Linear activated complex - degrees freedom

Question. Given that the total number of degrees of freedom for a polyatomic molecule is 3N, calculate the number of vibrational modes open to (a) an atom, (b) a diatomic molecule, (c) a non-linear polyatomic molecule with N atoms and (d) a non-linear activated complex with N atoms. 

The expression in Eq. (10.56) only becomes simple in the special case when the reactants A and B are atoms with no internal degrees of freedom. Then, in the second factor which is related to the gas-phase reaction, Va = Vb = 0, and with no interactions between the reactant molecules, Vint, = 0. The integral over reactant states is therefore simply equal to the square of the volume, V2. The reaction coordinate Q is the distance, r, between the atoms, so the five coordinates involve the center-of-mass coordinates and the rotational coordinates for the linear activated complex. The integrals over them cancel, since both p( ) and V/ t ra are independent of them. There are no other integration variables, so the integrands just take the value they have for <3 = 0. We find... 

Assume that the activated complexes are non-linear. Determine the temperature dependence for hv C fc///, corresponding to classical partition functions for the harmonic vibrational degrees of freedom, as well as for hv 3> ksT. 

Recent calculations (see Section 3.1) show that the activated complex is non-linear, that is, the average rotational energy is (3/2)ksT and Ea = Eq + (.E ib) — (E ib). %2 = 4395 cm-1 and the two vibrational frequencies associated with the activated complex are 3772 cm-1 and 296 cm-1, respectively (remember that the third vibrational degree of freedom of the non-linear triatomic molecule is the reaction coordinate which is not included in (/A). The thermal energies associated with the... 

Consider the reaction NO + CI2 NOCl + Cl. The values for 9 , 6, and for NO and CI2 are given in Table 29.1. Estimate the frequency factor for this reaction at 300 K using the Eyring equation. Assume that the activated complex is linear Cl—Cl—N—O and that the N—Cl distance is 200 pm while the Cl—Cl and N—O distances are the same as in the separated molecules. The degeneracy of the electronic state is the same in the initial and in the activated state. Assume that for all the vibrational degrees of freedom,/ = 1. 

With this brief introduction to partition functions, let us return to Figure 6.1 and derive a rate of reaction, as represented by step 6.2, which can be thought of as a very loose vibration resulting in bond rupture (or formation) and passage over the peak of the energy pathway to give the product. Equation 6.1 describes the quasi-equilibrated formation of the activated complex, which possesses either 3(Na + Nb) — 6 or 3(Na + Nb) - 5 degrees of vibrational freedom for a nonlinear or a linear complex, respectively. We note that... 

In our particular case, the activated complex is linear and therefore has two degrees of freedom of rotation, and the rotation contribution is ... 

The rates of the isothermal dehydration of these salts were studied at 130, 150 and 170"C using TGA (Saito 1988). Linear plots of [1 — (1 — versus time (a is the degree of reaction) reflect a surface-controlled dehydration mechanism. The activation energies calculated from these data, which agreed with those reported earlier (Nathans and Wendlandt 1962), are inversely proportional to the cationic radius r. Such a relation is expected for the ionic interactions dominant in lanthanide complexes. The entropy of activation AS for these solid phase reactions varies from — 226 J K mol for La to — 123 J K mol for Tm. The variation in entropy is related to changes in the degree of rotational freedom of water molecules in the activated states as a result of lattice expansion. The linearity of the plot of A versus AS was interpreted as evidence for a common dehydration mechanism independent of the particular lanthanide cation present. 

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