## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 5. The Transition from Microscopic to Macroscopic: Statistical Thermodynamics

### 5.5. Internal Contributions

So far, we have limited the discussion to molecules without internal degrees of freedom, that is, molecules without any other motion than translation. In Section 5.2 we saw that the contributions from internal mechanisms in molecules, as long as their energy is additive, can be incorporated in the overall partition function *Z* as a factor, for example, , where *z*_{tra}, *z*_{vib}, *z*_{rot} and *z*_{ele} denote the translation, vibration, rotation and electronic partition function, respectively. In this section we will address the vibration, the (external and internal) rotation, and (briefly) the electronic transitions that can occur in molecules. The pattern of how to calculate thermodynamic properties will be clear by now: find the energy expression, evaluate the partition function; and calculate from the partition function the thermodynamic properties.

**5.5.1 Vibrations**

The simplest molecule with internal structure is a diatomic molecule with *m*_{1} and *m*_{2} as the masses. The vibrations of a diatomic molecule can be described by the harmonic oscillator model with spring constant *a* and using the reduced mass *μ* = *m*_{1}*m*_{2}/(*m*_{1} + *m*_{2}) of the two atoms as the mass in the model. The energy of the oscillator using the circular frequency *ω*, as given in Section 2.2, is

Hence, for the average energy *U* = ⟨*ε*⟩ we find

If we use *β* = (*kT*)^{−1}, the partition function *z*_{vib} for a single vibrator can be written as

and, since this is a geometric series, we can evaluate the sum as

(5.41)

This expression cannot be used for a too-high temperature for two reasons. First, at higher temperature *T*, anharmonicity occurs and the harmonic oscillator model no longer applies. Second, in summing *z*_{vib} we assumed that *n* → ∞, but in reality *n* is finite. Knox [8] estimated that one should have *kT* < 0.1*D*, where *D* is the dissociation energy. For a typical value of *D* = 4 eV, this results in *T* < 4600 K.

Since *z*_{vib}⟨*ε*⟩ = ∑* _{n}* exp[−

*ħωβ*(

*n*+ ½)]

*ħω*(

*n*+ ½) = −∂

*z*

_{vib}/∂

*β*, we can write

(5.42)

The Helmholtz energy *F* is given by

(5.43)

so that the entropy *S* is given by

__(5.44)__

The energy can also be calculated from *U* = *F* + *TS*. The heat capacity *C _{V}* reads

__(5.45)__

At low and high temperature the expression for *C _{V}* expands to, respectively,

The behavior of the harmonic oscillator is characterized by *ω*. Equivalently, we use the *characteristic* (*vibration*) *temperature* *θ*_{vib}, given as *θ*_{vib} = *ħω* /*k*. When *T* >> *θ*_{vib}, the behavior can be classified as classical and the energy becomes *U* ≅ *kT*. When *T* << *θ*_{vib}, expansion of the expression results in *U* ≅ ½*ħω*. When *T* ≅ *θ*_{vib}, the full expression must be used. Since the characteristic temperature for vibration is typically hundreds of degrees, this implies that in most cases the quantum expression for the partition function must be applied. Values for *θ*_{vib} as well as the equilibrium distance *r*_{eq} and dissociation energy *D*_{eq} for some molecules are given in __Table 5.1__.

** Table 5.1** Vibration, rotation, and dissociation data for several diatomic molecules.

^{a)}

For a *N*-atomic molecule we can describe the vibrational behavior by 3*N* − 6 *normal coordinates* (3*N* − 5 for a linear molecule), each of which can be modeled as a harmonic oscillator__ ^{13)}__. These normal coordinates are independent, and hence the behavior can be described as the sum of the behaviors of the individual normal coordinates. In fact, they are an example of generalized coordinates. For example, for H

_{2}O the three vibrational temperatures are 2294, 5262, and 5404 K, respectively, while they read 960, 960, 132, and 3380 K for CO

_{2}.

As an aside, we note that the harmonic oscillator results are directly applicable to the *Einstein model* for solids in which each atom vibrates independently of the others with the same (Einstein) frequency *ω*_{E}. In this case, we have for *N* atoms , that is, without the factor *N*!^{−1} because, although the atoms are indistinguishable, the lattice sites distinguishable and the total number of configurations is *N*!.

Problem 5.17

Show that the entropy *S* for the harmonic oscillator is given by __Eq. (5.44)__.

Problem 5.18

Show that the specific heat *C _{V}* for the harmonic oscillator is given by

__Eq. (5.45)__. Show also that for

*T*>>

*θ*

_{vib},

*U*≅

*kT*and

*C*≅

_{V}*k*.

Problem 5.19

Show that at 300 K most molecules are in the vibrational ground state.

Problem 5.20: The anharmonic oscillator*

Anharmonicity changes the energy expression and selection rule for the oscillator to

where *ε _{n}*/

*hc*is the energy in cm

^{−1}, the vibrational wave number (see footnote 11; typically 100 to 4000 cm

^{−1}), and

*x*is the

*anharmonicity constant*(typically ∼0.01). Estimate the contribution to

*C*from anharmonicity.

_{V}**5.5.2 Rotations**

The rotation behavior of molecules is somewhat more complex than their vibration behavior. We start with homonuclear and heteronuclear diatomic molecules, and thereafter deal briefly with polyatomic molecules and internal rotation.

The energy of the diatomic rigid rotator, as given in Section 2.2, is

(5.46)

Here, the moment of inertia *I* = *μr*^{2} with the reduced mass *μ* = *m*_{1}*m*_{2}/(*m*_{1} + *m*_{2}) and *m*_{1} and *m*_{2} the masses of the particles. The constant *B* is the *rotational constant*^{12}, alternatively expressed as the *characteristic* (*rotation*) *temperature* *θ*_{rot} = *B*/*k*. Each of the energy levels *J* has a (2*J* + 1) degeneracy. At low temperature only the first few terms are contributing, and we have

At high temperature the summation can be approximated by integration and thus

__(5.47)__

Actually, we have to divide *z*_{rot} by the symmetry number *σ*, denoting the number of ways the molecule can be rotated into a configuration indistinguishable from the original configuration. Obviously, *σ* = 1 for AB and *σ* = 2 for AA molecules. The origin of this factor lies in the symmetry of the overall wave function and for the details of the derivation we refer to the literature (e.g., McQuarrie [9]). A more complete analysis using the Euler–McLaurin summation expression results in

which is good to within 1% for *θ*_{rot} < *T*. Replacing the summation by integration is thus only justified at sufficiently elevated temperature, say for *T* > 5*θ*_{rot}. For a lower *T*, one must sum *z*_{rot} term by term. Typical values for *θ*_{rot} are given in __Table 5.1__.

If a polyatomic molecule is linear, the same expression as for diatomic molecules applies, of course, with adjusted moments of inertia. For a nonlinear polyatomic molecule a similar, but more complex, reasoning leads to

(5.48)

where *I _{x}*,

*I*and

_{y}*I*denote the principal moments of inertia or, equivalently,

_{z}*θ*,

_{x}*θ*and

_{y}*θ*the rotational temperatures. The symmetry number has the same meaning as before. As examples we quote

_{z}*σ*(H

_{2}O) = 2,

*σ*(NH

_{3}) = 3,

*σ*(CH

_{4}) = 12,

*σ*(CH

_{3}Cl) = 3,

*σ*(C

_{6}H

_{6}) = 12, and

*σ*(C

_{6}H

_{5}Cl) = 2. For polyatomic molecules the semi-classical partition function is generally valid for

*T*> 100 K.

Internal rotations, such as the transition from the staggered to eclipsed conformation in CH_{3}–CH_{3}, also contribute. In the case of molecules for which the associated barrier is relatively large, say > 10*kT*, the two groups only vibrate with respect to each other and the harmonic oscillator approximation can be used. If the barrier to internal rotation is relatively low, say < *kT*, one could consider the internal rotation as a free rotation for which the partition function is

(5.49)

where *I*′ is the moment of inertia for the internal rotation. In the case of ethane, however, the barrier is ∼12.1 kJ mol^{−1}, while *N*_{A}*kT* at 300 K is ∼2.5 kJ mol^{−1} and thus the neither the harmonic oscillator nor the free rotation approximation may be used. The internal rotation contribution is often the main source of error in calculating thermodynamic data (see, e.g., Ref. [10] for a more detailed discussion).

**5.5.3 Electronic Transitions**

Generally, the spacing between electronic states in molecules is large as compared with *kT*. This implies that transitions from the ground state to an excited state are rare, and only the electronic ground state has to be taken into account. Hence the electronic partition function is generally closely approximated by

(5.50)

where *g _{i}* indicates the electronic state degeneracy, for the ground state often

*g*

_{0}= 1. Taking the zero level at the electronic ground state the approximation

*z*

_{ele}=

*g*

_{0}is in most cases sufficiently accurate for both atoms and molecules. One exception is formed by the systems atoms where the electronic ground state is split by spin-orbit coupling. For example, for the F atom the ground state is split in a lower

^{3}P

_{3/2}level with

*g*

_{0}= 4 and an upper

^{2}P

_{1/2}level with

*g*

_{1}= 2. The energy gap between these levels is 404 cm

^{−1}, while the gap with the next electronic level is much larger than

*kT*= 207 cm

^{−1}at 298 K. Another example is the NO molecule with an unpaired electron in a π* orbital giving rise to a

^{2}Π

_{1/2}(

*g*

_{0}= 2) and a

^{2}Π

_{3/2}(

*g*

_{1}= 2) level with a gap of 121 cm

^{−1}. Again, the gap to the first excited (in this case vibration) level is much larger than

*kT*.

Problem 5.21

Estimate the temperature above which the replacement of the summation in the calculation of the rotational partition function by integration is justified. Take typical numbers from __Table 5.1__.

Problem 5.22

Calculate the contribution of the ^{3}P_{3/2} and ^{2}P_{1/2} levels for the F atom to *C _{V}* at room temperature and compare the value obtained with the one for translation.

Problem 5.23

Show that for a diatomic molecule the contribution of rotation to the heat capacity *C _{V}* is given by and to the entropy

*S*by .

Problem 5.24

Show that the most probable rotational state for a diatomic molecule is given by *J*_{max} ≅ (*T*/2*σθ*_{rot})^{1/2} − 1/2 by treating *J* as a continuous variable (high temperature approximation).

Problem 5.25: The non-rigid rotor*

Diatomic molecules are not rigid rotors, and the energy is more appropriately described by

and the vibrational wave number. For the HCl molecule the rotational constant *B* and *centrifugal distortion constant* *D* are 10.59 cm^{−1} and 5.28 × 10^{−4} cm^{−1}, respectively. Calculate the *C _{V}* according the rigid rotor energy expression,

__Eq. 5.47__. Estimate the contribution of the centrifugal distortion on the energy levels and compare the associated

*C*with the total

_{V}*C*.

_{V}