Thermal partition function

Similar to a thermal partition function, let us define the generating function of the moments of H as... 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

Thermal averages in the ensemble with constant pressure p are given via the corresponding partition function dVexp[—/3pF]Z(A, F, T). 

In order to obtain the partition function for systems of this type (where the thermal energy and potential barrier are of the same magnitude), it is necessary to have the quantum mechanical energy levels associated with the barrier. Pitzer5 has used a potential of the form... 

Figure 10. Arrhenius plot of the thermal rate constants for the 2D model system. Circles-full quantum results. Thick solid (dashed) curve present nonadiabatic transition state theory by using the seam surface [the minimum energy crossing point (MECP)] approximation. Thin solid and dashed curves are the same as the thick ones except that the classical partition functions are used. Taken from Ref. [27].

The quantum Boltzmann distribution only applies between allowed energy levels of the same family and each type of energy has its own characteristic partition function, that can be established by statistical methods and describes the response of a system to thermal excitation. If the total number of particles N, is known, the Boltzmann distribution may be used to calculate the number, or fraction, of molecules in each of the allowed quantum states. For any state i... 

From the known partition function the thermal energy for each mode can be calculated as the sum over all energy levels. Each term consists of the number of molecules in a quantum state Ni multiplied by the number of states gt at that level and the energy of the level above the ground state, tt - e0. If the total energy is represented by the symbol U, the thermal energy is... 

The function g is the partition function for the transition state, and Qr is the product of the partition functions for the reactant molecules. The partition function essentially counts the number of ways that thermal energy can be stored in the various modes (translation, rotation, vibration, etc.) of a system of molecules, and is directly related to the number of quantum states available at each energy. This is related to the freedom of motion in the various modes. From equations 6.5-7 and -16, we see that the entropy change is related to the ratio of the partition functions ... 

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

The reaction described by Equations 14.15 and 14.16 is a thermal system so that the equilibrium constant is given in terms of a ratio of canonical partition functions Q of Chapter 4 with the partition functions for M canceling... 

Recently, the concept of thermal entanglement was introduced and studied within one-dimensional spin systems [64-66]. The state of the system described by the Hamiltonian H at thermal equilibrium is p T) = exp —H/kT)/Z, where Z = Tr[exp(—7//feT)] is the partition function and k is Boltzmann s constant. As p T) represents a thermal state, the entanglement in the state is called the thermal entanglement [64]. 

Now that the entanglement of the XY Hamiltonian with impurities has been calculated at Y = 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble p = jZ, where = l/k T, and Z = Tr is the partition function. The thermal density matrix is diag-... 

If the energy separating the vibrational quantum states is small relative to the thermal energy, the partition function for vibration is approximately given by... 

The key to a treatment of molecular clusters in situations of thermal equilibrium are the N-particle partition functions. Specifically, the classical two-particle partition function, Z2(T), is given by [183, 184, 377]... 

The vibrational frequency of the CS molecule is 1285.08 cm-1, or (multiplying by the speed of light) v=3.8526 x 1013 s 1. At 300 K, the factor in the exponent of Eq. 8.71 is x = 6.1632. Thus the partition function room temperature the fraction of vibrationally excited CS molecules is very small. However, at T — 5000 K, x = 0.3698, and <7vib=3.235. Thus, at very high temperatures, the thermal population of vibrationally excited molecules becomes significant. 

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

It is easily shown that, in the classical limit, Eqs. (41) and (42) are consistent with the thermal capture rate constants for the oscillator model of charge-permanent dipole capture. The relevant part of the activated complex partition function, instead of Eq. (11), can be written as... 

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