Rotating Rigid Polyatomic Molecules

H.2 Euler-Maclaur in Sum Formula

20.7 Rotating Rigid Polyatomic Molecules

In the approximation of classical statistical mechanics, we can calculate the partition function by integrating over canonical coordinates and momenta in phase space and dividing by appropriate powers of h. Such a partition function should agree with a quantum mechanical result at high temperatures. Rigid rotation of a polyatomic molecule is a three-dimensional problem for a body that can have three different principal moments of inertia,I1,I2, andI3. See Appendix F for details. As shown in Section F.6, the orientation of the molecule can be expressed in terms of three Euler angles,φ,θ, andψ, where we have adopted the notation and convention of Goldstein [60, p. 107]. As shown in Section F.7, the Hamiltonian can be written in the forms

H=1

2(I1ω21+I2ω22+I3ω23)= L21 2I1 + L22

2I2 + L23

2I3. (20.109)

Here,ω1,ω2, andω3are principal angular velocities andLi =Iiωiare the corresponding principal angular momenta. Theωi can be expressed in terms of the Euler angles and their time derivatives (see Eq. (F.59)). Then the canonical momenta,pφ,pθ, andpψ, can be calculated by differentiation and are given explicitly by Eqs. (F.61)–(F.63). Thus

z= 1 h3

exp(−βH)dpφdpθdpψdφdθdψ. (20.110) One could proceed by solving Eqs. (F.61)–(F.63) forL1,L2,L3and using the results to eliminate these quantities from Eq. (20.109). This results in a very cumbersome expression for Has a function of the canonical momenta and the Euler angles and poses a rather unwieldy integration. An alternative procedure is to transform the integration variables to L1,L2,L3,φ,θ,ψby means of a JacobianJpolyso that

z= 1 h3

exp(−βH)|Jpoly|dL1dL2dL3dφdθdψ, (20.111) where

|Jpoly| =

∂

pφ,pθ,pψ,φ,θ,ψ

∂ (L1,L2,L3,φ,θ,ψ) =

∂

pφ,pθ,pψ

∂ (L1,L2,L3)

. (20.112)

This isnota canonical transformation, so the magnitude of the Jacobian is

|Jpoly| = det

⎛

⎝sinθsinψ sinθcosψ cosθ cosψ −sinψ 0

0 0 1

⎞

⎠

= | −sinθ| =sinθ. (20.113)

The partition function therefore becomes8 z= 1

exp(−βH)sinθdL1dL2dL3dφdθdψ

=8π2 h3

exp −β

L21 2I1 + L22

2I2 + L23 2I3

dL1dL2dL3. (20.114) We are left with the product of three Gaussian integrals of the form

∞

−∞exp[−βL21/(2I1)]dL1=

2πI1kBT1/2

. (20.115)

We therefore obtain

z=π1/2

2I1kBT h¯2

1/22I2kBT

¯h2

1/22I3kBT

¯h2 1/2

. (20.116)

This result will be used in Section 21.3.3 in the context of a gas of polyatomic molecules that can also vibrate.

For a diatomic molecule, only two degrees of freedom are considered becauseI3 is essentially zero9and the two remaining moments of inertia are equal, say toI. Thus

H=I

2(ω12+ω22)=I

2(sin2θφ˙2+ ˙θ2)= 1

2I(L21+L22). (20.117) Now the only canonical momenta are10

pφ=Isin2θφ˙ =L1sinθsinψ+L2sinθcosψ (20.118) and

pθ=Iθ˙=L1cosψ−L2sinψ. (20.119) Therefore,

zdia= 1 h2

exp(−βH)dpφdpθdφdθ= 1 h2

exp(−βH)|Jdia|dL1dL2dφdθ, (20.120) where the magnitude of the Jacobian

|Jdia| =

∂ pφ,pθ

∂ (L1,L2)

=sinθ. (20.121)

The integrals overφandθgive a factor of 4πand we obtain zdia=4π

exp −β

L21 2I+ L22

2I '!

dL1dL2. (20.122)

8The ranges of the Euler angles are 0≤φ≤2π, 0≤θ ≤π, and 0≤ψ ≤2π. Landau and Lifshitz [7, p. 149]

give a verbal argument that an integral over three unspecified angles gives a factor of 8π2and then proceed to integrate over onlyL1,L2,L3, but no justification in terms of canonical momenta is given.

9As shown in Section F.8 of Appendix F, the quantum states associated withI3have such high energies that they are not excited at any reasonable temperature.

10Here, we continue to use the same Euler angles as for the polyatomic molecule for the sake of a parallel treatment.

358 THERMAL PHYSICS

Integration results in two equal factors having the form of Eq. (20.115) which yields the result

zdia=2IkBT

h2 , (20.123)

in agreement with the high-temperature quantum mechanical result given by Eq. (18.85).

In this simple case, the Hamiltonian can be written in terms of the canonical momenta in the form

H= 1 2I

p2φ sin2θ +p2θ

, (20.124)

so there is not much advantage in transforming to an integral overL1andL2.

On the other hand, whenI1,I2, andI3are all different, there is great simplification in transforming toL1,L2,L3. For example, a normalized probability distribution function for the angular momentaLiwould be

M(L)= β

2πI1

1/2 β 2πI2

1/2 β 2πI3

1/2

exp −β

L21 2I1 + L22

2I2 + L23 2I3

. (20.125)

The quantityM(L)dL1dL2dL3 is the probability of finding an angular momentum in a cube of infinitesimal volume dL1dL2dL3centered atL. The average square of the angular momentum is

L2 =

M(L)(L21+L22+L23)dL1dL2dL3=kBT(I1+I2+I3). (20.126)

Alternatively, we can transform the integration variables to ω1,ω2,ω3,φ,θ,ψ in the partition function Eq. (20.110) by means of the Jacobian

|Jω| =

∂

pφ,pθ,pψ

∂ (ω1,ω2,ω3)

=I1I2I3sinθ. (20.127) This leads to the same partition function as in Eq. (20.116) but we can also deduce that the normalized distribution function for theωiis

M∗(ω)= βI1

2π

1/2βI2

2π

1/2βI3

2π 1/2

exp −β

I1ω21 2 +I2ω22

2 +I3ω23

2 '!

. (20.128)

This leads to an average value ω2 =

M∗(ω)(ω21+ω22+ω32)dω1dω2dω3=kBT 1

I1 + 1 I2 + 1

. (20.129)

It is interesting and physically reasonable that average values of the squares of the principal angular velocities are inversely proportional to their respective moments of inertia.

Grand Canonical Ensemble

In Chapter 19, we derived the canonical ensemble by starting with the microcanonical ensemble. The microcanonical ensemble applies to an isolated system which therefore has a fixed energy; on the other hand, the canonical ensemble applies to a system that has a fixed temperature. The derivation is accomplished by considering the system of interest to be a subsystem of a total system that is isolated and to which the microcanonical ensemble applies. The remainder of the total system, exclusive of the system of interest, acts as a heat reservoir whose temperature is imposed on the system of interest.

In the present chapter, we introduce the grand canonical ensemble (GCE) which applies to a system having a fixed temperature and a fixed chemical potential, but not a fixed energy or a fixed number of particles. Other extensive parameters of the system, which we take to be only the volumeV in the development that follows, are fixed.1Our system of interest is again considered to be a subsystem of a total system that is isolated and therefore has a fixed energy and a fixed number of particles. In this case, the remainder of the total system, exclusive of the system of interest, acts as both a heat reservoir and a particle reservoir for the system of interest. Thus, it imposes its temperature and its chemical potential on the system of interest. But the system of interest will have an average energy,U, and an average number of particles,N, which together with its volumeVwill be sufficient for its thermodynamic description.

By using the GCE for which the number of particles is not specified, we gain the flexibility to treat systems that have quantum mechanical constraints on the number of particles that can occupy a quantum state. We shall use it to treat ideal Fermi and Bose gases whose wave functions must be antisymmetric or symmetric, respectively, when its identical particles are interchanged. For such quantum ideal gases, the grand canonical partition function factors, which is not the case for the canonical partition function. The classical ideal gas will be shown to be a limiting case of either a Fermi gas or a Bose gas.

Thus the approximations used to treat the classical ideal gas by means of the canonical ensemble with the Gibbs correction factor N! can be clarified. Accordingly we treat a classical ideal gas of molecules having internal structure. Dilute systems for which the constituents can be regarded as independent subsystems can also be treated by a grand canonical partition function that factors. We shall illustrate its use to treat adsorption from a gas that imposes its chemical potential on a surface having dilute adsorption

1These are usually the parameters that allow the system to do work. A system without a volume might have an area or a length that is relevant. A system could also have a fixed number of sites that can be occupied by particles.

Thermal Physics.http://dx.doi.org/10.1016/B978-0-12-803304-3.00021-1 359

360 THERMAL PHYSICS

sites. Finally, we use the same methodology as used to derive the GCE to develop a pressure ensemble that we illustrate by treating point defects in crystals under conditions of constant temperature and pressure.

Work Due to Expansion of an Ideal Gas

Statement of the Second Law