The Use of Log-Log Plots in the Determination of Reaction Orders
James P. Birk Arizona State University Tempe, 85281
One primary source of information necessary to postulate a mechanism for a reaction is the empirical form of the rate law for that reaction ( I ) . In order to determine the form of a rate law, a kineticist is faced with several problems. First there must he some means of determining the concentration of reactant and/or product species as a function of time. Any analytical technique amenable to the system of interest may be used. The next step is the conversion of this concentration-time data into rate-concentration data. This procedure generally involves making appropriate plots of the concentration-time data so as to give a straight line, and is usually quite straightforward for simple reactions. Thus if a reaction, A B, follows a rate law
-
-d[A]ldt = k[AIz integration of this eqn. (2) gives [A]= [Ale - kt if (where subscript 0 indicates the concentration a t t = 0) In [A]I[A]o= -kt
if
Since [BIoand [Dl0 are constants, the rate law can he written as
-d[A]ldt
= k'[AIx
where
k' = k[B]oY[D]02 Thus the reaction, which has an overall order of x y + z, will hehave as though it were simply xth-order (i.e., the reaction is pseudo-xth-order) and the graphical procedures described above can he used to determine the values of x and of h'. In principle it would he possible to determine the order with respect to each species by isolating each of the terms successively in this manner, but usually there are experimental restrictions which prohibit such a procedure. A more common technique is the determination of values of h' in a series of experiments in which [B]o is varied (hut always >> [AID)while [D]ois held constant (and also >> [AJo)in order to determine the value of y. The value of z is then determined by measuring h' while varying [Dl0 and holding [B]o constant (or determining h'l[B]o a t various [B]o and [DID).An elementary approach to the evaluation of the exponents is the variation of the concentration of one species by a factor of two and examination of the ratio of pseudo-xth-order rate constants to determine by which power of two (1,2,4,8, etc.) the value of k' has changed
+
x=l
[Aj-I = [Ale-'+ k t if r = 2
Linearity of one of these plots throughout the course of the reaction establishes the order of the reaction (i.e., the value of x) and the value of the rate constant (h) can be calculated from the slope of this plot. In certain types of systems, which occur only infrequently, it is possible to measure the rateof the reaction directly, so thevariation in -d[A]ldt withchanges in [A] could then he used to establish the values of x and of k, and this step would not he necessary. For a simple system, the procedure outlined ahove works well and a kinetic study is a relatively trivial matter. Most reactions have a rate which depends on the concentration of more than one species, however, so more complex equations must he derived to convert concentration-time data into rate-concentration data. Data processing can be simplified in such systems if the experiments are set up with the concentrations of all species large compared to the concentration of the species which is being measured as a function of time. This technique, variously called the isolation method or swamping method or pseudo-nth-order method, essentially reduces a more complex rate law to one of the simpler forms encountered above. For example, a reaction A B C may have a rate law of the general form
+
-
-d[A]/dt = k[Ajr[B]Y[DIa (where D is any other species such as H+ present in the system). If the experimental conditions [B]o >> [Ale and [Dlo >> [Ale (unless [Dl does not change since it does not participate in the reaction) are maintained, the concentrations of the species present in large excess do not change appreciably while the species A reacts completely; so the rate law reduces to -d[A]ldt = k[A]*[BIoY[D]o' 704 / Journal of Chemical Education
kl'lkz' = ([Blo,~I[Blo,z)Y([Dlo,~l~Dl~,z~~ =(ZP when [B]o,l = 2 X [BJo,zand [D]o,l= [D]o,zThis approach is inadequate since it examines only a factor of two variation in concentration while a good kinetic study should encompass as wide a range of concentrations as experimentally possible in order to establish that the observed rate law is not merely a limiting form of the complete rate law. Over a narrow concentration range the value of y may turn out to be non-integral if the rate law contains more than one term and if more than one term contributes appreciably to the rate of reaction under these conditions. A preferable approach is to plot the logarithm of the pseudo-xth-order rate constants versus the logarithm of the concentrations of the species being varied. If the rate law is of the form k' = h[B]oYID]ox + y log [B]o+ z log [D]o then a plot of log h' versus log [B]o for a series of experiments at constant [Dl0 will have a slope of y, and a plot of log h' versus log [Dl0 at constant [B]o (or a plot of log (h'/[B]oY) versus log [Dlo) will have a slope of 2 . If such a plot gives a straight line with integral slope, the rate law contains hut one term in the concentration of that species. For example, the reduction of HCr04- by Fe(CN)s4- was found to be first-order in [HCr04-] according to a linear plot of In [HCr04-] versus and [H+]o >> [HCrtime when [ F ~ ( C N ) B ~>>- ][HCrOa-lo ~ 04-lo(3).The pseudo-first-order rate constants (hl, sec-I) obtained from such plots are shown in Figure 1as a function of [Fe(CN)s4-lo at constant [H+]o,on a log-log scale. Over a wide range in concentration, the slope of such a log-log plot is exactly equal to +1,so the reaction is also first-order in [Fe(CN)fi4-1:k l = h1'[Fe(CN)6~-]o,where the new constant hi' = kll[Fe(CN)64-]o still contains a dependence on [H+] which can be extracted only by further experiments with various [H+]. The results of such experiments over a range of log k' = log k
Figure 1. [Fe4CN)B4-] dependence of kr (--') reaction. The slope is +I.
for the HCr04--Fe(CN)a4-
[Hfjo = 5 X 10-'M to [H+]o = 0.50Mare depicted in Figure 2. Below 0.03 M H + , the slope of this plot is +1, but a t higher values of [H+] the slope decreases to 0, then passes through -1 and finally approaches -2. Thus kl' can be adequately represented as kl' = klU[H+] at [H+] 5 0.03 M, but this equation does not accurately describe the [H+] dependence of kl'at higher values of [H+].A complete description of the [H+] dependence of kl' requires a more complex equation containing more than one term in [H+]. In this case, the simplest equation which can adequately represent the data is given by although it turns out that d l ..;0, so that tern could have been omitted. While this equation appears to be quite complex, it is verv simolv derived from the shaoe of the loe-loeolot. I t is precisely in cases where curvature of the log-log plot is observed that this plot becomes a powerful tool for the determination of the form of the rate law. The followine rules allow the determination of a rate law from the shapeLf a log-log plot. I t is assumed that the slopes of these olots will have inteeral values (exceot in rare swtems where ratios of small integers such a s % are allowedjor the slooes will be chaneine from one internal value to another. giving intermediate values, in the various portions of the curve. It is also assumed that both the numerator and the denuminau)~ of the rate law, when appropriate, will be in the h + cx + form of polynomials ot' increasing order (. . . a x - ' dx2 . .-.)or can be rearranged to such aform. When a curve is approaching some integral value, but does not reach that value, it is assumed that this behavior is due to failure to study the reaction over a sufficiently wide concentration range, or to the onset of another transition region to a new integral value of the slope. Any integral value of a slope which is actually observed or is aooroached or is auicklv throueh . . nassed . .. (but . not maintained over an appreciable concentration range) will he called a limitine r.alue ofrhe rlooe tor a limirine slooe). . . Rule I : 1ncrea;ng slope with &easing concentration (upward curvature) indicates a rate law with a sum of terms in the numerator. The number of terms in the numerator is given by the number of limiting (integral) values of the slope which are either observed or approached. Rule 2: Decreasina slope with increasing concentration (downward curvaturejindfcates a rate law wi& a sum of terms in the denominator. The number of limiting vnlues of the slope equals the number of terms in the denomkator. Rule 3: A plot with hoth upward curvature and downward curvature in different areas indicates a sum of terms in both the numerator and the denominator of the rate law. The number of terms in the numerator is given by the number of
. ..
u
-.
+
+
.
Figure 2. [H+] dependence of k,' (W' sec-') lor the HCr0,--Fe(CN)e4reaction. me slope changes h-om +I to -0 to -1 to -2.
limiting values of the slope associated with upward curvature, and the number of terms in the denominator, by the number of limiting values associated with downward curvature. Any limiting value of the slope associated with an inflection point counts as both a numerator and a denominator term. Rule 4.Each limiting \ d u e of the slope gives the order of a limiting form of the rate law:. i.e,. n furm oi the rate law. ohserved in a particular concentration range, in whichone member of each sum of terms predominates over all other. members of that sum. (See reference (1)for further discussion of limiting forms.) Rule 5: Starting at the left of the log-log plot (lowest concentration range), the first limiting value of the slope gives the ratio of the orders of the first term in the numerator and the first term in the denominator. (If there is no downward curvature. the denominator is alwavs eoual to l.) . Unon eoine to higher'concentrations, each successive limiting value of the slooe ekes the ratio of the new term in the numerator with the last predominant term in the denominator (for upward curvature), or the ratio of the last oredominant term in the numerator with the new term in the denominator (for downward curvature). Rule 6: The derived rate law should always be tested by rearranging the equation so that a linear plot can be constructed, if this iipossible. If not, it m a y b e necessary to evaluate the parameters in the rate law by appropriate data processing techniques such as computer curve fitting and to use the values of these parameters to construct a curve which can be comoared to the exoerimental data. These rules will be demonstrated hy a number ofexamples. Firure 3 shows a olot of the oseudo-first-order rate constant (hi, sec-') as a fknction of IH+] on logarithmic scales with IH+ln >> ICrCN2+lofor the reaction CrCN2+ H+ Cr3+ HCG( 4 ) : ~ h eslope of this plot changes through the limiting 0 +l.The increasina slooe with increasine values -1 concentration indicates a sum of terms in the numerator a n i the three limiting values indicate three terms having orders with respect to [H+] of -1, 0, and +l.At low [H+], the first slope of -1 gives the equation, kz = a2[H+]-'. On going to higher [H+],the limiting slope of 0 adds a term: k2 = az[H+]-1 bp; a t still higher [H+],the limiting slope of +1 gives
.
--
+
- -~~
-
+
+
kz = n2[Ht]-' + bp + c2[Hf] This equation cannot be rearranged into a linear form, but plots of kz versus [H+] a t high [H+] and of kz versus [H+]-1 a t low [H+] should hoth be linear with the same intercept, b p Votume 53,Number I I, November 1976 / 705
Figure 5. IH+] dependence of k4 (M'sec-') for the Ti3f-Ca(NH3)sN32t action. The slope changes from -0 to -1.
re-
Figure 3. [Ht] dependence of k2 (sec-') for l h aquation ~ reaction of CrCN2+. The slope changes from - 1 to -0 to +I.
Figure 6. [HgZC]dependence of ks (sec-') for Ihe ~g~+-cisCrC12+ reaction. The slope changes from +1 to -0.
Figwe 4. [HI] dependence of kg (M' see-') for the Fe(CNk4--Br03action. The slope changes from SO to +2.
re-
(see reference (4) for representative plots). This derived rate law suggests the following mechanism (1, 4)
+
CrCN2+ Hz0
K X> [Fe(CN)64-]o,with
(3-15.
-d[Fe(CN)e4-]/dt = ka[Fe(CN)s4-I[BrOs-]
This plot has a slope approaching 0 at low concentrations of H+ and a slope of +2 a t high concentrations. Although the slope must necessarily pass through a value of +1,it is possible to tell whether this is a true limiting value only by making other plots. The change in limiting slopes 0 +2 indicates a rate law
-
k3 = 0 3 + b3[H+I2 This equation is verified by linearity in a plot of hs versus [H+]2, with no evidence of deviations from linearity which would have indicated the necessity of a first-order term. Figure 5 shows the [H+] dependence of the Ti3+Co(NH&Ns2+ reaction ( 6 )with 706 / Journal of Chemical Education
-d[Co(NH&NP]ldt = ~ ~ [ C O ( N H ~ ) E N ~ ~ + ] [ T ~ ~ + I
and [H+]o and [Ti3+jo>> [CO(NH~)SN~~+]O. At low concentrations the slope is approaching 0,while at high concentrations the slope is -1. Decreasing slope (0 -1) with increasing concentration indicates a sum of terms in the denominator. Two limiting values denote one numerator term and two denominator terms. The limiting slope of 0 gives h4 = a41b4and the slope of -1 adds a term
-
k4 = a A b 4
+ [Ht1)
Note that a t low [H+], when b4 >> [H+],the limiting form h4 = a4/b4with order of 0 is observed and at high [H+] with [H+] >> bq, the limiting form is kq = aq[H+]-l with order of -1, as indicated by the log-log plot. This equation can be rearrearranged to a linear form by inverting k4-1 = br/a4 + [Htl/a4
A plot of h4-I versus [H+] should be linear. (Such a plot is given in Fig. 1of reference (6).) Another example of this type of behavior is given in Figure 6 for the reaction of Hg2+ with cis-CrClz+ (7) at constant [H+] and with [Hg2+1o>> [cis-CrClz+] and with -d[cis-CrC12+]ldt = ks[cis-CrC1zt]
The slope is t l at low concentrations (h5 = as[Hg2+]lbs) and approaches 0 at high concentrations k~ = adHg2+ll(bg+ [Hg2+l)
The limiting forms of the derived equation are ks = as[Hgz+]lbswhen bs >> [Hg2+]
ks = as[Hg2+]/[Hg2+]= as when bs > [HCr04:]o and where k7 is defined by
-
- -
-d[HCrOa-]/dt = k.i[HCrO
