Gases partition function

The theoretical ideal-gas partition function for a system of c components of composition N=NU N2,..., Nc contained in a volume V at temperature T is... 

The results might be improved by modifying the ideal gas partition function to allow for gas imperfections or by introducing the hindered rotation into the solidlike partition function. The success of significant structure theory in predicting the velocity of sound and the van der Waals constant a, which are dependent on the second derivatives of the partition function, is another piece of evidence for its general applicability. 

It is the genius of TST that only the saddle point of the PES need be examined. At low pressure, we can use the ideal gas partition functions for translational motion ... 

This is the final form of the pressure equation for a system of spherical particles obeyingH he assumption of pairwise additivity for the total potential energy. Note that the first term is the ideal gas pressure (i.e., starting with the ideal gas partition function with 1/ = 0). The second term carries the effect of the intermolecular forces on the pressure. Note that, in general, g(R) is a function of density so that this term is not the second-order term in the density expansion of the pressure. There is a... 

It is noteworthy that a statistical mechanical calculation of absolute values of entropy or free energy is not required for determination of thermodynamic properties of matter. The functional dependence of the partition function on macroscopic properties, such as the total mass, volume, and temperature of the system, is sufficient to derive equations of state, internal energies, and heat capacities. For example, knowledge that the ideal gas partition function scales as is adequate to define and explain the ideal gas equation of state. 

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

Note that there is not a unique separation of the partition function as Zq = trans vib jjowever, using the result for the ideal gas translational partition function... 

In an ideal gas there are no interactions between the particles and so the potential ener function, 1 ), equals zero. exp(- f (r )/fcBT) is therefore equal to 1 for every gas partic in the system. The integral of 1 over the coordinates of each atom is equal to the volume, ai so for N ideal gas particles the configurational integral is given by (V = volume). T1 leads to the following result for the canonical partition function of an ideal gas ... 

By combining Equations (8.4) and (8.6) we can see that the partition function for a re system has a contribution due to ideal gas behaviour (the momenta) and a contributii due to the interactions between the particles. Any deviations from ideal gas behaviour a due to interactions within the system as a consequence of these interactions. This enabl us to write the partition function as ... 

A consequence of writing the partition function as a product of a real gas and an ideal g part is that thermod)mamic properties can be written in terms of an ideal gas value and excess value. The ideal gas contributions can be determined analytically by integrating o the momenta. For example, the Helmholtz free energy is related to the canonical partitii function by ... 

The standard entropies of monatomic gases are largely determined by the translational partition function, and since dris involves the logarithm of the molecular weight of the gas, it is not surprising that the entropy, which is related to tire translational partition function by the Sackur-Tetrode equation,... 

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

Here Zint is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, fi, so that we can write the desorption rate in (I) as... 

To introduce the transfer matrix method we repeat some well-known facts for a 1-D lattice gas of sites with nearest neighbor interactions [31]. Its grand canonical partition function is given by... 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

Given the partition functions, the enthalpy and entropy terms may be calculated by carrying out the required differentiations in eq. (12.8). For one mole of molecules, the results for a non-linear system are (R being the gas constant)... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

We run into a complication when we attempt to write an expression similar to equation (10.56) for a combination of N molecules that are not distinguishable.0 When this happens, combinations such as (ea. i + eb. ) and (ea2 + et,. i) are the same and should not be counted twice. Thus, the total number of terms in the partition function should be decreased to eliminate such duplications. To determine how to correct for this duplication, consider three gas molecules a, b, and c with energy levels we will represent as ei, e2, and e3. A total of 3 = 6 different combinations of these energy levels can be written as follows... 

We now have equations for the partition functions for the ideal gas and equations for relating the partition functions to the thermodynamic properties. We are ready to derive the equations for calculating the thermodynamic properties from the molecular parameters. As an example, let us calculate Um - t/o.m for the translational motion of the ideal gas. We start with... 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Values of and Qb can be calculated for molecules in the gas phase, given structural and spectroscopic data. The transition state differs from ordinary molecules, however, in one regard. Its motion along the reaction coordinate transforms it into product. This event is irreversible, and as such occurs without restoring force. Therefore, one of the components of Q can be thought of as a vibrational partition function with an extremely low-frequency vibration. The expression for a vibrational partition function in the limit of very low frequency is... 

The notation K is used to emphasize that the scale for K% is based on activities, not simply concentrations. In the gas phase one has available partition functions that can be used to calculate Kt. In solution, however, partition functions are not available. The... 

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