JOURNAL O F CHEMICAL EDUCATION
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A GRAPHICAL METHOD FOR DETERMINING THE ORDER OF HOMOGENEOUS REACTIONS J. H. WRIGHT,' J. H. BLACK: and JAMES COULL University of Pittsburgh, Pittsburgh, Pennsylvania
INA given chemical reaction it is possible to calculate from classical thermodynamics the energy change and equilibrium conditions of a system undergoing chemical change; however, only actual experimental data will provide means for determining the kinetics or "order" of a reaction. The kinetics is usually studied by finding the order with respect to each reactant; the sum is then taken as the over-all order of the reaction. The order with respect to a particular reactant may be found by a variety of published methods, of which the two best known are: (a) substituting the kinetic data into the equations for integral ordered reactions; and (b) constructing a graph of the rate versus time, the slope being the reaction order. The first method is limited t o integral orders, whereas a great many reactions have been found to be of fractional order; thus, the method loses in versatility what was gained in simplicity. The graphical method referred to in (b) above is widely used at present, but its use requires an extensive amount of data and involves approximating the instantaneous rate of reaction a t each time. An interesting new graphical method was published recently by C. R. Noddings.3 In this method a graph of per cent reacted versus a function of time (the fraction of the amount of time required for 90 per cent conversion) was plotted with whole and half integral ' Present address: Multiple Fellowship of Gulf Research & Development Company, Mellon Institute, Pittsburgh, Pennsylvania. Applied Research Laboratory, U. S. Steel Corporation, Monraeville, Pennsylvania. a NODDNNGS, C. R., Chem. Processing, 16, No. 11, 63 (1953).
orders as parameters. Although this particular method is fast and accurate, it is necessary to have the times required for conversion of 90 per cent and some other consistent fraction. Obtaining data at exact percentage conversion will usually involve interpolating between successive determinations. The method proposed in this paper requires a knowledge of the fraction reacted at two different intervals of time, and that the fractions converted must differ by a t least 20 per cent. REVIEW O F CLASSICAL KINETICS
First Order. Two differential equations for a first order homogeneous reaction have appeared indiscriminately in the literature as follows:
It is obvious that, unless the reaction is conducted at constant volume, the two equations are not equivalent. Benton4 has clarified this point by showing that the proper form for the first order reaction is equation (1) above. Similarly, the differential equations for other integral orders with respect to a single component have been showns to be: BENTON, A. F.,J. Am. Chem. Soc., 53, 2984 (1931). 6 SHERWOOD, T. K., A N D C. E. REED,"Applied Mathematics in Chemical Engineering,'' McGraw-Hill Book Co., Ine., New York, 1939,pp. 53-67.
VOLUME 33, NO. 11, NOVEMBER, 1956
Second Order:
543
sion (at least 20 per cent apart). The method is explained and illustrated below: Experimental data:
Third Order:
DERIVATION OF REACTION ORDER EQUATION
I n general it is proposed that the differential equation for a homogeneous reaction, the rate of which is controlled by a single component, be written:
where n is the order of the reaction with respect to the limiting component. Substituting a function of the fraction converted for the number of molecules present, N = N o (l-f) and d N = -Nodj, the result is:
For a reaction conducted isothermally and at constant volume, equation (6) may be readily integrated t o obtain equation (7) :
Only two points of data are needed. These are the times required for two specific percentages of conver- .
,f 40 68
t 23.5 min 65 min.
(1) Assume the reaction is first ordrr and obtain mt, = 0.54 from the graph for f, = 40 per cent. (This value may d m be calculated from equation (71 .) (2) Next, the first approximation to rn is calculated from t, = 23.5 to he 0 023. (3) This first estimate of m is multiplied by tl (65 min.) to obtain mtx = 1.493. (4) The point of intersection is found between f = 68 and mtn = 1.493. The result is the first approximation to n (1.41). (5) This value of n is used to obtain a more accurate value of m from fl and t l using either the graph or equation (7). Now, m is found to be 0.0242. (6) The new value of m is used along with f, and t2 to obtain a better appproximsiion of n using the point of intersection between fa and mtz. The value of n. is found to be 1.58 in this example. (7) A third trial might be used if the assumed order of reaction and that found in Step 6 differ significantly. I n this example, a third trial gives n = 1.60 and a perfect check.
CONCLUSIONS
A method, combining accuracy, simplicity, and speed, has been presented for determining the order of isothermal, isometric, homogeneous reactions. This method should prove useful in determining the order of nonintegral as well as integral ordered homogeneous reactions. A graph is presented which facilitates the calculations.
If equation (7) is represented graphically by plotting j versus mt with n as a parameter, not restricted to integral values, the resulting chart may be used for the rapid and accurate determination of order. The function may be evaluated for the case of n = 1 by the application of L'Hopital's rule for indeterminates. The reaction rate constant may then be found knowing the values of n and m. It is important to note that k has the same dimensions as C&"; therefore, m is di- V f mensionless. DETERMINATION OF ORDER
Ezp. 1 2
Symbols
Volume Fraction of a single reactant converted to product k = Reaction rate constant m = Reaction modulus = kco"-' n = Reaction order or correlating parameter t = Ti" = Subscript refers to initial conditions of N , C, V,and t
O
= =
