Frozen-in systems

With this assumption we freeze-in the conversion L H at X = 0, hence X =X = 1/2, and we follow the BI of the equilibrated and the frozen-in system. We discuss the following three cases. 

Slow Relaxation in Pure and Binary Frozen-in Systems Hiroshi Suga... 

The first term on the right-hand side of (2.3.52) gives the temperature dependence of the volume of the frozen-in system, and can be written as... 

In this particular model, all the heat capacity is due to the conversion between H and the L species. Note again that the heat capacity is zero in the frozen-in system. 

Fig. 3.14 The schematic process of adding one solute to water in two steps. First, add to a frozen-in system then, allow the system to relax to its final equilibrium state. The corresponding changes in volume are Vs and (Vl - VH)(dNL/dNs)eq, respectively.

Note that ixl and [ah are definable only for a frozen-in system (i.e. we must keep Nh and Ny constant in the respective definitions of ixl and iih). On the other hand, the chemical potential of s is definable in both the equilibrated and the frozen-in systems. In general, /xl and ixh have different values the equality between the two holds only when we use the equilibrium values Nl = N and Nh = Nfj. 

Note the difference between partial molar (or molecular) quantities on the left-hand side of (3.6.17)-(3.6.19) and the molecular parameters of the model on the right-hand side. Furthermore, we note that all the partial molar quantities of and H are definable only in the frozen-in system. 

All the derivatives of the chemical potential on the right-hand side of (3.7.4) are the derivatives in the frozen-in system, i.e. a system of three independent components L, H, and s. 

Note that Kj in (3.7.11) is the isothermal compressibility of the mixture of L and H, in the frozen-in system, where Ni and Nh are assumed to behave as independent variables. Note also that Vl and Vh are definable only in the frozen-in system, and therefore there is no need to add a superscript to these symbols. 

It is important to note the different types of the derivatives on the two sides of equations (F.6), (F.8), and (F.9). On the left-hand side, we have a derivative along the equilibrium line (this is indicated by constant Nuj). On the other hand, on the right-hand side of these equations, we have derivatives in a frozen-in system, i.e. we view the system as a mixture of three components Na,Nb,Ns, and we freeze in the equilibrium between... 

We next pose the following question Can we find a frozen-in system, i.e., fixed numbers N and for which the partition function (5.75) is practically equal to (5.74) It turns out that this can, indeed, be done. Since the arguments that follow are of fundamental importance in many problems in statistical mechanics, we elaborate on them in considerable detail. In what follows, we always assume that N and Nb are sufficiently large that we can safely use the Stirling approximation in the form ... 

This is the case when we tend to a pure lattice L. The chemical potential [jiL tends to the enthalpy (see below) of pure L, whereas the chemical potential of H tends to minus infinity. Such a system, of course, cannot be at equilibrium it may be attained in the frozen-in system. (Note that in this case, H forms a dilute ideal solution in the system.)... 

Among the second-order derivatives of the free energy, we distinguish between two subgroups those that are definable only in the frozen-in system and those that may be defined in either the frozen-in or the equilibrated system. We start with quantities belonging to the first group. The partial molar entropies of L and H are obtained by differentation, with respect to temperature, of the corresponding chemical potentials in (6.47) and (6.48) ... 

First, we measure the change of volume in the frozen-in system (i.e., in the absence of the catalyst), and get the value (dV)Nj, NH next step, we introduce the catalyst to the system, which generally produces an additional change in the volume, given by the second term on the rhs of (6.63). In our particular model, (6.62) can be written explicitly. Note that in this model, the total volume V is N Vl, and is assumed to be temperature independent. The quantity dN dT in (6.62) can be evaluated directly from the equilibrium condition (6.37). Alternatively, we can use the identity (Section 5.10)... 

This is the general expression for the heat capacity in the mixture-model approach (two components). Note that the first term on the rhs of (6.68) is the heat capacity of the system that would have been measured in the frozen-in system. The second term is the contribution to the heat capacity due to relaxation to the final equilibrium state. The arguments leading to this interpretation are the same as those discussed following Eq. (6.63). In our particular model, the first term on the rhs of (6.68) is zero and, for the second term, we have... 

That is, we first bring the two solutes to a distance P in a frozen-in system, and then let the system relax to its final state. Since the second step is spontaneous, we must have... 

The second PF, Q T, Ml, Mh, N) defined in (2.10.53) pertains to a system that may be referred to as a partially equilibrated system (PES) or, equivalently, as a partially frozen-in system. Here Ml and Mh are fixed, but the ligand molecules are in equilibrium, i.e., Nl and Nh are not fixed. Only the sum Nl + Nh is constant. It is at this intermediate level of details that we shall be working in this section. 

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