Analysis of Binary Mixtures by Second Order Kinetics Using Equal Concentrations of Reactants The n-Butanol, sec-Butanol System CHARLES
N. REILLEY and
LOUIS J. PAPA
Deportment of Chemistry, University of North Carolina, Chapel Hill, N . C.
b Three methods are proposed for the simultaneous determination of binary mixtures based on their differential rates of reaction. The methods apply to second order irreversible kinetics under conditions where the total concentration of the binary mixture is equal to the concentration of reagent. A new procedure for plotting the rate data makes possible the determination uf either the faster or slower component by a linear extrapolation method. Less laborious single and double point methods applicable to either the faster or slower reacting component are also described. The three methods are illustrated by the determination of a mixture of 1 butanol and 2-butanol and are compared with one another. A criterion for selection of reaction time is presented.
R
ECENTLY Siggia and Hanna (2)
reported the simultaneous detcrmination of 13 binary alcohol and 14 binary carbonyl systems by use of differential rates of reaction. Treatment of the rate data involved classical second order plots of log ( b - z)/ (a - 2) us. t which yield the conventional straight lines for the slower reacting component. This treatment requires, as the authors have stated, that the initial reagent concentration, b (acetic anhydride in the case of alcohols) is not equal to a, the total initial sample concentration, if the desired plots are to be applicable. In this paper, three methods are proposed, each of which is applicable for irreversible second order kinetics in the case where a equals b. This equality necessitates a prior determination of total sample content. Each of the methods is illustrated by the determination of a mixture of 1-butanol and 2-butanol, this example being selected from the series studied by Siggia and Hanna. In the first method, an extrapolation is made of a plot of z us. (a - z)t. This avoids the conversion to logarithms and the subsequent reconversion; however, it requires a
larger sample in order to keep the time of analysis convenient and is subject to a larger error due to the presence of water. In this method, as in that of Siggia and Hanna, the straight line portion of the curve a t later times is extrapolated back to zero time. However, the resulting intercept in this case can be read directly as the amount of the faster reacting component (1butanol); hence, if 2 is plotted as milliequivalents, then 1-butanol can be read directly in milliequivalents. An alternative method of plotting is given which permits determination of the slower reacting component (2-butanol) from the zero time intercept. A single point and double point determination can also be used in lieu of the graphical method. In the case of the single point method, the results are temperature dependent, while the double point method is temperature independent. Both of these methods are less laborious since for the calculation of the results only one and two points, respectively, are needed. These shorter methods gave results with accuracy quite similar to that obtained graphically. These methods are based on the assumption that measurements are made after all of al, but not of a2, has reacted. In actuality there is of course no time when all of al has reacted,
but there are times when the concentration of al is sufficiently small so that it may be considered negligible. A criterion t o predict this time when component al has essentially reacted, but a2 is still present in appreciable quantity, is given. REAGENTS
Reagent grades of pyridine, acetic anhydride, 1-butanol, and >butanol were used. Titration indicator-a 2 to 1 mixture of 0.1% Nile Blue A (sulfate) in 50% ethyl alcohol and 1% phenolphthalein in 95% ethyl alcohol. PROCEDURE
-4 sample containing 0.20 mole of hydroxyl is transferred to a 250-ml. volumetric flask, using pyridine. More pyridine is added until the total volume is approximately 190 ml. Then 50 ml. containing 0.20 mole of acetic anhydride in pyridine is pipetted into the flask (the acetic anhydride reagent was made immediately prior to the analysis). The solution is rapidly diluted to volume with pyridine, the time is noted, and the solution is mixed by inversion. The flask is placed in a thermostated water bath a t 25.5" C. (as are all the reagents prior to analysis). At specified intervals of time (approximately 20 minutes), 5-ml. aliquots are withdrawn and pipetted into 'stoppered flasks containing 10 ml. of water. After a 10- to 20-minute wait, the samples are titrated with 0.17-0.2N alcoholic potassium hydroxide. A blank containing acetic anhydride and pyridine is treated in the same manner. PRINCIPLE
The rate expression for 1-butanol and 2-butanol reacting simultaneously with aeetic anhydride is, neglecting the back reaction,
- -d[Aczol= dt
kl[AczO] [l-BuOH]
+
Figure 1. Typical reaction rate curve for a binary mixture (fast reacting component method) Upper dotted line at right,a = 0.98 b Lower dotted line a t right, b = 0.98 a
VOL. 34, NO. 7,
J U N E 1962
801
= amount of al consumed; x2 = amount of a2 consumed; x = x1 22 = amount of ilc10 consumed; kl = rate constant of 1-BuOH; and k z = rate constant of 2-BuOH. Substituting
+
ax
=
kl(a - 5 ) (al -
(a2
-
22)
= (a, [kl(Ul
2,)
+ k:(a -
+ a2 -
51
-
- a ) + kn(a2
2)
X
52)
x
-
2211
When all of the al present has reacted, al = x l , and the expression then is : ( o - d t I 0-', meq -min I
which, upon integration, yields a? -
52
= k2t
Figure 2. Reaction rate curve for a mixture of 1butanol and 2-butanol (fast reacting component method)
+c
Since c = l / a z a t t = 0, the equation becomes 1
1
a2 - 5 2
a2
- adaz*' - 5 2 )
=
kpt
Extrapolation Method. I n a mixture of two reactants, the expression given by Equation 2 will hold only after all of al has reacted. Since az - a = al and al = X I , then a2 = a - X I , and x - al = xz. Equation 2 may now be written: (3)
which on rearranging becomes z = kzaz(a
- x)t
1 1.5% fast reacting component
X 21.3% fast reacting component 0 41.3%fart reacting component
(2)
DETERMINATION OF FASTER REACTING COMPONENT
5 - a1 kpt = ____ a4a - x)
0
+ a1
Thus, if x is plotted us. (a - x)t, after all of al has reacted, a straight line will result which has a slope of k2azand an intercept of al as shown in Figure 1. It is then a simple matter to extrapolate the line t o t = 0 and read the intercept a1 directly. As Equation 3 has been derived using the condition, a = b, it is necessary to know the magnitude of the error
introduced when a # b, as well as its effect on the shape of the x us. ( a - s)t plot. Figure 1 demonstrates this effect for b < a (lower dotted line a t right) and b > a (upper dotted line a t right). Example calculation showed that when a was 2y0 less than b, the resulting intercept was approximately 2% low and when a was 2% larger than b, the intercept was approximately 2% high. The latter case would occur when water is present as an impurity. This probably accounts for the positive errors obtained. I t should be noted that error lines (dotted) in Figure 1 deviated from linearity a t very large values of t (times greater than those necessary for practical analysis). From Equation 3, it is readily seen that x - a1 (a
- z)t
=
azkz
for all mixtures of two components. When the reaction is complete, s = al U Z .
+
Then
Table 1. Analysis of 1 -Butanol in Presence of 2-Butanol (Fast reacting component method) Found, yo Sinele Double ExtraDpoint olatidn Taken, po&t method, method method Yo 11.5
21.3
41.3
802
0
12.4 11.9 11.9 11.7 22.5 22.3 22.3 21.8 41.7 41.5 41.3 41.3
11.9 12.3 11.5 11.7 22.0 21.8 21.6 21.5 41.9 41.3 41.4 41.4
ANALYTICAL CHEMISTRY
12.0
22.0
(a - z)t
=
1 kz
The experimental curves for three different mixtures of 1-butanol and 2butanol are given in Figure 2. For this series of mixtures the resulting straight lines can then be extrapolated in both directions. The intercept will give the amount of al, and the results are shown in Table I. In the other direction, the point of intersection for the various straight lines will occur a t l / k s on the (a - x)t axis and a1 a2on the x axis. The three lines shown in Figure 2 intersected a t approximately the same point. Two corresponded to at Q = 199 meq. and the other to 197 meq. In all three cases, the actual
concentration was 200 mey. (per 250 ml.). Single Point Method. Substituting the equality a2 = a - al into Equation 3 and solving for al yields [x - kz(a - z)tal a1 = (4) [ l - k2(a - z ) t ] Equation 4 can serve as a means of performing the analysis by substituting in the value for x obtained a t a single time during the reaction. This time must be chosen after the first species has reacted but prior to complete reaction of the second species. To use this method, the rate constant must be known, and the technique is in this sense temperature dependent. Table I1 lists the rate constants for the slower reacting species obtained from the slopes plotted in Figure 2 for three different concentrations of a2 and from the intersection of the three lines a t a1 az where (a - s)t equals l/k2. Also included are the rate constants for both the faster and slower components determined by normal kinetic methods using pure solutions. The analytical results obtained by this method are listed in Table I. The four results for each concentration are points taken a t different times during a single run on the same sample. Double Point Method. If the values x and x' are taken a t two different times, t and t ' , both times occurring after the fast reacting component is exhausted but prior to complete reaction of the second component, then two equations result.
+
+
41.7
+
These may be solved simultaneously to
yield the amount of the faster reacting component, al.
I
~
Table II.
Reaction Rate Constants
(25.5"C.) Rate Constant
Hence, no temperature dependence occurs in this method in the sense that no rate constants need be determined. The reaction must proceed, of course, a t some fixed temperature. The results listed for each concentration in Table I were determined from values obtained a t different times during the reaction of a single mixture. DETERMINATION OF SLOWER REACTING COMPONENT
Equation 2 can be treated in a manner similar to t h a t employed for the faster reacting component. Equation 1 may be written: Extrapolation
Method.
= kd
a - x
+ aZ 1
This equation, like the others, will hold only after all of al has reacted. In this case if l / ( a - z ) is plotted us. t , a straight line having a slope of k2 and an intercept of l / a z will result, after all the al present has reacted (see Figure 3). The intercept of the line extrapolated to t = 0 is used to find the concentration of a2. Experimental plots are shown in Figure 4,and the results obtained are listed in Table 111. Single-Point Method. Solving Equation 6 for a2yields
for 1-
Butanol (kl),Liters/ tion Mole-'/Min. -1 By normal 5.68 X kinetic method From slope 4.86 x 10-3 Method of Determina-
,
c,
J
.$!V
Figure 3. Typical reaction rate curve for a binary mixture (slow reacting component method)
The results obtained experimentally are listed in Table 111. Double Point Method. As before, two values, x and x', taken a t t and t' can be used in Equation 6 to yield two equations which can be solved simultaneously to yield the quantity of slower reacting component az =
- z ) ( a - x')(t' - t ) ( a - z')t' - ( a - z)t
(a
(8)
The conditions noted for the faster reacting component also apply in this case. The experimental results are listed in Table 111.
(10% 1-
butanol) From slope butanol) From slope (40% 1butanol) From intersection
yo
The means for choosing an appropriate time interval were presented by Lee and Kolthoff ( I ) , who derived the differential form of Equation 1 in the case where a equals 6:
-
- gya-2
"
p(p - 1)ya-3 2!
+
. . . .]dY
4.95 x lo-'
(Slow reacting component method)
58.8
PY
5.16 X
Table 111. Analysis of 2-Butanol in Presence of 1 -Butanol
Taken,
J;"-LYa-1
4.89 x 10-3
(20% 1-
CRITERION FOR TIME INTERVAL
-klt = c
Rate Constant for 2-Butanol ( k 2 ) , Liters/ Mole-'/ Min. - 1 4.58 X
78.7
88.5
Found, yo Double Extrappoint olation method method 55.0 58.8 58.2 55.1 60.0 80.0 76.9 77.2 76.8 77.6 86.6 86.5 88.1 87.9 88.3 86.6 88.7 87.3
Single point method 58.7 58.5 58.7 58.8 79.7 77.8 78.0 77.8 88.0
(9)
in which Equation 7 can be used for an analysis by substituting in a value for z a t a time t after al has completely reacted. The same conditions hold here as in the previous case of the single point method for the faster reacting component.
-23
L 0
0
40
80
I20
160
200
t (mi n)
Figure 4. Reaction rate curve for a mixture of 1butanol and 2-butanol (slow reacting component method)
0
78.7% slow reacting component 58.8% slow reacting component
200
300
t (min)
Figure 5. Time interval curves-the interval, f, required to attain XI = of k l k 2 and a2/41 A. B.
88.5~osiowreacting component
X
IO0
C.
400
500
60'
variation of time a1 as function
99%
az/m = 9 , k l = 1 .O liter/min.-'/mole-', and k z varying from 0.33 to 0.033 liter/min.-'/mole-'. a2 = 0.09 mole/liter, a1 = 0.01 mole/liter oz/oi = 9, k i = 0.3 liter/min.-'/mole-', and k2 varying from 0.1 to 0.01 liter/min.-'/mole-'. 0 2 = 0.09 mole/liter, a1 = 0.01 mole/liter az/m = 1, kl = 0.3 liter/min.-'/mole-', and kz varying from 0.1 to 0.01 liter/min.-'/mole-'. a1 = 0.05 mole/liter = a2
VOL 34, NO. 7, JUNE 1962
803
ble over the time period required to reduce al to 1% of its initial concentrs tion. This time can be calculated from y = 1+
al - 5,’
and the integration limits are Y and Yo
. The integral form df Equation 9, alsd’derived by Lee and Kolthoff ( I ) , was not used for the example calculations below because it is not applicable a t integral values of 8. From Equation 9, it is seen that in order to calculate a time denoting a given degree of completeness of reaction of al, the quantities kl, kz, all and a2 must be known. To demonstrate the effects of the magnitude of thess various quantities on the value of t a t which x1 is 99% of a1 (hence where the plot of the experimental data becomes linear), example calculations were performed, and the results are shown in Figure 5 . The value of Y resulting from the condition set on x1 is obtained from the equation ( I ) : This time approaches a minimum value a t high ratios of kl/kz. This limiting case occurs when kz is negligibly smalI compared to kl; this is equivalent to stating that the reaction of a? is negligi-
The minimum time is therefore dependent on the quantities a, al, and kl. The influence of these quantities is demonstrated by the three curves in Figure 5 . Curves A and B both reptesent the case in which ul equals 0.01M and a2equals 0.09M. The rate constant in curve A , however, is three times that in curve B. Curves B and C have the same rate constant but both al and az equal 0.05M in curve C. Thus, Figure 5 shows the manner in which the time necessary for 21 to reach 0.99 al varies as a function of variables kl, kp, al,and a2. For smaller ratios of k~/kz, the time necessary for analysis becomes excessively long and as kl/kz approaches zero, the required time approaches infinity. The per cent of a2remaining is shown a t various points (in parentheses) on the calculated curves. This quantity is important because a sufficient concentration of a2 must remain a t time t so that an adequate length of the straightline portion of the plot may be secured. When a series of samples are to be determined, the single and double point methods are much more convenient and expedient than the graphical method. Nevertheless, the time required for ul
to be reduced to 1% of its initial value must be determined. This may be accomplished graphically or by calculation. The sample selected for this determination must contain the largest concentration of the faster reacting component of the series because this sample will require the longest time to reach this condition. Thus for a series containing 20-50010 of the fast reacting component, the 50% sample is chosen. The singIe and double point methods are quite comparable in their applications. The single point method is faster, but requires accurate knowledge of the rate constant kz, and this value of k2 must be maintained throughout the entire series of determinations. Hence, for example, the temperature of the water bath must be rigidly controlled. The double point method does not possess either of these limitations, because the rate constant kz need not be known. Therefore, the bath temperature can vary slightly from day to day and need only be constant during each single determination. LITERATURE CITED
(1) Lee, T. S., Kolthoff,I. M., Ann. N . Y . Acad. Sci. 53,1093 (1951). (2) Siggia, S., Hanna, J. G., ANAL.
CHEM. 33,896 (1961).
RECEIVEDfor review August 28, 1961. Accepted April 2, 1962. Research was supported in part by a grant from the Upjohn Co.
X-Ray Spectrometric Determination of Plate Metals on Plated Wires EUGENE
P.
BERTIN and RITA J. LONGOBUCCO
Electron Tube Division, Radio Corp. of America, Harrison,
b Three x-ray fluorescence spectrometric methods have been developed for the determination of plate metal on plated v;!res of small diameter: a ratemeter method, a net-intensity method, and an intensity ratio method. The first two methods are applicable when the wire i s received on spools that are small enough to fit in the x-ray spectrometer, all of the same diameter, machine wound with at least three layers of wire, and made of material that does not interfere with the analysis. The intensity ratio method i s applicable to wires submitted on spools which do not meet the requirements for the other methods. A specified number of turns of wire are wound on a small card or plastic spool. The net intensities o f a plate and of a base wire line are measured and their ratio i s calculated and applied to a calibration curve established 804
ANALYTICAL CHEMISTRY
N. J.
with cards or spools wound with standard wires. For any of the methods, a set of standards i s required for every combination of plate metal, base wire, base wire diameter, and card or type of spool. The ratemeter and net intensity methods consume no wire, the intensity ratio method about 10 meters. Analysis times per sample for the three methods are, respectively, 0.5, 5, and 10 minutes. The precision and accuracy of the net intensity and intensity ratio methods are comparable to those of the conventional chemical methods.
M
for determination of plate metal on plated wires require 0.1 t o 1 gram of sample (several hundred meters of wire) for a replicate determination, and analysis time of one or more hours. OST CHEMICAL METHODS
Because such methods are clearly unsuited to analysis of a sample from every spool, there is a need for an analytical method that is rapid and either nondestructive or applicable to relatively small samples of wire. X-ray methods held promise of meeting these requirements. However, a literature search for x-ray methods for determination of plating thickness (2) found only one paper dealing specifically with plated wires: Cameron and Rhodes (3) mention the determination of Ag plate on Cu wires by preferential excitation of CuKa by use of a radioisotope. The marked influence of preferred orientation, crystallite size, stress, and sample position on x-ray diffraction methods for plate thickness on plane samples is well known. These effects are often extremely pronounced in wires. Moreover, the intensities of
