Mass-moments of inertia

E = positive phase impulse, or mass moment of inertia... 

The secondary a-deuterium KIEs calculated for the uncatalysed reaction were in the range found experimentally for other SN2 methyl transfers. The calculated KIE was also analysed in terms of the zero-point energy (ZPE), the molecular mass-moment of inertia (MMI) and the excitation (EXC) contributions to the total isotope effect. The inverse KIE was found to arise from an... 

The MMI (mass moment of inertia), EXC (excitation factor), and ZPE (zero point energy) terms are defined on successive lines of Equation 4.145. For reactions involving heavier isotopes the effects are no longer concentrated in the ZPE term and it is convenient to apply the Teller-Redlich product rule (Section 3.5.1) and eliminate the moments of inertia by using Equations 4.79,4.79a, and 4.141, thus obtaining an equivalent relation... 

For historic and practical reasons hydrogen isotope effects are usually considered separately from heavy-atom isotope effects (i.e. 160/180, 160/170, etc.). The historic reason stems from the fact that prior to the mid-sixties analysis using the complete equation to describe isotope effects via computer calculations was impossible in most laboratories and it was necessary to employ various approximations. For H/D isotope effects the basic equation KIE = MMI x EXC x ZPE (see Equations 4.146 and 4.147) was often drastically simplified (with varying success) to KIE ZPE because of the dominant role of the zero point energy term. However that simplification is not possible when the relative contributions from MMI (mass moment of inertia) and EXC (excitation) become important, as they are for heavy atom isotope effects. This is because the isotope sensitive vibrational frequency differences are smaller for heavy atom than for H/D substitution. Presently... 

A AZPE = AZPEii — AZPEd AZPEt) corresponds to the terms for the reactions of monodeuteriated aldehydes. Terms defined by IE = MMl x EXC x EXP (IE is the Isotopic exchange equilibrium, MMl is the mass moment of inertia term representing the rotational and translational partition function ratios, EXC is the vibrational excitation term and EXP is the exponential zero point energy). 

The partition function of a molecule also contains torsional motions and the construction of such a function requires the knowledge of molecular mass, moments of inertia, and constants describing normal vibration modes. Several of these data may be acquired from infrared and Raman spectra (67SA(A)891 85JST( 126)25), but the procedure has not yet been extensively applied owing to experimental limitations. To characterize the barrier one also needs to know more than one constant, and these are often not available from... 

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

The expressions for c, ci and c2 are derived from statistical mechanics on the approximate assumption that the masses of N and O are equal and the masses, moments of inertia and frequencies of N2, NO and 02 are equal the differences in the symmetry and multiplicity of the main terms are taken into account. Comparison with the exact computations bears out the applicability of the simple formulas derived above. 

In the calculation of the thermodynamic properties of the ideal gas, the approximation is made that the energies can be separated into independent contributions from the various degrees of freedom. Translational and electronic energy levels are present in the ideal monatomic gas.ww For the molecular gas, rotational and vibrational energy levels are added. For some molecules, internal rotational energy levels are also present. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies are summarized in Appendix 6. 

Translational and electronic energy levels are present in the ideal monatomic gas. For the molecular gas, rotational and vibrational energy levels are added. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies follow. 

In this Chapter the polar moment of inertia of a cross-section area (dimensions L ) and the mass moment of inertia (dimensions ML ) are represented by I and J, respectively. 

The one-frequency model represented by Eqs. (11.3)-(11.8) shows single isotopic frequency expressions for the MMI (mass/moment of inertia), ZPE (vibrational zero-point energy), and EXC (excited vibrations) terms of the usual Bigeleisen equation [21]. The extra term tun is the truncated Bell tunnel correction [22], used here to provide a simple way to express a tunneling effect in terms of a reaction-coordinate frequency, vh... 

The transition between a normal and an inverse EIE reflects the fact that these systems are not characterized by the typical monotonic variation predicted by the van t Hoff relationship. Instead, as discussed in an edifying review by Parkin,194 the EIEs in these systems are zero at OK, increase to a value >1, and then decrease to unity at infinite temperature. This unusual behavior is, nevertheless, rationalized by consideration of the individual factors that contribute to the EIE. As discussed above, the EIE may be expressed in the form EIE = SYM x MMI x EXC x ZPE (where SYM is the symmetry factor, MMI is the mass moment of inertia term, EXC is the excitation term, and... 

Next is the layout of the micromechanical and electronic parts of the system. Even at this stage changes are being made, which need to be back-annotated to the component level and also to the system level, if necessary. In electronics, for example, the back-annotated parameters are the specific capacitances, which are not known before layout. In mechanical micromachining, the specific parameters include capacitances as well as masses, moments of inertia, thermal capacities, and conductivities. 

Problems connected with mean densities, centres of mass, moments of inertia, mean pressures, and centres of pressure are treated by the aid of the above principles. 

Kinetic analysis — measures or estimates forces produced by body segments and other biomechanical parameters (e.g., center of mass, moment of inertia, etc.) and physical parameters of objects (e.g., mass, dimension, etc.) (the data are used for subsequent biomechanical analyses)... 

In a separate study, kinetic isotope effects were determined for the individual steps in methane loss from tungstanocene methyl hydride. These results were combined with theoretical studies that produced a complete picture for understanding isotope effects in terms of the contributing factors (zero point energy, mass moment of inertia, and excitation terms). The conclusion was that it is possible to observe either inverse or normal isotope effects, depending on the temperature at which the study is conducted. ... 

Another approach was provided by Marcus and Ben-Naim (1985) 24 years ago, based on an earlier work of Ben-Naim (1975) and the following ideas. The molecular parameters of light and heavy water are very similar (Table 1.1), except those that depend on the mass (moments of inertia) (Marcus 1998). For H2O and D2O molecules the bond lengths 0-H and 0-D are 0.09572 and 0.09575 nm, the bond angles H-O-H... 

B. Dynamics 1. Linear motion (e.g., force, mass, acceleration, momentum), 2. Angular motion (e.g., torque, inertia, acceleration, momentum), 3. Mass moments of inertia, 4. Impulse and momentum applied to a. particles, b. rigid bodies, 5. Work, energy, and power as applied to a. particles, b. rigid bodies, 6. Friction... 

In Chapter 9, we will look at another similarly defined property of an object, mass moment of inertia, which provides a measure of resistance to rotational motion. 

The objective of this chapter is to introduce the concept of mass and mass related quantities encountered in engineering. We will begin by discussing the building blocks of all matter, atoms and molecules. We will then introduce the concept of mass in terms of a quantitative measure of the amount of atoms possessed by a substance. We will then define and discuss other mass-related engineering quantities, such as density, specific gravity, mass moment of inertia, momentum, and massflow rate. In this chapter, we tvill also consider conservation of mass and its application in en neering. 

Well, what does all this have to do with mass Mass provides a quantitative measure of how many molecules or atoms ate in a given object. The matter may change its phase, but its mass remains constant. Some of you will take a dass in dynamics where you will learn that on a macroscopic scale mass also serves as a measure of resistance to motion. You already know this from your daily observations. Whidi is harder to push, a motorcyde or a truck As you know, it takes more efibrt to push a truck. When you want to rotate something, the distribution of the mass about the center of rotation also plays a significant role. The further away the mass is located om the center of rotation, the harder it wiU be to rotate the mass about that axis. A measure of how hard it is to rotate something with respect ro center of rotation is called mass moment of inertia. We will discuss this in detail in Section 9-5. 

The mass moment of inertia of a point mass. Mass moment of inertia of a system consisting of three point masses. 

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