Collision, diameter theory

The intermolecular distance at which the reaction can occur may also be different from the collision diameter taken in the collision theory. 

Using gas kinetic molecular theory, show that under typical atmospheric conditions of pressure and temperature corresponding to an altitude of 5 km (see Appendix V) collisional deactivation of a C02 molecule will be much faster than reemission of the absorbed radiation. Take the collision diameter to be 0.456 nm and the radiative lifetime of the 15-/rm band of C02 to be 0.74 s (Goody and Yung, 1989). 

Based on kinetic-theory principles and the Eucken correction, develop a general expression for the thermal conductivity of diatomic gases. Collect and combine all the constants, such that the expression depends on the molecular weight (g/mol), temperature (K), collision diameter (A), and reduced temperature T (nondimensional). 

Compute the fall-off curve using QRRK theory. For this calculation, assume a collision diameter of 4.86 A. Assume that the average energy transfer per N2-C-C5H5 collision is -0.69 kcal/mol (needed to calculate the parameter /5 used in the model). Take the number of oscillators to be, v = actual, with the frequency calculated above. Assume the reaction barrier to be E0, given above. 

The molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

Figure 6. Logarithmic plot values of LUm/fc versus pressure for the Cope rearrangement of bullvalene (torr at the experimental temperature of 356 K). Experimental values are signified by solid circles. Pressures are the total sample pressure at 356 K. Errors in frUni/fc are reported to 2o. The solid (upper) line represents the values calculated from RRKM theory using the biradicaloid transition state model. The lower line represents calculated rate constants using the aromatic transition-state model. The collision diameter was 3.6 A in both cases.

Liu and Ruckenstein [Ind. Eng. Chem. Res. 36, 3937 (1997)] studied self-diffusion for both liquids and gases. They proposed a semiem-pirical equation, based on hard-sphere theory, to estimate self-diffusivities. They extended it to Lennard-Jones fluids. The necessary energy parameter is estimated from viscosity data, but the molecular collision diameter is estimated from diffusion data. They compared their estimates to 26 pairs, with a total of 1822 data points, and achieved a relative deviation of 7.3 percent. 

Since hvfkT will generally be greater than unity, we see that the steric factors computed by the detailed theory will generally be about an order of magnitude greater than those computed from the transition-state theory and may in fact be somewhat greater than unity. Unfortunately, collision diameters are not well enough known to check such calculations with any reliability. 

Unlike the elementary kinetic theory, the three collision integrals ( 2D AB, Qu and T2a) are introduced in the Chapman-Enskog theory. Moreover, the collision diameter (er ) is used instead of the molecular diameter (d ). 

Although the Enskog theory is formally valid for rigid spheres only, fairly accurate results have been obtained for real gases as well provided that the effective collision diameter is appropriately adjusted. 

Stefan-Boltzmann constant, = 5.67 x 10 W/rn K ) alternative total (scattering) collision cross section rn ) collision diameter used in kinetic theory (m) differential scattering cross section rn ) surface tension N/m)... 

Example 2-2 Use the collision theory to estimate the specific reaction rate for the decomposition of hydrogen iodide, 2HI -> H + H2-Assume that the collision diameter a is 3.5A (3.5 x 10 cm), and employ the activation energy of 44,000 cal/g mole determined experimentally by Bodenstein. Also evaluate the frequency factor. 

Using the collision theory, calculate the rate constant at 300°K for the decomposition of hydrogen iodide, assuming a collision diameter of 3.5A and an activation energy of 44 kg cal (based on a rate constant in concentration units). To what entropy of activation does the result correspond ... 

Kinetic theory predicts the self-diffusion coefficient, tbei is, the diffusion coefficient for a gas mixture in which all molecules have identical molecular weights and collision diameters, to be12... 

In the simplest version of the kinetic theory of gases, molecules are treated as hard spheres of diameter d which make binary collisions only. In this approximation the mean distance traveled by a molecule between successive collisions, the mean free path I, is related to the collision diameter by ... 

Using the same model and the same standard pressure, the collision diameter can be calculated from the viscosity t] by the kinetic theory relation ... 

For the sake of simplicity we shall take the composition of air as 4 parts Ng to 1 part Og. We shall assume collision diameters as follows (Chapman and Cowling, The mathematical theory of non-uniform gases , Cambridge University Press, 1939, p. 229)... 

Our theory is done, then, except for some details concerning p and cr. In general these quantities can be only estimated, so that often an effective collision diameter, f7g, is used. This is defined by... 

Using the appropriate collision-theory rate constant, determine the effective collision diameter for this reaction based on the data above. Is this quantity a function of temperature ... 

Our examinations yield for the first time an exact correction to the simple classical collision theory. This theory is based on the concept of the "collision diameter" rigorously defined by equation (22.IV). The actual physical meaning of this concept in the framework of an accurate collision theory becomes clear if we recall our discussion in Sec.4.1.III concerning the conditions at which 1 (or... 

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