Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be [Pg.407]

It is now necessary to examine the partition function in more detail. The energy states for translation are assumed to be given by the quantum-mechanical picture of a particle in a box. For a one-dimensional box of length a. [Pg.607]

Finally, the generalization of the partition function q m transition state theory (equation (A3.4.96)) is given by [Pg.783]

Kaufman B 1949 Orystal statistics II. Partition function evaluated by Spinor analysis Phys. Rev. 65 1232 [Pg.556]

Of these, A are indistinguishable since the molecules are not labeled, and the complete partition function for N molecules becomes [Pg.607]

Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written [Pg.606]

All molecules in the second and subsequent layers are assumed to behave similarly to a liquid, in particular to have the same partition fimction. This is assumed to be different to the partition function (A2.2) of molecules adsorbed into the first layer. [Pg.1874]

Thennodynamics of ideal quantum gases is typically obtained using a grand canonical ensemble. In principle this can also be done using a canonical ensemble partition function, Q =. exp(-p E ). For the photon and [Pg.424]

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. [Pg.993]

Since other sets of constraints can be used, there are other ensembles and other partition functions, but these tliree are the most important. [Pg.375]

There is an inunediate coimection to the collision theory of bimolecular reactions. Introducing internal partition functions excluding the (separable) degrees of freedom for overall translation. [Pg.780]

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. [Pg.783]

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. [Pg.609]

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function [Pg.375]

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites [Pg.120]

Otlier expressions for tire diffusion coefficient are based on tire concept of free volume [57], i.e. tire amount of volume in tire sample tliat is not occupied by tire polymer molecules. Computer simulations have also been used to quantify tire mobility of small molecules in polymers [58]. In a first approach, tire partition functions of tire ground [Pg.2536]

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