Partition function internal energy states

In the limit of high pressure, collisions maintain the thermal distribution of reactant molecules over their internal energy states and consequently TST can be used to determine the thermal rate constants for dissociation and association. However, in the case where there is no maximum in the reaction path leading from reactants to products, it is necessary to take account of angular momentum (/) constraints as well as internal energy. The transition state is not found at a single separation but rather it depends on Eint and J. Then, in the language of the statistical adiabatic channel model (SACM), the partition function for the transition state can be expressed as ... 

LS now consider the problem of calculating the Helmholtz free energy of a molecular 1. Our aim is to express the free energy in the same functional form as the internal that is as an integral which incorporates the probability of a given state. First, we itute for the partition function in Equation (6.21) ... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

For molecular desorption, laser spectroscopic studies of the desorbing molecule can give full internal state distributions, Df Ef, 6f, v, J, f M ), Ts), where f M ) is some distribution function describing the rotational orientation/alignment relative to the surface normal. For thermal desorption in non-activated systems, most atoms/molecules have only modest (but important) deviations from a thermal distribution at Ts. However, in associative desorption of systems with a barrier, the internal state distributions reveal intimate details of the dynamics. Associative desorption results from the slow thermal creation of a transition state, with a final thermal fluctuation causing desorption. Partitioning of the energy stored in V into... 

In practice, it proves more convenient to work within a convention where we define tire ground state for each energy component to have an energy of zero. Thus, we view 1/eiec as the internal energy that must be added to I/q, which already includes Eeiec (see Eq. (10.1)), as the result of additional available electronic levels. One obvious simplification deriving from this convention is that the electronic partition function for the case just described is simply eiec = 1, Inspection of Eq. (10.5) then reveals that the electronic component of the entropy will be zero (In of 1 is zero, and the constant 1 obviously has no temperature dependence, so both terms involving eiec are individually zero). 

Zt represents the partition function for the internal degrees of freedom of the molecule E2 is the energy of the adsorption state... 

Equations (5) to (8) involve the partition function for the energy levels internal to a molecule, and the zero of energy for each molecule is the u = 0 level. The exponential term of Eq. (2) is then required to provide a common external energy reference point for all four molecules. A convenient reference state for the four-molecule system is that corresponding to complete dissociation to free atoms with no kinetic energy, as shown in... 

Once the partition function is known, thermodynamic functions such as the internal energy U and Helmholtz free energy A may be calculated according to For each of the partition functions the sum over allowed quantum states mns to infinity however, since the energies become larger, the partition functions are finite. Let ... 

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