ∂V
N
e−βEj^. (13c)
We therefore have (17.32)
p =
β
∂ln Q ∂V
N,T
= kBT
∂ln Q ∂V
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 ). (15)
In principle we can calculate the equation of state, p = p(N, V, T ) from the {Ej }.
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
q1D(V, T ) =
[
mkBT 2 πℏ^2
] 1 / 2
L (22)
For a particle of mass m in a 3D volume V at temperature T ,
qtrans(V, T ) =
[
mkBT 2 πℏ^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 ,... (24)
B = ℏ^2 / 2 I. The rotational partition function is
qrot =
J=0, 1 ,...
(2J + 1)e−ΘrotJ(J+1)/T^ McQ&S, eq. (18.33) (25)
where Θrot = B/kB. For T Θrot we can replace the sum over rotational quantum number J by an integral,
⇒ qrot =
T
Θrot
2 IkBT ℏ^2 McQ&S, eq. (18.34). (26)
In general qrot =
T
σΘrot
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 = ℏω,
qvib =
n=0, 1 ,...
e−ℏω(n+
1 2 )/kBT^ (28a)
e−βℏω/^2 1 − e−βℏω^
McQ&S, eq. (18.23) (28b)
e−Θvib/^2 T 1 − e−Θvib/T^
McQ&S, eq. (18.24). (28c)
with Θvib = ℏω/kB. For T Θvib, qvib =
T
Θvib
