Partition function ideal gas

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. 

For an ideal gas, the functional F[p(r)] is known exactly. Because V = 0, the partition function and density profile are given, respectively, by... 

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 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]... 

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... 

In the second line, we have carried the integral over the ideal gas part, which results in the temperature-dependent de Broglie wavelength, A. The final expression is similar to the familiar casting of the canonical partition function,... 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

The chemical potential pa on the left is the full chemical potential including ideal and excess parts. In this chapter we will scale the chemical potentials by (3 and often refer to this unitless quantity as the chemical potential. 3/ia yields the absolute activity. The first term on the right is the ideal-gas chemical potential, where pa is the number density, Aa is the de Broglie wavelength, and q 1 is the internal (neglecting translations) partition function for a single molecule without interactions with any other molecules. 

Statistical mechanics enables one to express the chemical potential i, for an ideal gas phase system in terms of the spectroscopic properties of individual gas phase molecules. The reader is referred to standard statistical mechanics texts (e.g. D. A. McQuarrie Statistical Mechanics , reading list) for the development of the relationship between the system Helmholtz free energy, A , and the corresponding canonical partition function Qi... 

In Equations 4.51 and 4.52 k is Boltzmann s constant, T is the absolute temperature and the Eis s are the energy states of the molecules i. The statistical mechanical considerations in this book will refer to an ideal gas unless explicit mention is made to the contrary. For an ideal gas, a gas of non-interacting molecules, one can express the partition function Q of a collection of N molecules of species i in terms of the single molecule partition functions q as follows1... 

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

When treating polyatomics it is convenient to define an average molecular partition function, In = (lnQ)/N, for an assembly of N molecules. In the dilute vapor (ideal gas) this introduces no difficulty. There is no intermolecular interaction and In = (In Q)/N = ln(q) exactly (q is the microcanonical partition function). In the condensed phase, however, the Q s are no longer strictly factorable. Be that as it may, continuing, and assuming In = (In Q)/N, we are led to an approximate result which is superficially the same as Equation 5.10,... 

We begin by assuming that our ensemble is an ideal gas. The first consequence of this assumption, since ideal gas molecules do not interact with one another, is that we may rewrite the partition function as... 

A noteworthy aspect of Eqs. (10.16) and (10.18) is that they are altogether free of any requirement to carry out an electronic structure calculation. Equation (10.16) is well known for an ideal gas and is entirely independent of the molecule in question, and Eq. (10.18) can be computed trivially as soon as the molecular weight is specified. Note, however, that the units chosen for the various quantities must be such that the argument of the logarithm in Eq. (10.18) (i.e., the partition function), is unitless. 

The translational partition function is a function of both temperature and volume. However, none of the other partition functions have a volume dependence. It is thus convenient to eliminate the volume dependence of 5trans by agreeing to report values that use exclusively some volume that has been agreed upon by convention. The choices of the numerical value of V and its associated units define a standard state (or, more accurately, they contribute to an overall definition that may be considerably more detailed, as described further below). The most typical standard state used in theoretical calculations of entropies of translation is the volume occupied by one mole of ideal gas at 298 K and 1 atm pressure, namely, y° = 24.5 L. 

A point of occasional confusion arises with respect to units. In Eq. (15.22), all portions are unitless except for k T/h, which has units of sec , entirely consistent with the units expected for a unimolecular rate constant. In Eq. (15.23), the same is true with respect to the r.h.s., but a bimolecular rate constant has units of concentration" sec", which seems paradoxical. The point is diat, as with any thermodynamic quantity, one must pay close attention to standard-state conventions. Recall that die magnitude of die iranslational partition function depends on specification of a standard-state volume (or pressure, under ideal gas conditions). Thus, a more complete way to write Eq. (15.23) is... 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

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