Degree of advancement

Figure A2.1.9. Chemically reacting systems, (a) The entropy. S as a fiinction of the degree of advancement of the reaction at constant U and V. (b) The affinity Aas a fiinction of for the same reacting system. Equilibrium is reached at 0.623 where tiis a maxuniim and A= 0.

The reaction coordinate, s, is also call the molar extent or degree of advancement. 

In order to measure the progress of a reaction it is necessary to define a parameter, which is a measure of the degree of conversion of the reactants. We will find it convenient to use the concept of the extent or degree of advancement of reaction. This concept has its origins in the thermodynamic literature, dating back to the work of de Donder (1). 

The first step in the analysis is to determine if the chemical equations A to C are independent by applying the test described above. When one does this one finds that only two of the reactions are independent. We will choose the first two for use in subsequent calculations. Let the variables a and B represent the equilibrium degrees of advancement of reactions A and B, respectively. A mole table indicating the mole numbers of the various species present at equilibrium may be prepared using the following form of equation 1.1.6. 

Thus the equilibrium condition is expressible in terms of the initial concentrations and the single variable, e, which is called the degree of advancement. Also, all concentrations can be expressed in terms of a single one of the others, say,... 

When several reactions occur simultaneously a degree of advancement is associated with each stoichiometric equation. Problem P4.01.26 is a application of this point. Some processes, for instance cracking of petroleum fractions, involve many substances. Then a correct number of independent stoichiometric equations must be formulated before equilibrium can be calculated. Another technique is to apply the principle that equilibrium is at a minimum of Gibbs free energy. This problem, however, is beyond the scope of this book. 

In terms of the individual degrees of advancement the material balances are... 

The starting mixture contained 3 mols hydrogen per mol C02. The temperature is 1073 K and the pressure is 500 atm. The degrees of advancement are recorded below the chemical symbols in the equations above. The material balances are,... 

In a parallel experiment, the extent of the reaction a is measured using the partial heat to a particular time divided by the total heat of the isotherm plus the residual heat of a subsequent 10°/min ramp. Figures 5 and 6 show the observed relationship of In a and In t to a. As expected for a similar degree of advancement a, the ionic mobility a increases with temperature. Similarly for the same value of a, the dipolar mobility increases. An increase in dipolar mobility corresponds to a shorter relaxation time. Thus t decreases as temperature increases. Somewhat unexpected, both In a and In t exhibit a nearly linear dependence on a. Curvature in the In a and In t versus a plot is most pronounced for small values of a and at the highest temperature. There is no evidence of a break in the In <7 or In T dependence on a which would indicate gel. 

The extent to which any given direct mechanisms may be combined without cycle formation can be determined by noting whether such combinations contain irreducible cycles. The latter are the cycles with a minimal number of steps which characterize a given system. They can be determined by a procedure that is analogous to that for finding direct mechanisms [Sellers (9a). For a multiple overall reaction, the relative degrees of advancement for each of the simple overall reactions chosen as a basis introduce additional restrictions on the allowable cycle free combinations) [Sellers (9b)]. 

Figure 4.4 Log(i) versus l/T (°K) at comparable degrees of advancement between 0.70 and 0.95...

Each parameter takes the integral values — 2, —1,0,1,2,... and is called degree of advancement , because it indicates how far the reaction p has advanced.510 The values of the p are limited by the requirement rij 0 for all j, but the next section will show that this limitation creates no difficulties. 

Exercise. When only a single reaction is possible, the master equation (2.4) represents a one-step process on the chain of accessible lattice points. Exhibit this fact more clearly by writing the master equation in terms of the degree of advancement f. 

Evidently this is not a Poisson distribution. Alternatively, in terms of the degree of advancement... 

The effect of polar groups on the diimide reaction is sensitive to the configuration of the attached groups. For example, fumaric acid (trans) is ten times as reactive as maleic acid (els ) and the ratio of reactivities of the geometrical isomers of cinnamic acid, trans/cis. is 10 3 (ref. 21b). In comparison, cis- and trans-2-butene have almost identical reactivities. The difference may be explained by a change in the degree of advancement of the transition state towards the saturated product where the eclipsed conformation would result in a greater non-bonded repulsive interaction between the cis-substituents than the trans. 

Introducing the degree of advancement of the reaction, , which is defined by... 

The general relationships involved for a single chemical reaction in a closed system are shown schematically in Figure 1, where the degree of advancement at point e corresponds to chemical equilibrium. Point t represents a state of the system corresponding to spontaneous chemical reaction. While the invariant condition of the closed system considered is the equilibrium state, e, this generally is not the case for a thermodynamic system open to its environment. For such a system, the time-... 

Equations 27 and 28 permit a simple comparison to be made between the actual composition of a chemical system in a given state (degree of advancement) and the composition at the equilibrium state. If Q K, the affinity has a positive or negative value, indicating a thermodynamic tendency for spontaneous chemical reaction. Identifying conditions for spontaneous reaction and direction of a chemical reaction under given conditions is, of course, quite commonly applied to chemical thermodynamic principle (the inequality of the second law) in analytical chemistry, natural water chemistry, and chemical industry. Equality of Q and K indicates that the reaction is at chemical equilibrium. For each of several chemical reactions in a closed system there is a corresponding equilibrium constant, K, and reaction quotient, Q. The status of each of the independent reactions is subject to definition by Equations 26-28. 

In contrast to Clausius, Gibbs did not discuss uncompensated heat, as he started directly with the total differential of entropy. Gibbs presentation appealed very much to De Donder, However, he wanted to find the meaning of this mysterious uncompensated heat. He considered a system whose physical conditions, such as pressure and temperature, were uniform and which was closed to the flow of matter. Chemical reactions, however, could go on inside the system. De Donder first introduced what he called the degree of advancement, , of the chemical reaction so that the reaction rate v is the time derivative of . 

Of course, De Donder was not the only thermodynamicist who was interested in nonequilibrium conditions at this time. The work of the French school, notably Duhem and Jouguet, and the Cracow school with Natanson should also be recalled. (A short survey of the history can be found in my Etude Thermodynamique des Phinomines Irreversibles, De-soer, Liege, 1947.) However, his contributions to the introduction of the degree of advancement and of affinity are now classical. 

When A is positive, the reaction should proceed to the right. When A = 0, chemical equilibrium prevails. Formulated in this way, A suits our perception of chemical affinity perfectly. Another characteristic property is the degree of advancement of reaction , [4]. The chemical reaction velocity and the degree of advancement are related by... 

In a not too complicated case, e.g., in the case of the hydrogen-bromine reaction investigated so thoroughly by the Bodenstein school, it can be shown with certainty that its well known mechanism is really the only one which conforms with the (very accurate) experiments by Bodenstein and Lind and later experimenters. In this and often in more complicated cases, too, the calculations permit expressing the time from the beginning of the experiment by means of the sum of a number of known functions of the degree of advancement x, each multiplied by a constant. 

If the constants can be chosen so that if the measured values of x are inserted the sum becomes equal to the corresponding measured times, this is a strong evidence that the proposed mechanism is the right one, provided of course that the partial reactions in the proposed mechanism add up to the actual total reaction. The relation between time and the degree of advancement should, however, hold over a very wide range from the beginning of the reaction nearly to completion, and should also hold when the original concentrations (but not temperature or pressure) are varied. 

Concerning the possibility of the existence of another mechanism which should simultaneously conform with the measured kinetics, it must be noted that if the assumptions concerning the mechanism are altered the relation between time and degree of advancement will be altered qualitatively, i.e., the number, the nature, and the dependence on the original concentrations of the functions will be altered simultaneously or... 

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