A Unified Equation
for Chemical Kinetics
Xinyi Tan and Siegfried Lindenbaum Department of Pharmaceutical Chemistry, 3006 Malott Hall, The University of Kansas, Lawrence, KS 66045
Noel Meltzer Hoffmann La Roche, 340 Kingsland Street, Nutley, NJ 07110
Chemical kinetics is generally taught by applying separate equations for zero-, first-, and higher order reactions. The purpose here is to present, in addition, an equation
from which those various chemical kinetic equations can be readily derived. Also this new equation may be applied to more complex situations not amenable to simple rate law expressions. For any reaction, the rate equation can be written as da dt
da
—^(a)
/(a)
where F(a) is the integrated form of 17/la) with respect to The integration constant is obviously zero. Equation 3 indicates that the product of the apparent rate constant k and t„, the time corresponding to a,,, can be calculated from Fla,,). ktn=F(an)
For example, for a* tQ !
=
a
a
0.1
is the shelf life of the product
an = 0.5 t0 5 is the half life
Then 1
=
first-order reaction, f(a)
1
=
-
a
F(0.1)
(5)
^0.5 =F(0.5)
(6)
f(a)
m Integrating eq 2,
566
we
=
—
Equation 4 is the unified equation that implies all the kinetic equations for the calculation of shelf life and half-life. For the zero-order reaction,
Sometimes, the form off(a) is not readily determined. Exchanging dt and /la) in eq 1, we get da
(4)
pharmaceutical product,
kt$ i
and for
fef
a.
(1)
a
=
(3)
kf(a)
is the fractional reaction progress at time t; k is the apparent rate constant; and f is a function representing the expression of the rate law. Often, the form of /la) is known. For example, for a zeroorder reaction,
where
r
J
=
l an
k dt
get
Journal of Chemical Education
(2)
ktn=F(an)
=
jia
=
an
(7)
^0.5
5-5
-
f°-5“_fr5 k
For
a
(.g-l
first-order reaction, /(a)
=
kt n
=
kt0 i
=
t 01 ktQ
5
k
(l-o.)
into eq 4,
F(a„) =
=
J
-In (1
-
=
°'5_
(9)
get
Hu„)"e E/RT Z
In
tn
=
In
are
the
where x is any real number except 0 and 1. Obviously, the equation relating the rate constant and shelf life or halflife can be readily derived from eq 4 because a)1
-
be easily integrated. In some cases, the expression of Han) is unknown. Although eq 4 cannot be directly applied in that case, it can further lead to other useful equations. This is specially useful for solid-state reactions. Ng has put forward a rate equation to describe the decomposition kinetics of many solid-state reactions (7).
In t o.i
where x and y are constants characteristic of the reaction rate law. In this case, even if a k value is known, the shelf life and half-life cannot be directly calculated because Ha) is unknown. No textbook deals with the kinetic parameters for this kind of reaction. Now eq 4 can be applied. Because a1_st(l
a)1"3,
-
constant independent of temperature. ktn
=
E 1 R T
(13)
In fo 5
For
a
,0.1
El
.0.5 In ^
FI ^ ^
An^ + RT
.
F(an)
=
krn
(ii)
where k' and are the rate constant and fractional decomposition time at another temperature. Therefore, the room temperature shelf life or half-life could be calculated from the rate constant at room temperature, the rate constant and the shelf life or half-life at another temperature. An example of this calculation has been shown (2).
(14) (15)
first-order reaction,
In t 0.5
a
+
Equation 13 implies that the plot of In tn versus 1/T will a straight line because F(a„) is independent of temperature as long as Zis a constant independent of temperature. After the tn data at higher temperatures are measured, the activation energy can be determined from the slope of the straight line, and the t„ values at room temperature can be extrapolated from the straight lines. The application of eq 13 to such a case was recently demonstrated in a paper from this laboratory (3). This equation can also be applied to literature data (4), and straight lines are obtained. Equation 13 can be applied not only to complicated situations, but also to simpler cases. In fact, it is easier to understand eq 13 when Ha) is known. For a zero-order reaction,
In l 0.1 :)n
F(a) is
Z
be
can
=
Hex,,)
(10)
Equations 6-10, derived from the unified equation, same as those in textbooks. For reactions of any other order,
/(a)
(12)
The logarithm of eq 12 can be used to determine the activation energy and to extrapolate the t„ value at room temperature.
0.693
=
0,693 k
/(a) = (1
Ze'EIRT
=
f
-In 0.9= 0.105
-In 0.5
we
a„)
5-105 k
_
Sometimes the degradation curve cannot be simulated by one single equation. This is the case when/fa) and Ha) are unknown. In this situation, the Arrhenius equation is not applicable because rate constants are not available. Once again, eq 4 can be used to deal with this kind of problem. Inserting the Arrhenius equation,
0.105
E 1 R T
0.693
E
]a^ ,
+
(16)
1
R T
(17)
Equations 13-17 yield straight lines for the plot of In tn 1/7’. They are the standard equations used to characterize zero- and first-order decomposition reactions (5).
versus
Summary Equation 4 covers all the rate equations when /fa) and its integrated form are known. This one equation leads to all of the kinetic equations. Under complex situations when the Arrhenius equation cannot be used, eq 4 can yield useful equations for the calculation of shelf lives and half-lives. Literature Cited 1. 2. 3. 4. 5.
Ng, W. L,.Ausl. J. Chem. 1975,28, 1168-1178. Tan, X; Meltzer, N.; Linden baum, S. Pharm. Res. 1992, 9, 1203-1208. Tan, X; Meltzer, N.; Linden baum, S. J. Pharm. Biome. Anal. 1993, 0, 0000-0000. Chen, S. M.; Chafetz, L. J. Pharm. Sci. 1987, 76, 703—706. Carstenson, J. T. Drug Stability: Principles and Practices; Marcel Dekker: New York and Basel, 1990; p 32.
Volume
71
Number 7
July 1994
567
