Rotational partition functions, calculation

Here is the rotational partition function calculated at the transition state geometry and kj=o is the thermal rate calculated only for J=0. Thus, within the J-shifting approximation, the thermal rate constant can be obtained from a dynamical simulation for J=0 only. 

To calculate the rotational partition function for the molecule we need to be careful and check whether the assumption under which Eq. (56) has been derived is valid. 

The rotational microwave spectrum of a diatomic molecule has absorption lines (expressed as reciprocal wavenumbers cm ) at 20, 40, 60, 80 and 100 cm . Calculate the rotational partition function at 100 K from its fundamental definition, using kT/h= 69.5 cm" at 100 K. 

Isotope (H (deuterium), discovered by Urey et al. (1932), is usually denoted by symbol D. The large relative mass difference between H and D induces significant fractionation ascribable to equilibrium, kinetic, and diffusional effects. The main difference in the calculation of equilibrium isotopic fractionation effects in hydrogen molecules with respect to oxygen arises from the fact that the rotational partition function of hydrogen is nonclassical. Rotational contributions to the isotopic fractionation do not cancel out at high T, as in the classical approximation, and must be accounted for in the estimates of the partition function ratio /. 

To calculate the molecular rotational partition function for an asymmetric, linear molecule, we use Eq. 8.16 for the energy level of rotational state /, and Eq. 8.18 for its degeneracy. As discussed in Section 8.2, rotational energy levels are very closely spaced compared to k/jT unless the molecule s moment of inertia is very small. Therefore, for most molecules, replacing the summation in Eq. 8.50 with an integral introduces little error. Thus the... 

The translational contribution to the molecular partition function, which is calculated using Eq. 8.59, clearly makes the largest contribution. (In obtaining this value, we also made use of the ideal gas law to calculate the volume V = 0.02479 m3 of a mole of gas at this temperature and pressure.) The rotational partition function is evaluated via Eq. 8.67, and the vibrational partition function for each mode is found via Eq. 8.71. Only the very... 

The activated complex partition function has contributions from translation (with total mass w,4 +mj) and from rotation of the (linear) activated molecule. Assuming that the bond length of C is the sum of the atomic radii r a and rg, the rotational partition function for the activated complex can be calculated from Eq. 8.65, the moment of inertia / = m 2(rA + re)2, where m 2 is the A-B reduced mass (Eq. 10.38). 

Rotation-vibration interactions, if present, make the calculation more difficult, because then the vibration and rotation partition functions are coupled and cannot be separate factors. 

The classical approximation for the rotational density of states of a molecule is familiar from elementary statistical mechanics, where it is common to assume that the rotational states form a continuum in calculating the rotational partition function. For the external rotations of most molecules this approximation is very good. For example the classical approximation for the rotational partition function of an asymmetric top is... 

Relative CO2-HCI and C02-HBr concentrations were measured under conditions similar to those used in the photoinitiated reactions, but in a completely separate apparatus that used pulsed slit nozzle expansions and tunable infrared diode laser spectroscopy. This was achieved by recording fully rotationally resolved CO2-HXIR absorption spectra using the CO2 asymmetric stretch chromophore, as described in Section III.A. The rotational distributions fit temperatures quite well, usually 3K. Rotational partition functions were calculated for the measured rotational temperatures and the areas under the absorption lines recorded for all transitions. From this, relative concentrations were determined. Details are available elsewhere [139, 140]. 

The moment of inertia can be derived from spectroscopic data or it can be calculated from the dimensions of the molecule, so that the rotational partition function can be determined. The values of the moments of inertia of a number of diatomic molecules in their ground states are given in Table VIII. ... 

Problem Calculate the rotational partition function of (i) hydrogen gas, (ii) iodine chloride gas, at 300 K. 

Derive the value of the universal constant a in the expression Qr aa IT for the rotational partition function of any diatomic (or any linear) molecule I is the moment of inertia in e.g.s. units and T is the absolute temperature. Calculate the rotational partition function of carbon dioxide (a linear symmetrical molecule) at 25 C. 

Calculated from translation/rotational partition functions (9). b Estimated activation energy for amide substrate of chymotrypsin (38). s 25 cal/mole/A2 of buried surface area. 

The ratio of the translational partition functions is virtually 1 (because the masses nearly cancel explicit calculation gives 0.999). The same is true of the vibrational partition functions. Although the moments of inertia cancel in the rotational partition functions, the two homonuclear species each have er = 2, so... 

Similarly, most terms cancel in the calculation of the ratio of rotational partition functions. For diatomic molecules and linear polyatomic molecules, this ratio is given by fjl ... 

For calculations of rotational partition functions the moment of inertia (e.g. the molecule structure) should be known. In case of a diatomic moleule the rotation partition function is... 

Calculated Vibrational-Rotational Partition Functions and Free-Energy Contributions of HC1 as a Function of Temperature1 ... 

Vibration-rotation partition function for HC1 obtained via Fourier path-integral AOSS-U Monte Carlo calculations from Topper et al. [46]. Error bars are given at 95% confidence level (2w ). Unless otherwise noted, all calculations used = 128 Fourier coefficients per degree of freedom and n = 100000 Monte Carlo samples. 

This is the value of the rotational partition function for unsymmetrical linear molecules (for example, heteronuclear diatomic molecules). Using this value of we can calculate the values of the thermodynamic functions attributable to rotation. 

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