General Discussion Constant-Volume Systems

As noted earlier, the assumption of constant volume is valid for a vast majority of liquid-phase reactions. This is because the mass density (mass/volume) of most of the liquids is not very sensitive to either composition or temperature. 

The constant-volume assumption is also valid for certain gas-phase reactions. One obvious example is when the dimensions of the vessel are fixed. In this case, the pressure in the reactor may rise or fall as the reaction proceeds. The exact change in pressure will depend on the change in the number of moles on reaction, and on the change in the temperature of the system as the reaction proceeds. 

Writing the design equation, Eqn. (3-5), for reactant A and substituting Eqn. (4-5) gives 

In order to integrate Eqn. (4-6), Na and each of the three concentrations on the right-hand side must be expressed as functions of a single variable that measures the progress of the reaction, i.e., that defines the composition of the reactor contents at any time. Moreover, if the volume V is not constant (independent of time and composition), we will need to express V as a function of either time or composition. First, we will revisit the constant-volume case. Then we will extend our analysis to the case of a variable-volume reactor. 

---

In this section we discuss the mathematical forms of the integrated rate expression for a few simple combinations of the component rate expressions. The discussion is limited to reactions that occur isothermally in constant density systems, because this simplifies the mathematics and permits one to focus on the basic principles involved. We will again place a V to the right of certain equation numbers to emphasize that such equations are not general but are restricted to constant volume batch reactors. The use of the extent per unit volume in a constant volume system ( ) will also serve to emphasize this restriction. For constant volume systems,... 

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. 

The free energy difference methods reviewed in this chapter, unless specified otherwise, are discussed for conditions of constant volume and constant temperature (NVT). The extension to ensembles of other types is straightforward.The classical canonical partition function is determined by the classical Hamiltonian 3 6(p, q ), describing the interactions of all N particles in the system in terms of the set of generalized coordinates and conjugated momenta p. For a system with N particles at temperature T, the canonical partition function can be written as... 

As to free energy, below we shall use the Helmholtz free energy F = U — TS (U, S and T are total energy, entropy and temperature, respectively) that is more appropriate for discussion of the systems in terms of temperature and volume V (or density p) at constant pressure p. In a more general case, the thermodynamic potential (or Gibbs free energy) (b = F + pV appears to be more suitable for an expansion, e.g. when varying pressure p. 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

Flow processes for wliich the accumulationtermof Eq. (2.28), d mU v/dt, is zero are said to occur at steady state. As discussed with respect to tire mass balance, tliis means tliat tire mass of the system within the control volume is constant it also means that no changes occur with time in tire properties of tire fluid witliin the control volume nor at its entrances and exits. No expansion of the control volume is possible under these circumstances. The only work of the process is sliaft work, and the general energy balance, Eq. (2.28), becomes ... 

---