General Discussion Variable-Volume Systems

An example of a variable-volume batch reactor is treated below. This example will introduce the methodology that will be required later, for variable-density flow reactors. 

This reaction takes place in a variable-volume batch reactor at constant total pressure. 

If the pressure is constant, the volume of the reactor may change because either (1) the number of moles in the reactor changes as the reaction proceeds, and/or (2) the temperature changes as the reaction proceeds. Both of these phenomena occur in the automobile engine. There is an increase in the number of moles in the cylinder as the fuel is burned, and the temperature of the burning gases increases because heat is not removed through the cylinder walls as rapidly as it is produced by combustion. 

In the present example, we will ignore any temperature change and assume that the reactor is isothermal. 

The initial volume of the reactor is Vo and the reactor initially contains Nao moles of A, Njo moles of an inert gas, and N o moles of D. There is no B and no C in the reactor initially, so Ngo = Nco = 0. The sum of Nao, Mo. and Mdo will be designated Afro- We will assume that the mixture obe % the ideal gas laws over the whole course of the reaction. 

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Many choices of independent variables such as the energy, volume, temperature, or pressure (and others still to be defined) may be used. However, only a certain number may be independent. For example, the pressure, volume, temperature, and amount of substance are all variables of a single-phase system. However, there is one equation expressing the value of one of these variables in terms of the other three, and consequently only three of the four variables are independent. Such an equation is called a condition equation. The general case involves the Gibbs phase rule, which is discussed in Chapter 5. 

The heat capacities that have been discussed previously refer to closed, single-phase systems. In such cases the variables that define the state of the system are either the temperature and pressure or the temperature and volume, and we are concerned with the heat capacities at constant pressure or constant volume. In this section and Section 9.3 we are concerned with a more general concept of heat capacity, particularly the molar heat capacity of a phase that is in equilibrium with other phases and the heat capacity of a thermodynamic system as a whole. Equation (2.5), C = dQ/dT, is the basic equation for the definition of the heat capacity which, when combined with Equation (9.1) or (9.2), gives the relations by which the more general heat capacities can be calculated. Actually dQ/dT is a ratio of differentials and has no value until a path is defined. The general problem becomes the determination of the variables to be used in each case and of the restrictions that must be placed on these variables so that only the temperature is independent. 

Since the system is homogeneous, the functiony() is invertible for volume. (This is not true for systems that may split into two or more phases. For instance, H2O at 100°C and 1 atm may have, at equilibrium, any volume between that of the liquid and that of steam.) This in turn implies that the state of the system, which in principle is the set V, T, q, may also be taken to be p, T, q. The latter is more generally useful, and the following discussion is based on that choice. Partial derivatives are indicated as /6, with the understanding that the independent variables are indeed p, T, and q. 

According to Zwanzig, in dealing with a system intrinsically nonlinear, it should still be possible to build up a generalized Langevin equation such as that of (3.S4). However, in such a case it can be proved that the memory kernel (p(t) is no longer independent of the state of the variable of interest. This forces us to face the problem of preparation in a completely different way, as discussed elsewhere in this volume. 

Values for tte internal variabtes in thetmodynamic, internal equilibriwn are generally uniquely defined by the values for the external variables. For instance, in a simple, thermomechanical system (i.e. one that reacts mechanically solely volume-elastically) the equilibrium concentrations of the conformational isomers are uniquely described by temperature and pressure. In this case the conformational isomerism is not explicitly percqitible, but causes only overall effects, for example in the system s enthalpy or entropy. Elastic macroscopic effects may, however, occur when the relationship between internal and external variables is not single-valued. Then the response-functions of the system diverge or show discontinuities. The Systran undergoes a thermodynamic transformation. The best-known example of sudi a transformation based on conformational isomerism is the helix-coil transition displayed by sonte polymers in solution. An example in the scdid state is the crystal-to-condis crystal transition discussed in this paper. The conditions under which such transformations occur are dealt with in more detail in Sect 2.2. 

The statistical interpretation of the trend towards equilibrium has been discussed in 1 17 and 1 18, and the significant points will be briefiy summarized. Consider any closed macroscopic system of chosen energy and volume. In general, the statement of the values of these variables is sufficient to fix neither the thermodynamic state of the system nor the value of Cl. Thus we need to know in addition whether the volume is divided internally by means of impermeable or non-conducting partitions, and also what catalysts are present. In brief, we need to know what quantum states are accessible to the system. 

Landell-Ferry Free Volume fwFL> Fluctuation Free Volume uc. ittid Free Volume for Thermal Expansion fa p- The effect of system variables on solubility is discussed in Handbook of Solvents, 2001. Miller (1968) noted that the concept of free volume is easy to grasp, but, quantitatively, its definition runs into snares. Is free volume the specific volume of the liquid (solution) minus the volume of the molecules computed from Van Der Waals radii, or minus the volume swept out by the segments as they rotate, or is it some other volume The free volume is generally, but necessarily, about 2.5% for all polymers. 

To make an emulsion, oil, water, surfactant and energy are needed. The composition of the system and the way of processing then determine emulsion type (oil-inwater or water-in-oil), droplet volume fraction ( ), droplet size and composition of the layer of surfactant around the droplets. These variables determine most emulsion properties, notably physical stability. Consequently, knowledge of emulsion formation is of considerable importance. In this chapter, a review is given, with some emphasis on newer developments. Some aspects are left out, because they have been sufficiently discussed in earlier reviews. " For the convenience of the reader, however, important general points are recalled. Some aspects are not discussed, such as the preparation of high-internal phase emulsions, double emulsions, microemulsions and emulsions with very coarse drops. Typically, the emulsions considered have droplets of, say, a micrometre in diameter. Some specialized methods of emulsion formation will also be left out. 

It is the purpose of this paper to establish a statistical thermodynamical formalism for dilute polyelectrolyte solutions which may serve to discuss the assumptions which are usually introduced in the theoretical treatment of such systems. Use is made of a model which is kept as general as possible, and certainly is more realistic than a Kuhn-like chain, without being too complicated to be handled by ordinary statistical procedures. Canonical ensemble statistics are used, the external thermodynamic variables being the volume V, temperature T and composition of the system. Of course, for the problem thus outlined no exact solution of practical nature is presented which is in the present stage still beyond reach. It is hoped that such a formal treatment may help a better understanding of the problems underlying the theoretical approach to polyelectrolyte systems. It will also help to discuss the generality of theoretical description as presented by Marcus [7]. 

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