Rotation and rigidity

To end this chapter, we deal with one more question which MOs allow us to answer how flexible is a molecule The answer depends on the molecule of course, but more importantly it depends on the type of bond. You may be aware that many alkenes can exist in two forms, cis and trans, also called Z and E (see Chapter 17). These two forms are not usually easy to interconvert—in other words the C=C double bond is very rigid and cannot rotate. 

Maleic and fumaric acids were known in the nineteenth century to have the same chemical composition and the same functional groups, and yet they were different compounds—why remained a mystery. That is, until 1874 when van t Hoff proposed that free rotation about double bonds was restricted. This meant that, whenever each carbon atom of a double bond had two different substituents, isomers would be possible. He proposed the terms cis (latin meaning on this side ) and trans (Latin meaning across or on the other side ) for the two isomers. The problem was which isomer was which On heating, maleic acid readily loses water to become maleic anhydride so this isomer must have both acid groups on the same side of the double bond. 

Compare that situation with butane. Rotating about the middle bond doesn t break any bonds because the c bond is, by definition, cylindrically symmetrical. Atoms connected only by a a bond are therefore considered to be rotationally free, and the two ends of butane can spin relative to one another. 

The same comparison works for ethylene (ethene) and ethane in ethylene all the atoms lie in a plane, enforced by the need for overlap between the p orbitals. But in ethane, the two ends of the molecule spin freely. This difference in rigidity has important consequences throughout chemistry, and we will come back to it in more detail in Chapter 16. 

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If the elements of the structural units hinder the rotation and rigidity of the chain, the macromolecules will take linear conformations, (Figure 3.227a). An experimental proof was presented for the globular structures as well as for the linear ones. 

We will presently give the explicit expressions for the /> and the ( pj pX y for the linear momenta of the translational motion of the center of mass. The corresponding expressions for the rotational motion will be determined separately for the case of rigid linear rotators and rigid symmetric tops in the next sections. 

In any case, the absorption of heat over a considerable temperature range agrees with conclusions from Optical rotation and rigidity measurements that gelation involves an equilibrium which can be progressively shifted by changing the temperature. 

Figure 3 Upper/lower bounds calculation (a) shortest path between target atoms, (b) rotatable and rigid portions (rigid portions treated as a long bond), (c) all-franr rotations gives upper bound, and (d) lower bound from depth-first conformational analysis...

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

It is straightforward to introduce active and adiabatic treatments of K into the widely used RRKM model which represents vibration and rotation as separable and the rotations as rigid rotors [41,42]. Eor a synnnetric top, tlie rotational energy is given by... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

Thus far, exaetly soluble model problems that represent one or more aspeets of an atom or moleeule s quantum-state strueture have been introdueed and solved. For example, eleetronie motion in polyenes was modeled by a partiele-in-a-box. The harmonie oseillator and rigid rotor were introdueed to model vibrational and rotational motion of a diatomie moleeule. 

In Chapter 3 and Appendix G the energy levels and wavefunctions that describe the rotation of rigid molecules are described. Therefore, in this Chapter these results will be summarized briefly and emphasis will be placed on detailing how the corresponding rotational Schrodinger equations are obtained and the assumptions and limitations underlying them. 

At low energies, the rotational and vibrational motions of molecules can be considered separately. The simplest model for rotational energy levels is the rigid dumbbell with quantized angular momentum. It has a series of rotational levels having energy... 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

In Section 5.2 the set of internal state variables k was introduced. In the referential theory, a similar set of referential internal state variables K will be introduced in the same way without further physical identification at this stage. It will merely be assumed that each member of the set K is invariant under the coordinate transformation (A.50) representing a rigid rotation and translation of the coordinate frame. 

A proper orthogonal tensor represents a rigid-body rotation, and R is called the material rotation tensor. It has the properties... 

It is expected that constitutive equations should be invariant to relative rigid rotation and translation between the material and the coordinate frame with respect to which the motion is measured, a property termed objectivity. In order to investigate this invariance, the coordinate transformation... 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

When dealing with the motions of rigid bodies or systems of rigid bodies, it is sometimes quite difficult to directly write out the equations of motion of the point in question as was done in Examples 2-6 and 2-7. It is sometimes more practical to analyze such a problem by relative motion. That is, first find the motion with respect to a nonaccelerating reference frame of some point on the body, typically the center of mass or axis of rotation, and vectorally add to this the motion of the point in question with respect to the reference point. 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Anharmonicity and Nonrigid Rotator Corrections With the rigid rotator and harmonic oscillator approximations, the combined energy for rotation and... 

By starting with this partition function and going through considerable mathematical manipulation, one arrives at the following equations for calculating the corrections to the rigid rotator and harmonic oscillator values calculated from Table 10,4, U... 

Data summarized in Tables 10.1 to 10.3 can be used to solve the exercises and problems given in this chapter. Unless specifically stated otherwise, the rigid rotator and harmonic oscillator approximations (and hence. Table 10.4) and the assumption of ideal gas can be used. 

E10.6 For the diatomic molecule Na2, 5 = 230.476 J-K-1-mol" at T= 300 K, and 256.876 J-K-,-mol-1 at T= 600 K. Assume the rigid rotator and harmonic oscillator approximations and calculate u, the fundamental vibrational frequency and r, the interatomic separation between the atoms in the molecule. For a diatomic molecule, the moment of inertia is given by l pr2, where p is the reduced mass given by... 

Table A4.5 summarizes the equations for calculating anharmonicity and nonrigid rotator corrections for diatomic molecules. These corrections are to be added to the thermodynamic properties calculated from the equations given in Table A4.1 (which assume harmonic oscillator and rigid rotator approximations).

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