T H E THEORY OF GASEOUS EXPLOSIONS AND THE OXIDATION OF HYDROGEN SULPHIDE BY H. AUSTIN TAYLOR
H. W. Thompson’ in a paper under the above title has recently criticised the results of some investigations upon the oxidation of hydrogen sulphide2 and of ethaneTconsidered from the point of view of the theory of gaseous explosions developed by Semenoff. I t is claimed that the use of the Semenoff results in the form of an equation: log p = A/T B instead of: log p / T = A’T B
+ +
has “invalidated much of the interpretation of the results of the oxidation of hydrogen sulphide” while a t the same time it is admitted that precisely the same conclusions would have been drawn had the second equation been used, as were drawn from the results of the first equation. I t is the object of the present paper to show that both equations are approximations of a more general equation and while it must be admitted that the log p / T form is the more accurate, due to a fortuitous cancellation of errors, thevalueof A whichit yields is higher than the truevalue. The employment of the log p equation can hardly therefore be judged as a “misunderstanding of the essential results of Semenoff’s theory.” It is hoped further to show that other criticisms are unwarranted on the basis of the evidence presented. In order to demonstrate this it will be necessary to consider the Semenoff theory in detail and enumerate the many approximations that are involved in the deduction of the final expression in either of the above forms. The actual magnitude of these approximations has not yet apparently been considered even by Semenoff. Semenoff’s theory of thermal explosions with which we are concerned here assumes a production of heat, by reaction between molecules activated in the “classical Arrhenius” sense, a t a rate which is greater than that of the loss of heat by conduction, or by radiation from the walls. Considering a mixture of a molecules of type A and b molecules of type B the rate of reaction is proportional to a b e-’ I” where E is the energy of activation. The rate of production of heat by the reaction assumed exothermal is then given by: q, = B a b e-cinT (1) where B is a constant involving the heat of reaction, size of vessel, collision J. Phys. Chern., 35, 3639 (1931). ?Taylor and Livingston: J. Phys. Chem., 35, 2676 (1931). Taylor and Riblett: J. Phys. Chern., 35, 2667 (1931).
H. AUSTIN TAYLOR
I052
frequency and so on. The rate of loss of heat by the system is supposed as a first approximation to be proportional to the temperature difference between the reacting mixture and the walls, that is: qz = k(T-T,) (2) where T is the temperature of the reacting gases and To that of the walls of the containing vessel. I n the limiting case for explosion the rate of production of heat equals the rate of loss of heat and the rates of change of each of these with temperature must be equal. For a gas mixture entering a vessel a t To and exploding a t a temperature Tz we have:
B a 6 e-E’RTz = k(TP-T,)
(4)
and
B a b e-E’RTz = k(RT:/E)
(5)
whence
Tz Tz
and
- To = RT:/E =
E/2R.
(I
(6)
41-4RTJE)
The positive sign before the root would make TPso much greater than To as to be impossible. Hence we may take:
TP = E/2R. ( I - ~ I - ~ R T J E ) From ( 5 ) substituting this value of TZwe have ab
(- I-dI:4RT0/E )
=
(7)
~E/~R.(I-~/I-~RT,/E)~
If n is the total number of molecules and CY is the fraction of type A then a = CY n and b = (I -a)n. If no is the number of molecules in the vessel a t N.T.P. and 6 and 6, the densities under reaction conditions and N.T.P. respectively then n = n,S/6, where 6, = 760/R273 and 6 = p/RT,, p being the observed pressure of the gas mixture. Making these substitutions and taking logarithms we obtain: In p/To = I / ( I - ~ I - - ~ R T , , / E ) ln Ex ( I - ~ I - ~ R T J E ) K (8)
+
+
where K is a constant involving B, no, CY etc., that is, a constant for a given reaction mixture. This is the exact form of the equation which Semenoff’s theory yields before any immediate approximations are included. The principal approximation made in an attempt to simplify the above equation is that 41 - 4RT,/E = I - zRT,/E. That this approximation is not justifiable can be seen by referring to equation (7) where Tz = E/PR.(I - 41 -4RT,/E) which with the approximation becomes Tz = To indicating either no reaction or reaction with no heat change, in neither of which cases could explosion possibly occur. Overlooking this momentarily however if we assume this approximation and introduce it into the general equation (8) we obtain: In p/T,
=
E/zRT,
+ In 2RT,/EM + K
(9)
THEORY O F OXIDATION O F HYDROGES SULPHIDE
1053
and assuming with Semenoff that lnT, does not change much in the average temperature range studied, the expression may be written:
+B
In p/T, = A/T,
There appears to be an error in Semenoff’s paper1 which has been perpetuated in all papers employing his relation. The relation between A and E is given by Semenoff as log e A = E - = E (.II) 2R
whence E = 9.9 A This latter is obviously in error, and should be, as it stands 0.09. The real value of zR/log e is however nearer 9.2. The neglect of the variation of lnT, with temperature is confined in the above solely to the right hand side of the equation. If simultaneously neglected on the left hand side we would have: In p = A/T,
+B
an equation which will therefore involve twice whatever error is involved in the first form and which might a t first sight therefore also be considered negligible. This unfortunately is not so since as may be shown the neglect of the variation of lnT, with temperature is almost balanced by the simultaneous neglect of the higher powers in the binomial expansion of d~ -4RTJEwith the result that the In p/T, form is fortuitously more accurate than the In p form. The difficulty of demonstrating this in the general case of equation (8) owing to its complexity may be circumvented by calculations using the actual data available in the hydrogen sulphide oxidation. The value of A using the In p/T equation which is given by Thompson as “about 2 0 5 0 ” is by actual calculation 2 0 1 0 whether all of the data quoted by Taylor and Livingston are used or merely those used in the plot by Thompson. The energy of activation according to Semenoff ought thus to be very close to 2 0 , 0 0 0 calories. Assuming a value of E = 20,ooo calories we may by equation (8) calculate the values of In p/T a t say 500 and j joo K which is within the range studied. A substitution of these values in the equation log p/T = A / T
+B
will give for A the correct value it should have to satisfy the observed data by this form of an equation. Calculation shows this value of A to be 195 0 , that is somewhat less than the observed value 2010. This deviation is due however to two separate approximations as may be seen by taking merely the next term in the binomial expansion of the root. Taking ( I - ~ R T J E ) ” = I - 2RT,/E - 2RzTZ,/EZ ‘2.Physik, 48, 571 (1928).
n. AUSTIN
I054
TAYLOR
the general equation (8) becomes:
Comparison with equation (9) shows that the use of E/2RT0 as equal to A/T" will yield a value for A which is too great, whilst neglect of the variation with temperature of the second term on the right due to its incorporation in B will yield a value that is too low. These errors approximately neutralise each other since if we repeat the above calculation of A, assuming E = 20,000 calories, between 500 and 550' K using however (9) instead of (8) a value of 1945 is obtained which may be compared with the 1950 derived from the exact equation. It is furthermore actually possible to evaluate each of these variations separately and confirm the above result. There appears then no especially good reason to attempt to simplify the general equation (8) even if such were possible since the results of log p/T = A/T B are fortunately quite good despite the fact that it can theoretically hold only under conditions where no explosion could occur. There remains only to emphasize one or two points in the paper by Taylor and Livingston in connection with certain criticisms by Thompson. It is apparently tacitly assumed by Thompson that the conditions prevailing in the two sets of experiments are similar. Taylor and Livingston however pointed out that using gases which had been dried over phosphorus pentoxide, the reaction rates were erratic but consistently slower the longer the period of drying and in consequence used throughout the work, gases saturated with water vapor a t 21OC. Reproducibility was then quite good provided a thorough cleansing of the system was made before each run. Again, the experiments of Taylor and Livingston were carried out in a spherical bulb of 300 ccs. capacity and approximately 8 crns. in diameter. This would correspond with a ratio of surface to volume of about 0 . 7 5 as compared with the smallest ratio used by Thompson and Kellandl of 1.3 increasing to 7.8 for another vessel used by them. Accepting the chain nature of the reaction this ratio is of considerably less importance than the fact of the vessel diameter being 8 crns. in comparison with 3.2 in their widest vessels. Such a difference would mean for a chain reaction, less frequent breaking of chains by the walls and a consequently greater proportion of reaction in the gas phase. The fact that the minimum explosion pressures observed by Thompson and Kelland are considerably higher a t a given temperature than those found by Taylor and Livingston confirms this point. Hence with moist gases as opposed to extremely dry ones (Thompson and Kelland fractionated the hydrogen sulphide having frozen it in liquid air) and with a wider reaction system, it would appear a gratuitous assumption to postulate that the proportion of heterogeneity in the reaction was the same in the two cases merely because the temperature was the same. The energy of activation measured need not then correspond to that of a surface reaction particularly since in addition to the
+
J. Chem. SOC.,1931,
1809.
THEORY OF OXIDATION OF HYDROGEN SULPHIDE
I055
above, the data used were obtained a t temperatures and pressures only just below the explosion limit. At the same time an attachment of any special importance to the value 18,000 calories obtained by Thompson and Kelland is uncalled for owing to the ‘‘extensive averaging,” actually over a range from about 9,000 t o 40,000 calories in individual experiments. I n conclusion the statement by Thompson that “the experimental evidence on the effect of ultraviolet light upon the system a t room temperatures offered by Taylor and Livingston would apparently speak against such a mechanism (a primary dissociation of hydrogen sulphide) since no oxidation occurred despite dissociation of the hydrogen sulphide” would not appear to be a logical conclusion in view of the fact (as pointed out by Taylor and Livingston) that the photochemical oxidation of hydrogen is veryodifficult since oxygen does not commence to absorb appreciably above 1950 A whilst hydrogen only below about 1030 R. Nichols Chemical Laboratory, NEW York University, New York, N . Y .
