Inertia, moment of

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

Equation XVI-21 provides for the general case of a molecule having n independent ways of rotation and a moment of inertia 7 that, for an asymmetric molecule, is the (geometric) mean of the principal moments. The quantity a is the symmetry number, or the number of indistinguishable positions into which the molecule can be turned by rotations. The rotational energy and entropy are [66,67]... 

Molecular moments of inertia are about 10 g/cm thus 7 values for benzene, N2, and NH3 are 18, 1.4, and 0.28, respectively, in those units. For the case of benzene gas, a = 6 and n = 3, and 5rot is about 21 cal K mol at 25°C. On adsorption, all of this entropy would be lost if the benzene were unable to rotate, and part of it if, say, rotation about only one axis were possible (as might be the situation if the benzene was subject only to the constraint of lying flat... 

The rotational states are characterized by a quantum number J = 0, 1, 2,. .. are degenerate with degeneracy (2J + 1) and have energy t r = ) where 1 is the molecular moment of inertia. Thus... 

The same expression applies to the transition state s rotational energy Ei(J) except that the moment of inertia /... 

The only tenn in this expression that we have not already seen is a, the vibration-rotation coupling constant. It accounts for the fact that as the molecule vibrates, its bond length changes which in turn changes the moment of inertia. Equation B1.2.2 can be simplified by combming the vibration-rotation constant with the rotational constant, yielding a vibrational-level-dependent rotational constant. 

Rotational transition frequencies acquired in the THz region expand upon and complement those acquired in the microwave. Two types of molecules undergo rotational transitions that fall in the FIR molecules witli rotation about an axis having a small moment of inertia, and molecules in high-J states. FIR spectra of the first type of molecules are... 

The effective moment of inertia / and the friction coefficient / could easily be estimated. The force constant k associated with the relative motion of the lobes was determined from an empirical energy function. To do so, the molecule was opened in a step-wise fashion by manipulating the hinge region and each resulting structure was energy minimized. Then, the interaction energy between the two domains was measured, and plotted versus 0. 

In the case of a polyatomic molecule, rotation can occur in three dimensions about the molecular center of mass. Any possible mode of rotation can be expressed as projections on the three mutually perpendicular axes, x, y, and z hence, three moments of inertia are necessar y to give the resistance to angular acceleration by any torque (twisting force) in a , y, and z space. In the MM3 output file, they are denoted IX, lY, and IZ and are given in the nonstandard units of grams square centimeters. 

The balance wheel of a ehronometer is eonstiueted so that its entire mass of 0.100 g may be eonsidered to be eoneentrated in a ring of radius 0.600 em. What is its moment of inertia ... 

Given the bond distances and intemuclear angle in Problem 9, what is the moment of inertia of the H2O molecule about its principal axis through the oxygen atom (the y-axis in File 4-5) ... 

Calculate the moment of inertia about the -axis of ethylene. 

Convert the three moments of inertia in File 4-3 to MKS units. 

What is the moment of inertia of acetylene about its C—C axis ... 

Here the total moment of inertia I of the molecule takes the place of iRe2 in the diatomic molecule case... 

Moleeules for whieh all three prineipal moments of inertia (the li s) are equal are ealled spherieal tops. For these speeies, the rotational Hamiltonian ean be expressed in terms of the square of the total rotational angular momentum P ... 

Moleeules for whieh two of the three prineipal moments of inertia are equal are ealled symmetrie top moleeules. Prolate symmetrie tops have la < Ib = Ic J oblate symmetrie tops have la = Ib < Ic (it is eonvention to order the moments of inertia as la < Ib Ic ) ... 

The rotational Hamiltonian ean now be written in terms of and the eomponent of J along the unique moment of inertia s axis as ... 

When the three principal moment of inertia values are identical, the molecule is termed a spherical top. In this case, the total rotational energy can be expressed in terms of the total angular momentum operator J2... 

Molecules for which two of the three principal moments of inertia are equal are called symmetric tops. Those for which the unique moment of inertia is smaller than the other two are termed prolate symmetric tops if the unique moment of inertia is larger than the others, the molecule is an oblate symmetric top. 

Again, the rotational kinetic energy, which is the full rotational Hamiltonian, can be written in terms of the total rotational angular momentum operator J2 and the component of angular momentum along the axis with the unique principal moment of inertia ... 

The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a so-called asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the J, M, and K quantum numbers. However, given the three principal moments of inertia la, Ib, and Ic, a matrix representation of each of the three contributions to the rotational Hamiltonian... 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

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