Constant volume system

Set a hypothesis as to the mathematical form of the reaction rate function. In a constant volume system, the rate equation for the disappearance of reactant A is... 

The kinetics of many decompositions are conveniently studied from measurements of the pressure of the gas evolved in a previously evacuated and sealed constant volume system. It is usually assumed, and occasionally confirmed, that gas release is directly proportional to a, so that the method is most suitable for reactants which yield a single volatile product by the irreversible breakdown of a substance that does not sublime on heating in vacuum. A cold trap is normally maintained between the heated reactant and the gauge to condense non-volatile products (e.g. water vapour) and impurities. The method has found wide application, notably in studies of the decomposition of azides, permanganates, etc., and has been successfully developed as an undergraduate experiment [114—116]. 

In a constant-volume system in which neither expansion work nor nonexpansion work is done, we can set w = 0 in Eq. 7 (At/ = tv + q) and obtain... 

In constant volume systems it is convenient to employ the extent per unit volume... 

First-Order Reactions in Constant Volume Systems. In a first-order reaction the reaction rate is proportional to the first power of the concentration of one of the reacting substances. 

Third-Order Reactions in Constant Volume Systems. Third-order reactions can be classified into three primary types, according to the general definition. 

Fractional and Other Order Reactions in Constant Volume Systems. In chemical kinetics, one frequently encounters reactions whose orders are not integers. Consider a reaction involving only a single reactant A whose rate expression is of the form... 

In experimental kinetics studies one measures (directly, or indirectly) the concentration of one or more of the reactant and/or product species as a function of time. If these concentrations are plotted against time, smooth curves should be obtained. For constant volume systems the reaction rate may be obtained by graphical or numerical differentiation of the data. For variable volume systems, additional numerical manipulations are necessary, but the process of determining the reaction rate still involves differentiation of some form of the data. For example,... 

Thus, for both variable and constant volume systems, one can manipulate concentration versus time data to obtain values of the reaction rate as a function of time or as a function of the concentrations of the various species present in the reaction mixture. The task then becomes one of fitting this data to a reaction rate expression of the form of equation 3.0.13. 

Differential procedures are illustrated schematically in Figure 3.1. The first diagram indicates how the rate may be determined from concentration versus time data in a constant volume system the second schematic illustrates the method just described. The third diagram indicates the application of our general differential method to this system. 

These expressions apply to constant volume systems in which vA = — 1. The sums are taken from i = 1 to i = n. Although the use of these equations is somewhat laborious for hand calculations, they are easily handled by even the simplest types of computers. 

Equations 3.3.28 and 3.3.29 indicate that plots of log tf versus log CA0 are linear, and reaction orders may be determined from the slopes of such plots (slope = 1 — n). For constant volume systems there is another method based on equations 3.3.29 and 3.3.30 by which one can obtain preliminary estimates of the reaction... 

Fractional Life Relations for Constant Volume Systems... 

Equation 3.1.32 applies to a constant volume system that follows nth-order kinetics. If we take vA = — 1 it can be rewritten as... 

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,... 

In a constant volume system the rate may be written in terms of the extent per unit volume as... 

The analysis is very similar to that employed in proceeding from equation 5.1.12 to equation 5.1.16, but the physical situation is somewhat different. The reaction is first allowed to come to equilibrium with Ae and Be representing the equilibrium concentrations of species A and 5, and the equilibrium extent of reaction per unit volume in a constant volume system. Under these conditions the net rate of reaction is zero. 

The reaction is assumed to occur in a constant volume system. A relaxation analysis of the type employed for the first-order reaction leads to the following analog of 5.1.49. 

If the kinetic parameters for the upper reaction are denoted by the subscript 1 and those for the lower reaction by the subscript 2, the appropriate rate expressions for constant volume systems may be written as... 

This section discusses the kinetic implications of series reactions. We will be concerned only with those cases where the progress of the various stages of the overall transformation is not influenced by either parallel or reverse reactions. The discussion will again be limited to constant volume systems. 

The corresponding rate equations for constant volume systems can be written as... 

For a constant-volume system, an infinitesimal change in temperature gives an infinitesimal change in internal energy and the constant of proportionality is the heat capacity at constant volume... 

Such a can is thrown away when it contains no more air freshener, although it certainly still contains much propellant. Incineration of the can leads to an increase in the kinetic energy of the remaining propellant molecules, causing them to move faster and faster. And as their kinetic energy increases, so the frequency with which they strike the internal walls of the can increases. The force of each collision also increases. In fact, we rediscover the ideal gas equation, Equation (1.13), and say that the pressure of the gas (in a constant-volume system) increases in proportion to any increase in its temperature. In consequence, we should not incinerate an old can of air freshener because the internal pressure of any residual propellant increases hugely and the can explodes. Also note the additional scope for injury afforded by propane s flammability. 

The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31-37], The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms. 

Filled-bulb temperature sensors are also widely used. An inert gas is enclosed in a constant-volume system. Changes in process temperature cause the pressure exerted by the gas to change. Resistance thermometers arc used where accurate temperature or diflcrcntial-temperature measurement is required. They use the principle that the electrical resistance of wire changes with temperature. 

By following the change in total pressure of a constant-volume system. 

Analysis of Total Pressure Data Obtained in a Constant-Volume System. For... 

Suppose that is the initial amount of A in the reactor at time t = 0, and that Aa is the amount present at time t. Then the conversion of A in the constant volume system is given by... 

The transfer of autoclave pressure to the resin in the laminate does not occur hydrostatically because the resin is not enclosed in a constant-volume system. Flow can occur initially both vertically (thickness direction) and horizontally. Furthermore, the network of fibers can also eventually act as a network of springs to which the vacuum bag and bleeder assembly transfer the stress from the autoclave pressure. This stress can then be transferred... 

Note that the material balances for fixed beds are valid for die case of constant-density (constant volume) systems. The important term here is the one including the fluid velocity, i.e. the term uJdCIdz. For a variable volume system,... 

The integrated form of the material balance for the case of a constant-volume system becomes (Levenspiel, 1972)... 

Comparing this result with that of a constant-volume system, we see that the fractional conversion at any time is the same in both cases. However, the concentration of materials is not the same. 

---