Relation to the partition function

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 ... 

The Helmholtz free energy of each component in reaction is related to the partition function Q. 

If a system contains two types of species, but the membrane is permeable only to species number 1, the natural variables for the system are T, K //, and N2, where N2 is the number of molecules of type 2 in the system. The thermodynamic potential for this system containing two species is represented by U[T, //,]. The corresponding ensemble is referred to as a semigrand ensemble, and the semigrand partition function can be represented by P(71 K /q, N2). The thermodynamic potential of the system is related to the partition function by... 

We insert Eq. (5.2) into Eq. (5.1), and note that the factors in front of the integral are related to the partition function associated with the relative translational motion of the reactants, see Eq. (A. 14) ... 

The thermodynamic functions of one mole of compound consisting of No identical molecules in standard state are related to the partition function and its derivatives, thus [44] ... 

This perspective suggests to determine the correlation function C t) via calculation of AF t) = Fab (t) — Fa, the difference of the free energies related to the partition functions in the numerator and denominator. The free energy difference AF can also be viewed as the work WAB t) necessary to convert the two ensembles reversibly into each other,... 

In an NPT (constant mass, pressure and temperature) ensemble of clay-water systems the Gibbs free energy, G, is related to the partition function, Q, by the formula ... 

It can also be shown that Kct the equilibrium constant in terms of concentrations, is related to the partition functions evaluated per unit volume... 

It is well-known that in a polymer melt as in a semi-dilute solution, the size exponent has the trivial value v = 1/2. However, for a melt, it is also possible to define exponents aM as was done for dilute solutions (see Section 3.3) as before, these exponents are directly related to the partition functions associated with a network [see eqns (12.3.64) and (12.3.65)]. The values of aM in the present case are not obvious. However, the exact values in two-dimensions... 

The canonical ensemble corresponds to a system of fixed N and V, able to exchange energy with a thermal bath at temperature T, which represents the effects of the surroundings. The thermodynamic potential is the Helmholtz free energy, and it is related to the partition function Q ryT as follows ... 

The molecular field is thus related to the partition function and so it is possible to generate a self-consistent value of the molecular field,. Thermodynamic properties can then be calculated from the partition function. For example, Marcelja calculated the pressure as a function of the area per polar head group for surface monolayers at a variety of temperatures. His results showed good qualitative agreement with experimental results for such systems. 

Before the significance of the various factors expressed in the equations for In is discussed further, an alternative method of deriving the equilibrium constant will be considered. In this it is related to the partition functions of the various molecules taking part in the reaction. It will suffice to deal with Kj. ... 

According to the exact definition (22,IV), the collision diameter is related to the partition function therefore, it has the meaning of a statistical mean value of the distance of closest approach of molecules A and B in a very fast "non-adiabatic" collision. 

It is worth noting that within the PI implementation, we are mainly interested in evaluating the trace of the density matrix, as it is directly related to the partition function. Also when using the primitive approximation, we neglect terms that are of the order x. To improve the precision of results in MC simulations and to achieve faster convergence as P increases, higher order corrections (or propagators of the density matrix) have been developed. 

The density matrix of a quantum particle at inverse temperature P is then related to the partition function of the polymer as... 

Statistical thermodynamics relates the partition function to the measurable properties of the system as in Eq. [3] where, for example, the Helmholtz free energy is related to the partition function by... 

The free energy. A, is related to the partition function, Q, by Eq. [3]. If one considers a relative change in free energy between two well-defined states 1 and 2, it is possible to express the free energy difference in terms of a ratio of the two partition functions describing each state. For example, letting Qj and Q2 be the partition functions for states 1 and 2, the difference in free energy between these states is ... 

Here A° is a constant related to the partition function for the adsorbate and usually assumed to be temperature dependent and independent of the adsorption energy this approximation will be discussed in Section 3. 

Notice that both A and G are directly related to q, rather than related to a derivative of q. Although this is an artifact of the mathematics and the definition of the ensemble, there should be some wonder that the important state functions (G and A) are so intimately related to the partition function q, which becomes the central focus in statistical thermodynamics. 

The previous section made it clear that all thermodynamic state functions are in some way related to the partition function q. This means that in order to know these state functions, we need to know what q is. How ... 

The partition functions depend on the spacings of the energy levels - the more closely spaced they are, the higher is <3 at a particular temperature. The molar entropy, S, of the reactants and transition state are also related to the partition functions, while the change in molar enthalpy, A//, is related to Eq. Since AG = A// — T AS (1.20 and 1.21), where G is the molar Gibbs energy, (3.13) leads to ... 

As discussed above, one of the vibrational degrees of freedom of t in mechanism (6.III) became a translation along the reaction coordinate in mechanism (6.II). This can be factored out of the transition state in mechanism (6.II), <2. and related to the partition functions of the normal molecule, Qp... 

Experimental access to the energetic j roperties of proteins can be obtained by several methods. However, none of the.se methods provide sucli a direct and powerful approach as differential scanning calorimetry. DSC monitors the response function heat capacity which is directly related to the partition function of the system. Knowledge of the partition function is sufficient to derive all the thermodynamic information on the system. 

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