ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
77
Measurement of Chemical Equilibrium Constants by Continuous Mass Spectrometry of Transient Compositions at the Outlet of a Pulsed Reactor Daniel Schweich * and Jacques Villermaux Laboratoire des Sciences du G6nie Chimique, CNRS-ENSIC, 1. rue Grandville, 54042 Nancy, France
This paper reports a sophisticated method of measuring the equilibrium constant of reactions using only a very small amount of reactant. The method has been found to be very accurate and especially suitable for expensive reactants. It requires a programmed mass spectrometer linked to a microreactor and a computer. The method is illustrated for the example of the dehydrogenation of cyclohexane but more realistic systems are discussed together with other applications of the pulsed reactor.
T h e determination of the composition of a complex reacting mixture at equilibrium is the best way of calculating a chemical equilibrium constant. Unfortunately most available methods consume a lot of reactant. This is the case of the continuous operation of a n open reactor which requires a great deal of reactant t o reach steady state. A solution consists in using a batch microreactor and t o wait for the attainment of steady-state equilibrium composition. Nevertheless t h e amount of mixture is seldom less than 1 cm3 of diluted reactant. When a few tens of milligrams of reactant or less are available, the only way t o measure the chemical equilibrium constant is to use a pulsed reactor. However, the detection system must: (1) measure simultaneously t h e concentration of each species, (2) store the measured values, (3) respond quickly to instantaneous concentration changes (no time lag), and (4) be sensitive t o any kind of molecule. T h e only detection device which fullfills these requirements is a mass spectrometer linked t o a minicomputer. We shall first describe the use of such a sophisticated device and then deal with its potential applications with respect to t h e high investment involved.
EXPERIMENTAL Apparatus. The experimental apparatus, proposed in this study, is shown schematically in Figure 1. The reaction chosen to test the method is the dehydrogenation of cyclohexane (C) to benzene (B) catalyzed by platinum. The reactor consists of a 1-m length, 0.495-cm i.d., stainless steel tubing packed with 14 g of a 0.370 by weight platinum on alumina catalyst. The reactor is fitted in a Varian 2760-30 chromatograph. The cyclohexane is injected with a syringe with nitrogen as the carrier gas. A constant flow rate of argon is added to the stream flowing out of the reactor so that any change in the volumetric flow rate due to the vaporization or the reaction of the cyclohexane becomes a concentration change of argon. The quadrupolar mass spectrometer consists of a gas analyzer (Balzers QMG 111) and a scan programmer (QPT 101) which allows the user to periodically observe several isolated masses. W'e have chosen to observe specific ions of each of {he following substance: hydrogen, mle = 2; benzene, m l e = 78; cyclohexane, m / e = 57; and argon, m l e = 40. The spectrometer is fed with a small fraction of the gas mixture by a continuous gas sampling system (Balzers GES 10) which allows high pressure reduction without any change in the chemical composition. Finally the mass spectrometer is connected to a small computer (CII Mitra 15) which stores the values representing the concentration profiles of each species. 0003-2700/79/035 1-0077$01.OO/O
Procedure. A few tens of seconds before the injection of the reactant, data acquisition is initiated to locate the base line for each substance at the beginning of the experiment. Data acquisition is stopped when all the concentration profiles have returned to their respective base lines. The experiment is repeated with different amounts of reactant (From 10 pL up to 50 gL). At the end of these experiments, the relative sensitivity of the mass spectrometer with respect to cyclohexane and benzene is measured with a mixture of known composition injected at the outlet of the reactor. The physical parameters (temperature and volumetric flow rate Q(0) of nitrogen) are measured and a new set of experiments can be performed at a new temperature. Finally, the recorded concentration profiles are processed by a computer program which calculates the best fit for the chemical equilibrium constant.
THEORETICAL CONSIDERATIONS Let us consider t h e stoichiometric relationship C s B + 3H2 together with the chemical equilibrium constant defined by
If the reaction rate is fast, expression 1 is true a t any time and the concentration profiles may be used to compute K at any data point. The available information consists of three apparent (i.e., measured) concentration profiles y , ( t ) (arbitrary unit) connected with the true profiles C',(t) (mol/cm3) by t h e sensitivity coefficient s, defined by, y,(t) = s,C,(t)
(2) The value of s, need not be known for all the substances. Let us show that the ratio S = sB/sc is the only coefficient required. Assuming that the volumetric flowrate Q ( t )is known, a mass balance allows us t o write: _(;lmQ(t)C,(t)dt =
JcQit)CH2(l)dt = nco -
where nrois the amount of cyclohexane injected. Relations 2 and 3 together with the definition of S lead to: 1 sg = Qit)[rB(t) + S y c ( t ) l d t (4)
nco
S,
T h e right-hand side of Equation 4 is computed from t h e previously stored profiles y , ( t )and from the measured value of S. Then it is easily shown from Equation 2 and 3 t h a t
T h e volumetric flow rate Q ( t ) is computed from the concentration profile of argon -yA(t). Since the volumetric flow 1978 American Chemical Society
78
ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979 1
Ti"
MINUTES
U
0 Figure 1. Experimental layout. (1) Nitrogen cylinder. (2) Pressure and flow regulator. (3) Feed. (4) Isothermal reactor. (5) Oven. (6) Argon cylinder. (7) Heated capillary. (8) Continuous gas sampling system. (9) Vacuum pump. (10) Quadrupolar mass spectrometer. (11) Electrical interface. (12) On-line computer
rate of pure argon added to the stream flowing out of the reactor is constant, the molar flow rate of argon is constant: Q ( ~ ) Y A ( ~=)
Q(o)r~(o)
(6)
If Q(0) is known
Thus, the knowledge of y , ( t ) , y B ( t ) , * ( ~ , ( ty)A, ( t ) . S, nco and Q(0)is sufficient to compute C,(t), CB(t) and CH,(t). In order to compute an average value of K using all the data points y,(t), we minimize the functional with respect to K
J = jm[Kc,(t) 0 - C ~ i t ) ( C , , ( t ) ) ~ ] dt '
(8)
0 1 . . 0
,
,
,
5
,
,
,
,
10
,
,
,
,
.
15
,
,
,
MINUTES
20
Flgure 2. Typical experimental resutts: 25.5 pL of Cyclohexane injected at 250 "C. Top diagram: concentration profiles (H2 profile is divided by 3). Middle diagram: experimental (continuous curve) and computed (dotted curve) C,H, profiles. Bottom diagram: volumetric flowrate deduced from the argon concentration profile
This method avoids division by a concentration close to zero as could be the case if Equation 1 was used directly to compute
K.
RESULTS AND DISCUSSION Figure 2 shows a typical experimental result. It can be seen that the three components are separated from each other (top diagram). This is the consequence of the chromatographic properties of alumina in the temperature domain (187 up to 245 "C). In spite of this chromatographic effect, the reaction and the separation occuring simultaneously, chemical equilibrium is achieved a t any point so that the concentrations a t a specified time obey the mass action law. T h e second diagram shows the experimental cyclohexane profile icontinuous line) and the calculated profile (dotted line) from the experimental values of CB and CH2and the optimized value of K . T h e agreement is found to be excellent. The third diagram shows the variation of the volumetric flowrate, which is high at the beginning of the experiment; this is a consequence of the vaporization of the cyclohexane. The weak variation which follows is due to the reaction itself. Figure 3 shows a Van't Hoff plot of K(atm3) between 245 "C and 205 "C. A linear regression gives: AH = 50960 cal/mol and AS = 91.86 cal/mol/K. The results are very close to those proposed by Stull et al. ( I ) : Z0 = 51620 cal/mol a t 500 K and A S o = 92.91 cal/mol/K a t 500 K. The accuracy of the proposed method has to be compared to the accuracy obtained in the more standard method which consists in waiting for steady-state composition at the outlet of the reactor. To prove the usefulness of the pulse method, we must operate the reactor during stationary runs with the same amount of cyclohexane as used in pulsed runs. For this
MI18
.OM0
DO22
Figure 3. Van't Hoff plot of K(atm3). Squares: experimental results in pulsed runs. Vertical segments: experimental resuits in steady-state
runs. Between 245 and 205 OC, chemical equilibrium composition is reached at the outlet of the reactor. The values of K a t 187 OC (bottom right) which are meaningless prove that kinetic limitations occur at this temperature for Q(0) = 0.44 cm3/s STP
purpose, we have used a saturator which gives a constant mole fraction of cyclohexane in nitrogen. In the physical conditions of Figure 2, about 5 min are required to reach the steady state, so that 25 pL of cyclohexane pass into a feed stream containing xco = 5.5% of reactant (Q(0)= 0.32 cm3/s STP). The vertical segments on Figure 3 represent the values of K obtained in stationary runs and show the poor accuracy of the standard method. These results may be explained by considering the sensitivity of K to the extent of reaction X and to the inlet
ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
mole fraction xco. From Equation 1 we get:
K=
27 x3cox4 3XCoX)3 1 -
CT3
(1
+
x
where CT is the total concentration of the gas phase. By differentiating
-=3-+-
dK K
dCT CT
dx( x
4-.-- 9 x c o x 1 3X&X
+
The method proposed above has been illustrated by the measurement of chemical equilibrium constants because no mathematical model is required. B u t it is obvious that this method can be used to investigate any time-dependent physicochemical phenomena. David and Villermaux (2, 3 ) have shown that kinetic rates and kinetic schemes may be determined with a similar apparatus by studying the behavior of a pulse of reactant and the related pulse of products. With minor modifications, the method could be used to investigate periodically operated reactions (for instance, chromatographic reactors (4-6) or self-oscillating reactions (for example the oxidation of carbon monoxide on a platinum catalyst (7)). The great number of people working in the field of pulsed microreactors (8,9) shows the interest in transient analytical methods. The use of a programmed mass spectrometer connected to a transient chemical reactor is probably one of the best solution.
+L)+ 1- x
dxro(3-9 1 +xcOx 3xcox XCO
79
)
(10)
When the reactor is operated with a low mole fraction of reactant, X is close to 1 and X / ( 1 - X)is high, resulting in poor accuracy as can be seen from (10). T h e main assumption made throughout this paper is that chemical equilibrium is reached instantaneously. What happens when kinetic effects occur? Firstly, they are easily recognized because the computed values of K are meaningless and highly dependent on the amount of reactant injected (see for instance data points a t 187 "C on Figure 3). Secondly, kinetic limitations may be eliminated by decreasing the volumetric flowrate, so that the proposed method can be applied. As a consequence, the only assumption which must be kept in mind is that the mixture should be ideal.
CONCLUSIONS T h e proposed method was found to be accurate even with a very small amount of reactant (a few tens of microliters). Moreover, if no chromatographic effects had been present, the amount required would have been smaller (perhaps, only a few microliters), but this study raises three important questions: (1) For which reactions is the method efficient? (2) Is the proposed apparatus the best available? (3) Can the method be extended to other physicochemical problems? Indeed. the method is of no use for reactions such as the dehydrogenation of cyclohexane to benzene because statistical thermodynamics and thermochemistry allow the computation of very accurate values of K . Conversely, the method may be of great interest for reactions involving complex and expensive molecules. Nowadays, it is possible to connect a mass spectrometer to a liquid chromatograph and thus to study the thermodynamic properties of reactions involving complex and heavy molecules such as amino acids or steroids. T h e proposed apparatus is composed of two main parts: the mass spectrometer and the reactor. Till now, we have proposed the use of a chromatograpic column or more generally a plug flow (integral) microreactor. It would be of interest to test the method with another reactor, for instance a continuous stirred tank reactor (differential) which would give broader concentration profiles.
NOMENCLATURE C
K
mle n
Q S
S t X
X Y AH AS
concentration (mol/cm3) chemical equilibrium constant mass n u m b e r number of moles volumetric flowrate sensitivity coefficient ratio sB/sc time mole fraction extent of reaction measured concentration (arbitrary unit) enthalpy of reaction entropy of reaction
Subscripts B benzene C cyclohexane H2 hydrogen 1 B, C or H2 Co cyclohexane a t t h e inlet LITERATURE CITED (1) R. D. Stull, E. F. Westrum, G. C.Sinke, "The Chemical Thermodynamics of Organic Compounds", Wiley, New York, 1969. (2) R. David and J. Villermaux, Can. J . Cbem. Eng.. 51. 630 (1973). (3) R . David, J. Villermaux, 4tb Int. Symp. Cbem. React. Eng., 1-41, Heidelberg, (1976). (4) S. H. Langer, J. Y. Yurchak, and J. E. Padon, Ind. Eng. Cbem., 61,10 (1969). (5) J. M. Matsen, J. W. Harding, and E. M. Magee. J . Phys. Cbem., 69, 522 (1965). (6) S.2. Roginskii, M. I. Yanovskii, G. A. Gaziev, Kinet. Kafal., 3, 520 (1962). (7) M. Steintuch and R. A. Schmitz, Cafal. Rev., 15, 107 (1977). (8) T. Hattori and Y. Murakami, J . Cafal., I O , 114 (1968). (9) F. Tokehiko, S. Motoyuki, and J. M. Smith, Catal. Rev., 13 (1976).
RECEIVED for review June 20, 1978. Accepted September 19, 1978.
