COMMUNICATION
T O T A L PRESSURE METHOD OF K I N E T I C ANALYSIS
A general method of expressing the partial pressure of the selected reactant in terms of the total pressure, p t , or total pressure difference, Apt, is suggested. It is based on the stoichiometry of reaction, initial pressures, and measurements of the total pressure with time. Its application, however, is limited to the kinetic analysis of simple gas phase reactions a t constant T and V when the number of moles of reactants is different from that of products.
IN
THE KINETIC A N A L Y S I S of gas phase reactions at constant temperature and volume, partial pressures may be conveniently substituted for concentrations in the rate equations according to the following relationship for the perfect gas behavior:
CA
=
PA/RT
(1)
This modification seems to be fully justified since the partial pressure of the selected reactant may be easily determined by : Adsorption, absorption, or independent chemical reaction. Measurement of some physical property of reactant A , proportional to its partial pressure. Measurement of the total pressure or its change with respect to the initial total pressure of the reaction system. The suggested method uses directly the measurements of the total pressure or of the total pressure difference of the reaction system in the rate equation of the scrutinized reaction. This is done by expressing the partial pressure of the key reactant in terms of either of these measurements, conversion ofp, into C, in accordance with Equation 1, and substitution of the resulting expression for C, in the proper rate equation in t e r m of concentrations.
Application of the Method
The total pressure method may be applied to all types of elementary gas reactions involving reactants in the pure state or diluted by an inert gas when the number of moles of reactants is different from that of products. However, for brevity, only its application to the monoreactant reaction of the type aA +.qQ is shown. For this method, it is more convenient to write the stoichiometric equation per one mole of the selected reactant. Consequently, the above reaction may be represented as: A=’Q
Sow, following the recommended procedure, the total pressure of the reaction system a t time t after the start of the reaction is first expressed as the sum of the partial pressures of unreacted A and formed Q.
Solving for p A:
(3) a
or in the differential form: Procedure
The following procedure for the application of the total pressure method is recommended. (The procedure for the total pressure difference, Abt, follows the same steps.) From the known initial condition and stoichiometry of reaction, derive an equation for the total pressure of the system, p l , a t time t , in terms of the total initial pressure PO and partial pressure of the key reactant, pa. Solve the resulting equation forp, in terms ofpo and p l . Convert the expression thus obtained for p a into C, by means of Equation 1 and substitute into the proper rate equation. Kow, the only variables involved in the rate expression are time and total pressure or total pressure difference. I n conclusion, the equation may be used for the kinetic interpretation of experimental data just in terms of the directly measured variables.
If one uses the rate equation in terms of partial pressures, the conversion into concentrations, obviously, is not needed. 220
I&EC FUNDAMENTALS
(4) a
Respective equations in terms of the total pressure difference are :
a
(44 a
Then, in accordance with the third step of the procedure, Equations 3 and 4 or 3a and 4a may be used in combination with Equation 1 as follows:
They may be substituted into the proper differential form of the rate equation and applied directly for the interpretation of kinetic d a t a ; dp,jdt and d(Apt),'dt may be determined analytically ( 2 ) , numerically, or graphically from the experimental total pressure-time curves. From Equation 3 or 3a, PA may be substituted into the proper integrated rate equation which then may be used for the interpretation of kinetic data in terms of total pressure or total pressure difference. The application of the two approaches is now shown for the reaction A = ( q / a ) Q. Assume that the reaction is of the first order. Then its rate equation in terms of concentration may be written as:
Since dC, = dp,'RT. Equation 5 simplifies to :
The reaction proceeds stoichiometrically as follo\vs :
+ CO
CH3CHO + CHa
Calculate the rate constant and order of reaction. Solution. For this reaction q / a = 2. Assume first order reaction. Substitution of p a from Equation 3a into the firstorder integrated rate expression yields:
from which k , can be calculated. The results of calculation a t several characteristic times are tabulated below:
(5)
t , See.
then in terms of partial pressures,
42 190 384
k , X 707,Mole/(Sec.) (Liter)( M m . Hg)
0,475 0,403 0,344 0,264 0.215
a40
1440
Substituting for dp, ,and p , in Equation 6 their values from Equations 3 and 4, a rate equation is obtained in terms of total pressures :
which can be used for the determination of the rate constants k , or k , in the same way as when the progress of reaction is followed by a change in concentrations. I n the second approach? p;I from Equation 3 is directly substituted for C, in the integrated form of the first-order equation in terms of concentrations, giving:
The falling trend in the k , indicates that the reaction is not first order. S o w the second order is tried. In this case the rate equation in terms of concentrations is :
Substituting for C, and C,, their values in terms of Equations 1 and 3a, one obtains:
The values of k , based on Equation 11 are shown belo\r.
t , Sec. 42
I t is apparent that Equation 8 may also be obtained by direct integration of E:quation 7. It'hen the total pressure differences are available, Equations 3a and 4a should be used. The same procedure may be applied to orders higher than one. However, the interpretation of the rate equations in such cases must lead to the determination of the rate constant and order of reaction. Moreover, the method can be easily extended to bireactant and trireactant reactions, as well as to reactions with an inert gas present in the system. Problem. Practical application of the method is shown for the thermal decomposition of acetaldehyde at 51 8 C. with an initial pressure of acetaldehyde equal to 363 mm. of Hg. This reaction was investigated by Hinshelwood and Hutchison (7), who obtained the following data for the total pressure difference in a constant volume reactor. f, See.
P t , M m . Hg
0
0
42 73
34 54 74 114 134 154 174 194 224 244 264 284
105
190 242 310 384 480 665 a40 1070 1440
19, 384 a40 1440 Av.
1,376 1.346 1.392 1,363 1.394 1.374
The constancy of k , indicates that the reaction is second order. Nomenclature
number of moles of reactant A concentration of reactant A k, reaction rate constant in terms of concentrations k, reaction rate constant in terms of partial pressures n order of reaction pa partial pressure of reactant A a t time t pa, = initial pressure of reactant A p , = total initial pressure p t = total pressure of the system a t time t S p , = total pressure difference, Apt = p , - ,bo q = number of moles of product Q R = gas constant t = time T = absolute temperature, K. a
C,
= = = = = =
Literature Cited (1) Hinshelwood, C. N., Hutchison, \V. K., Proc. Roy. Soc. (London)
111A, 380 (1926). ( 2 ) Pasfield, W. P., \Varing, C. E., J . Am. Chem. SOC.78, 2696
(1956).
LEON S. KOWALCZYK L'niuersity of Detiozt, Detroit, Mich.
RECEIVED for review November 27, 1961 ACCEPTEDMay 21, 1962 VOL. 1
NO. 3
AUGUST 1 9 6 2
221
