Entropy partition function

We have stated our definitions of entropy, partition function, and other quantities entirely in terms of summations. Often, however, the quantity changes only slowly from cell to cell in this case it is convenient to replace the summations bv integrations over the phase space. We recall that all cells are of the same volume, if there are n coordinates and n momenta in the phase space. Thus the number of cells in a volume element dqi. . . da dv. dvn of phase space is... 

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 ... 

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 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,... 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

The assumption that the energy can be written as a sum of terms implies that the partition function can be written as a product of terms. As the enthalpy and entropy contributions involve taking the logarithm of q, the product thus transforms into sums of enthalpy and entropy contributions. 

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)... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

A macroscopic quantity X, such as the enthalpy or entropy, can be calculated from the partition function Q, as discussed in Chapter 12. For easy reference we reproduce eqs. (12.6-12.8) here. 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

Quite similar equations can be formulated for AG and AH by use of the partition function f of the activated complex. It follows from equations (6) and (7) that AEp can only be evaluated if the partition functions and AEz are available from spectroscopic data or heat capacity measurements. However, if AG = AH, the entropy change AS equals zero, and if AEz also equal to zero, either AG or AH can then be identified with the potential energy change. If... 

This distribution follows automatically if we require that the entropy of a system with many members that is in equilibrium is at maximum. The denominator in the Boltzmann distribution ensures that the frequencies P are normalized and add up to unity, or 100%. This summation of states (Zustandssumme in German) is called a partition function ... 

Other thermodynamic quantities such as chemical potential and entropy also follow directly from the partition function, as we demonstrate later on. However, to illustrate what a partition function means, we will first discuss two relatively simple but instructive examples. 

At very high temperatures, however, the excited state will also be occupied. Entropy maximization requires that both levels be equally populated. The high-temperature limit of the partition function is... 

The above two examples illustrate that the value of the partition function is an indicator for how many of the energy levels are occupied at a particular temperature. At T = 0, where the system is in the ground state, the partition function has the value q = 1. In the limit of infinite temperature, entropy demands that all states are equally occupied and the partition function becomes equal to the total number of energy levels. 

Clearly, the sticking coefficient for the direct adsorption process is small since a considerable amount of entropy is lost when the molecule is frozen in on an adsorption site. In fact, adsorption of most molecules occurs via a mobile precursor state. Nevertheless, direct adsorption does occur, but it is usually coupled with the activated dissociation of a highly stable molecule. An example is the dissociative adsorption of CH4, with sticking coefScients of the order 10 -10 . In this case the sticking coefficient not only contains the partition functions but also an exponential... 

The probability distribution is normalized by ZM( p, t), which is a time-dependent partition function whose logarithm gives the nonequilibrium total entropy, which may be used as a generating function. 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

As suggested previously, the density of states has a direct connection to the entropy, and, hence, to thermodynamics, via Boltzmann s equation. Alternately, we can consider the free energy analogue, using the Laplace transform of the density of states - the canonical partition function ... 

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