Let us now consider the right-hand side of Equation 2 and write:
Reaction Rates of Nonisothermal
I
=
f Kdt
(3)
As a first approximation the dependence of rate constant K on temperature is given by the equation: l n K = - aT+ b K = e
or
-;+b
(4)
(5)
Constant a is proportional to the “activation energy” of the process, and Equations 4 and 5 are valid if this activation energy is i n d e p e n d e n t of the temperature. For many reactions this condition is nearly true over a considerable temperature range. In any case, Equations 4 and 5 may be considered valid over a s m a l l t e m p e r a t u r e interval. Introducing the value of K from Equation 5 into 3:
J. SHERMAN The Pennsylvania State College, State College, Pa.
~
Reactions which proceed nonisothermally are discussed for various timetemperature relations. The pertinent integrals are evaluated for calculating the amount of reaction when the timetemperature change is linear, exponential, sinusoidal, and polynomial. Examples are given of the application of the results to specific problems.
In order to carry out the integration, the temperature must be known as a function of the time, or vice versa, Writing t = F(T) T = G(t)
(78) (Th)
Equation 6 becomes
QT
H E rate of a chemical or physical change a t constant temperature may, in general, be written as dt
= Kj(A)
A denotes the amount of product formed (or consumed) at time 1, and f ( A ) some function of A (e. g., in a first-order reaction, f ( A ) = - A ) . K is the reaction rate constant and is a function of the temperature and pressure. Rewriting Equation 1 as
Here F’(T) denotes the derivative of F ( T ) with respect to T , and TIand TZare the temperatures a t the times tl and h, respectively. I n any given problem, either Equation 8a or 8b may be used; the choice will depend on mathematical convenience.
Linear Time-Temperature Dependence THEORETICAL. Let us now consider a reaction which occurs as the temperature varies linearly with the time: the right-hand side may be integrated directly for a reaction which occurs a t constant temperature and pressure. However, for many important industrial processes, the reaction proceeds under varying conditions of temperature and pressure, and it is therefore of interest to calculate the amount of product formed under these conditions. In this paper the evaluation of the right-hand side of Equation 2 is discussed for various time-temperature relations. The application of the results are illustrated by specific examples.
T=ct+d
and Equation 8a becomes
a
Discussion The left-hand side of Equation 2 can be integrated when f(A) is known; that is, when the kinetics of the reaction are known. It will not be discussed further in this paper.
(9)
where x = T / a
Equation 11may be evaluated by a transformation of variable z = l/y, and integration by parts. The result follows: 1026
SEPTEMBER, 1936
INDUSTRIAL -4SD ENGINEERING CHEMISTRY
reaction T / a should be greater than 0.175, which may be the case for reactions a t very low temperatures, Table I could not be used. However, H ( r ) can be calculated from the equation relating it to the integral logarithm. This equation can be derived from 13 and 15:
(z)
K T and ~ K T , are the rate constants a t the temperatures T2 and T I ,respectively, as given by Equation 5 . Ei and
Ei
1
H ( r ) = r lo-;
(2)are the integral logarithm functions defined by the
+
1
lmeq
1
H ( r ) = r lo-;
Numerical values of the integral logarithm are given by Jahnke and Emde (6) for values of the argument from 0 to -15. For most physical and chemical problems, however, values of the argument -a/T are in the range -10 to -30. Hence, for numerical calculations, extension of Jahnke and Emde's data is necessary. However, it would be simpler to proceed more directly as follows: Equation 11 may be written -2.303T2 1
1
- 0.05 X 10-0.05 2.303 [Ei( - 2 . 3 0 3 / ~ )- Ei ( - 2 . 3 0 3 / 0 . 0 5 ) ]
For r > 0.175, 0.05 X 10-0.05, and Ed (-2.303/0.05) are 1 entirely negligible compared to r lo-; and Ei (-2.303/7), respectively; therefore we may write:
identity:
- Ei(-z) =
1027
+ 2.303 Ei ( - 2 . 3 0 3 / r )
(16)
For r = 0.175, Ei (-2.303,'~) = Ei (-13.16); therefore numerical values of Ei (-2.303/r) for T > 0.175 may be found in the tables of Jahnke and Emde (2).
-_
10 Z d x
2.303c- 2.303T1
[.(+) 2 303T
=
-H
(e')] 2 303T
(14)
2.303~
where H(r) is defined by the equation
0 O M
The transformation of Equation 11 into 14 and the value of 0.05 for the lower limit in Equation 15 were chosen purely for practical convenience in computation. The function H ( r ) is given in Table I for values of r varying from 0.05 to 0.175, by increments of 0,001. In Figure 1, H ( r ) is plotted against T . It is interesting to note that log H is nearly a linear function of 1/r (Figure 2). The arguments of the H function in Equation 14 depend only on the ratio T / a and not on T or a alone. For a given reaction, T / a changes slowly over a wide temperature range, and hence Table I should be useful for numerical calculations for both high- and low-temperature reactions. If for a given
0. 050
,051
,052 ,053 ,054 0.055 056 ,057 058
059 0 060 061 ,062 ,063 ,064 0.065 066 067 ,068 069 0.070 ,071 ,072 ,073 074 0.075
0 x 10-2s 1.629 5.569 14.78 35.65 0.8154X 10-21 1.796 3.837 7.977 16.18 3.204 X 10-10 8.209 11.78 21.92 40.01 7.174X 10-19 12.64 21.91 37.38 62.82
1.041 X 10-17 1.700 2.741 4.361 6.859 1.066X 10-16
0 075
076 077 ,078 ,079 0.080
,081 ,082
,083 ,084 0.085 ,086 ,087 ,088
,089 090 ,091 092 093 ,094 0.095 .096 097 0
,098
099
0.100
1.066 X 10-1' 1.639 2.491 3.748 5.582 0.8235X 10-16 1.203 1.743 2.503 3.564 5.034 X 10-1' 7.056 9.816 L3.56 18.59 2.532 X 10-14 3.427 4.608 6.158 8.181 1.081x 10-1: 1.419 1.854 2.410 3.115 4.008 X 10-LJ
0.100 .lo1 .lo2 .lo3 .lo4 0.105 .lo6 ,107 ,108 .lo9 0.110 .111 .112 ,113 .114 0.115 .116 ,117 .118 ,119 0.120 .121 ,122 ,123 .124 0.125
006
I
008
(
1
010
013.
012
016
018
r
F I G ~ X1~. E PLOTOF H ( r ) AS A FUNCTION OF r
APPLICATION. As an example of the way in which the results of the foregoing section may be applied, let us consider the size of a limestone sphere that would be completely calcined in the blast furnace a t a given temperature. Kinney (3) showed that the temperature of the stock increases 6.76" C. per minute along the walls of the blast furnace. Furnas (1) showed that the rate of calcination of calcium carbonatethat is, the rate a t which the phase boundary between the calcium carbonate and lime advances from the outside to the
4.008X 5.131 6.539 8.296 10.48 1.318 X 1.651 2.059 2.558 3.166 3.905 X 4.797 5.874 7.167 8.716 1.056 X 1.276 1.538 1.846 2.211 2.640 X 3.143 3.732 4.419 5.219 6.148 X
10-18
10-1:
lo-"
10-11
~~~
10-11
10-11
0.125 .126 .127 .128 .129 0.130 .131 .132 ,133 .134 0.135 .136 .137 ,138 .139 0.140 ,141 ,142 ,143 .144 0.146 .146 .147 ,148 ,149 0.150
6.148X 7.225 8.470 9.907 11.56 1.346 X 1.563 1.812 2.096 2.419 2.787X 3.204 3.676 4.210 4.813 ... 5.491X 6.255
10-11
10 - 1 0
10-10
0.150 ,151 .152 .153 ,154 0.155 156 .157 ,158 .159 0.160 ,161 ,162 .163 -164
10-10
7.112 8.073 9.148 1.035X 10-9 1.169 1.318 1,485 1.669 1.874 X 10-'
0.165 .166 ,167 .168 ,169 0.170 .171 ,172 ,173 .174 0.175
1.874 x 10-9 2.101 2,352 2.629 2.936 3.272X 10-9 3.643 4.050 4,498 4,989 5 . 5 2 7 :< 10-9 6.115 6.757 7.459 8.226
x
0.9058 l ( 1 - 8 0.9966 1.095 1,202 1.319 1.445 :< 10-8 1.581 1,729 1.888 2.061 2.247 X 10-8
INDUSTRIAL AND ENGINEERING CHEMISTRY
1028
inside-is constant a t a given temperature. It was also shown that this mechanism holds for a limestone containing varying amounts of magnesium carbonate, silica, etc.; hence Equation 2 may be written
VOL. 28, NO. 9
and Equation 8a becomes (18) This can be integrated to give
where y is defined as twice the thickness of the calcined layer in inches, and K is in inches per minute. Equation 14 may then be written =
aeb
2.303 X 6.76
eb [H(2.303T~/a)- H(2.303Ti/~)]2.303~
[H(2.303T2/a) - H(2.303Tda) 1
Ei ( - a / T ) and H
Let us suppose that constants a and b for a particular stone are 30,000 and 17, respectively, and that i t is desired to find the size of stone that would be completely calcined a t 1500°K. T I ,the temperature at which appreciable calcination begins, may conveniently be taken as 1100" K. Substituting these values of the constants in the above equation,
'
E-
30'000e'7 [H(2.303 X 1500/30,000) 2.303 X 6.76 H(2.303 X 1100/30.000) 1 . . ,. = 4.83 X 10" [H(0.115) - H(0.084)] = 4.83 X 1011 (1.056 X lo-" - 3.564 X 10-'6) = 5.10 inches =
The values of H(0.115) and H(0.084) are found from Table I. This result states that a sphere of limestone (for which decomposition constants a and b have the values given above) 5.10 inches in diameter would be completely calcined a t 1500" K. as it moved along near the walls of the blast furnace where the temperature rise was assumed to have the uniform value of 6.76" c. per minute.
(2.303 T/a) have been previously deihed. APPLICATION.Let us apply Equation 19b to a hypothetical case. h s u m e To = TI = 1000" K., Tz = 500" K., a = 15,000, b = 16, c = 1. Equation (19b) becomes: 500
el6
I = - 2.303 [H(0.077) - H(0.154)] - 15.000 el6
- 2.935
(2.491 X 2.303 = 0.1428 e-
X lo-")
500
+
$.
-3 e-" 0 + 15 e
FromEquation 17 the time taken to go from 1000" to 500°K. is seen to be -In 500/1000 = 0.6931. If the reaction had proceeded isothermally, I would have been equal to Kt = 0.6931 K . Equating 0.6931 K to 0.1428, it is found that T = 854" K. I n other words, the amount of reaction which takes place in a given time as the temperature decreases exponentially from 1000" to 500" K. is the same as if the reaction had proceeded isothermally a t 854" K.
Sinusoidal Time-Temperature Variation THEORETICAL. Let us now consider a reaction in which the temperature fluctuates periodically about some value To. A general treatment of the problem would necessitate a Fourier series expansion of the temperature as a function of the time. This would be too difficult to evaluate. Instead, we shall assume that the temperature varies sinusoidally with the time: T = To(1 cy sin 2 4 (20)
+
Introducing this value into Equation 8b:
I: FIQURE 2.
-LOO H(r)
us. l / r
According to Equation 14 the size of a limestone sphere that would be completely calcined a t a given temperature is inversely proportional to c, the rate at which the temperature rises. Thus, if in the previous numerical problem this rate were only 3.38" per minute instead of 6.76", then the size of stone that would be completely calcined a t 1500" K. would be (2 X 5.10 =) 10.2 inches. The foregoing example illustrates the way in which Table I may be used to calculate the amount of product formed in a reaction which proceeds as the temperature changes linearly with the time. Similar calculations could, of course, be made for any other reaction for which the mechanism was known.
This integral is still too difficult to be directly evaluated. However, it can be integrated over an integral number of cycles when the amplitude of the temperature fluctuations is small compared to the mean temperature-that is, when the constant a is small compared to unity. In this case the exponent of the integrand may be expanded into the series:
Neglecting terms in a higher than the first power, Equation 21 becomes: n
Exponential Time-Temperature Dependence THEORETICAL. Let us consider a reaction which proceeds as the temperature decreases exponentially with time, according to Newton's law of cooling: T = Toe-ct
(17)
Jo
The integration is carried over n cycles. Equation 22 can be integrated to give the following:
SEPTEMBER, 1936
INDUSTRIAL AND ENGIKEERING CHEMISTRY
Here K T , is the rate constant of the reaction a t the temperature To, i = and J o is the zero'th order Bessel function of the first kind. Numerical values of this function are tabulated in many books and by Jahnke and Emde (6). I n Figure 3, J o (is)is plotted as a function of 2. .-IPPLICATIOX.Let us apply the foregoing result to a reaction for which the temperature fluctuates sinusoidally about 400" K., the amplitude of the fluctuations being 20"; i. e., the temperature varies between 380" and 420" K. Suppose, further, that the fluctuations occur twenty times per minute and that a = 10,000. Equation 23 then becomes:
42
I
=
-
(f$)
n (10,000 X 0.05i -K 4 0 0 JO 400 20 1.431 nK400 20
If the integration is carried out over twenty cycles-that if the reaction proceeds for 1 minute-then I = 1.431 Kioo
Furthermore, in problems of practical interest the temperature may not be expressible as a simple function of the time a t all. However, by the method of least squares it would be possible to express the temperature as a polynomial function of the time to any desired degree of accuracy, or, if the change were periodic, as a Fourier series in time. This latter case has already been discussed.
I
01 0
05
I
15
I O
20
25
30
x
is,
If the reaction had proceeded isothermally for 1 minute, I would have been equal to KMO. Hence under the above conditions, the reaction proceeds as far in 1 minute as it would have proceeded in 1.431 minutes under isothermal conditions. Suppose now that the amplitude of the temperature oscillations were only lo", the frequency remaining the same as above. Then
I
1029
n 1.100n K 4 0 0 (0.6252') = _ _ 20 20 = 1.100 K4oofor 1 minute
= - Ktoo J o
This result shows that the "effect" of temperature oscillation is about only one-fourth as large as i t was in the preceding case when the amplitude was twice as great. Let us now consider that the amplitude of the temperature fluctuations is 20°, as originally assumed, and that To, the temperature about which these fluctuations take place, is 800" instead of 400" K. The value of the integral becomes: 20 I = - Keoo Jo (0.6253') 20 = 1.100 Ksao Thus a 20' oscillation about 800" K. causes only a 10 per cent increase in the amount of reaction as compared to 43.1 per cent when the temperature fluctuated about the mean value of 400" K. At 2500" K. the increase in the amount of reaction is only 1 per cent. This shows that a t high temperatures small fluctuations in temperature cause only a slight increase in the amount of reaction. From Equation 23 i t is seen that the amount of reaction for a given time interval is independent of the frequency of temperature oscillations. It depends only upon n/v. Thus, if the frequency were 40 oscillations per minute in the preceding example, n/v would have been n/40. To calculate the amount of reaction in one minute, the number of oscillations would be equal to 40, and so n/v would have been 40/40 instead of 20/20.
FIGURE 3. ZERO'TH ORDER BESSELFUXCTIOX FOR PURELYIMAGINARY ARGCXENT
Let us consider, then, that the temperature may be represented by the polynomial: T To dit d2t2 d3L3 . . . d n t n
+ +
+
+
+
Introducing this value into Equation 8b: I =
eb
"re To f
a
dit f d i t 2 f . . . -I- d,t"
Except for the linear case already considered, the evaluation of this integral would be extremely difficult or impossible. The only alternative in the general problem is t o express the time as a polynomial function of the temperature. Writing t = ma
+ mlT + m2T2 + . . . +m,T"
Equation 8a becomes
. . . +nm,Tn-l)dT
(24)
Obviously it is necessary to consider only the general term:
where with n ) an integer, 2 1, we obtain:
L?e-'
T2
I' =
gn+l
1
2" dx
-
a
Let us now evaluate the indefinite integral,
It may be evaluated by successive integrations by parts. The first four expressions are:
Polynomial Time-Temperature Dependence
THEORETICAL. In the preceding cases the temperaturetime relations were taken to be linear, exponential, and sinusoidal. Obviously many other relations might be considered.
A
S, =
s - 120 go
2 2
e-; (1
- 22 + 6x2 - 24zs)
1030
INDUSTRIAL AND ENGIXEERING CHEMISTRY
\-OL. 28, NO. 9
Se-; 1
So denotes the indefinite integral, ds, and is equal to [ H ( r ) ] / 2 . 3 0 3when the integration is carried out beheen the limits 2.303 X 0.05 and 2.303 T . Higher members of the series may be obtained by means of the recursion formula, (n
+ 1)Sn + (1 - nx)Sn-1 - XSn-2
=
0
(26)
or by means of the formula, n-1
Let us now consider the time t o be expressible as a polynomial function of 1/T. Corresponding to Equation 25 we consider the indefinite integral
Assuming a = 16,000, T I = 1000" K., T , = 500" K., and b = 16, we obtain 1 = 0.589. The time taken to go from 1000" to 500°K. is 109/(,!jOO)3 109/'(1000)3= 7 . If the reaction had occurred isothermally, the amount of reaction would have been 7 K . Equating this to 0.589, the temperature turns out to be 813' K. In other words, the amount of reaction which occurs in a given time interval as the temperature cools from 1000" to 500°K. is the same as if the reaction had proceeded isothermally a t 813" K.
Summary This may be evaluated explicitly for n 2 2. (For n = 1, the problem is just that of the exponential temperature-time relation already considered.) The first four expressions are
The integral J K d t = Jj&, where K is the rate constant for a reaction, has been evaluated for the following timetemperature relations:
1
T=ct+d
_ _1
T = T = To(1
S-z = . e - ; s-3
=
+ - _1 (1 + 2 s + 2x2)
5 (1
S-4 =
__ 1
S-6 =
(1
(A) (13) (C)
+ a sin 2rvt)
2)
+ 3x + 6 x 2 + 6 x 9
The values of the integral
Kdt, corresponding to the
above relations, are: Other expressions for S-, may be obtained by means of the recursion formula,
xS-, -
(TLX
- 2s
1)S-n+1
+ (n - 3)s-n+z = 0
C
% + aeb [Ei(-a/T,) - E i ( - a / T 1 ) ] = C
(29)
ueb
[H(2.303Ttz/U) - H(2.303Ti/a) (A')
2.303~
or by means of the formula,
eb
- [Ei(-a/T,)
- Ei(-a/T1)]
=
where F (a,b, c, x) is the hypergeometric series defined by the equation, F(a, b, c, 2) = 1
a(a + 1 M b + 1) + ab IC 2 + 1 x 2c(c + 1)
x2 + ,
.. 1
n-1
(31)
APPLICATION. I n order to illustrate the application of these formulas to a numerical problem, let us consider the temperature to be dependent upon the time in accordance with the formula,
eb an-l [S-n(a/TZ)
- S-n(a/Tl)I,
1
T = 1000t-Zor t
=
109/~3
Equation 8a becomes
E--
a3
[S-c(Tz/a) - S - i ( T l / a ) l
From the above expression for 8-4,
S-, =
52-n
2 2
e - : lim F ( 2 E+=
(E')
- n, 1, 8,
-6%)
The foregoing expressions permit the computation of = fo d A -that is, the amount of reaction for the appropriate time-temperature relation relative to the amount that would be obtained under isothermal conditions. However, the calculation of the absolute amount of product A a t a given time cannot be carried out unless f(A) is known dA and the integration - performed. f(A) depends on the f(A) kinetics of the reaction and is discussed in textbooks of physical chemistry, including that of Taylor (4). The formulas of this paper are valid only under the as-
pat
a
3 X lOQ eb
where
1
SEPTEMBER, 1936
103 I
IKDUSTRIAL -4UD ENGINEERIKG CHEMISTRY
sumption that the reaction rate constant is determined by lii K = -a/T b. Finally, it must be realized that the results presented here apply only to individual reactions and not to a complex heterogeneou. process which is the resultant of several reactions. In the latter case the dependence of the over-all rate constant on temperature may not be given by In K = -a’T +b.
+
Acknowledgment
E+)
e = base of natural logarithms = integral logarithm of 2
F (a, b, c, 2) = hypergeometric function 1
H(r) =
integration.
Nomenclature
+
a , b = constants in the equation In K = -a/T b, expressing the rate of a reaction as a function of the temperature
ds
” L 5
i=m Ja(z) = aero’th order Bessel function KT = rate constant at temperature T Y = frequency of temperature oscillation T = absolute temperature t = time
Literature Cited
The author wishes to express his appreciation to Selson
K. Taylor a t whose suggestion this paper was written, and to P. L.Smith and B. Ellefson for checking the f0rmUkis O f
lo-;
F,lmas, c, , I;vD, ESG. CHEII,, 23,534 (1931), (2) Jahnke, E,, and ~ ~ F,, “Tables d ~ of , ti^^^," znd ed,, pp. 83-5, 278, €3. G. Teubner, 1933. ( 3 ) Kinney, 6. P., Bur. Mines, Tech. Paper 442 (1929). (4) Taylor, H. S., “Treatise on Physical Chemistry,” 2nd ed., Tol. 2, Chaps. XI\-and XV, New York, D. Van Nostrand Co., 1931. RECEIYED
>ray 29. 1936.
ANESTHESIA Clinical Application of Recent Chemical Contributions JOHN S. LUNDY Mayo Clinic, Rochester, Minn.
C
HERIISTS hare supplied clinical anesthetists with a number of anesthetic agents in the last few years which have interesting physiologic effects and are meeting a long-felt need in medicine. S o doubt there will be some variation of opinion among medical men concerning the relative value of various anesthetic agents in clinical practice. The ideas expressed here are those of the author and do not represent general opinion on the subject. There has been a progressive demand in the last few years for special agents and methods of anesthesia over ordinary agents and methods; this is illustrated by experience a t the Mago Clinic (Table I). A great deal of progress has been made in the development of new anesthetic agents to meet this demand. Table I1 shows these special agents and methods used a t the clinic, when they appeared, and when certain of them were supplanted by others or when their use was
discontinued. Regional anesthesia produced by local anesthetic agents has continued to be a popular method, in spite of some unsatisfactory results which have been obtained with certain agents introduced in the period here reviewed.
Local Anesthetics
SUPERCAISE. Of the local anesthetic agents used in recent years, nupercaine or “percaine” (a-butyloxy cinchoninic acid y-diethylenediamide hydrochloride) is one of an unusual type; it was applied clinically as a spinal anesthetic 18) in a concentration of 1 t o 200. The dose of nupercaine was measured directly in the syringe, and spinal fluid was then aspirated into the syringe to secure a dilution of 1 to 800. Injection was carried out at a rate of 0.5 cc. per second. The use of nupercaine for spinal anesthesia, however, did not prove as satisfactory a t this clinic as it evidently did later in the experience of Jon& ( 3 ) . He injected a solution of nupercaine and spinal fluid in a coiicentration of 1 to 1500. The desirability of Attention is called to trends in the use of special anesthetic injecting a local anesthetic agent agents and methods of administering them in the last ten which is also a surface anesthetic intrigues the clinician into trying such years, including certain new local anesthetics, general inhalaagents, always in the hope that one tion, anesthetics (both gaseous and volatile), intravenous will finally become a v a i l a b l e to anesthetics (especially those of *the thiobarbituric acid series supply all needs. Then full attenand evipal soluble), and rectal anesthesia. The author extion can be deiroted to a complete presses his own ideas and does not pretend that they repreand thorough understanding of the physiologic action of the agent so that sent the general opinion of clinicians. The paper contains a it may become generally used and tinclinical evaluation of the drugs only. The author’s reasons derstood. Since, however, nuperfor liking or disliking agents is expressed briefly together with caine is essentially a quinine derivahis ideas of their clinical usefulness. tive and may produce e x t r e m e l y toxic effects, I have discarded it.
