Thermodynamics and Kinetics of Chemical Equilibrium in Solution Multipurpose Experiment in the Physical Chemistry Course I. A. Leenson Moscow State University, Moscow, USSR The relative availability of 2-methyl-2-nitrosopropane ( t BuNO),' as well as the physical and chemical characteristics of its solutions in inert organic media2 make this compound convenient for physical chemistry laboratory experiments in the spectrophotometry, thermodynamics, and kinetics of equilihrium in solution. The following are examples of studies that can he made: 1) Lamhert-Beer's law and causes of deviation from it; determina-
tion of the true molar absorption coefficient of the substance in solution. 2) The study of thermodynamic characteristics of an equilibrium in solution;determination of the equilihrium constant, enthalpy, and entropy of the reaction. 3) The study of kinetic characteristics of a reaction approaching equilihrium; determination of rate constant, pre-exponential factor, and activation energy for forward and reverse reactions. It ispussiblefora pairofstudents tocomplete the first and sewnd itrms in nhout 2 h a n d the third in about 4 h. Theory Thermodynamics of the Equilibrium In Solution Solid t-BuNO has the structure of an azodioxy compound
that corresponds to a dimer formula (t-BuN0)n. The dissolution of the dimer in organic solvents results in equilihrium D e 2M hetween the dimer and monomer. Aliphatic Cnitroso compounds absorb at X = 300 nm, E ;.: lo4M-'cm-' (dimer) and a t X a 700 nm. r ;.: 1-20 M-'em-' ( m ~ n o m e r ) . ~ ~ o n s r q u e n ttlh~r monomc.; makes the solutiol~.bri~ht hlue. 'l'h(! dilution of the soldtion and or a rise in the temperature shift the equilibrium rightwnrd, while in concentrated solutions and undrr coolinr the eau~lihriumshifts leftward. The equilihrium is described by K = [MIZIIDl
material is in the dimer or monomer form: [Dl, = % [MI,. The degree of dimer dissociation a is then given hy the expression a = ([Dl, - [D])/[D], = [M]/[M],, whence [Dl = [DI,(I-aI
(3)
Substituting eqs 2 and 3 into eq 1gives K = 4aZ[D]J(1 - a )
= 2aZ[M]J(1 - a)
(4)
An absorbance of the monomer solution in its absorption maximum is determined in accordance with Lambert-Beer's law: where CM is t h e molar absorptivity of t h e monomer (M-'cm-') and 1 is the length of the optical path (cm). I t is impossible, however, to calculate CM by this equation since the equilihrium concentrations of the monomer in solution are unknown. It is convenient to introduce the "effective" absorption coefficient4, which neglects partial association of the monomer and is determined by equation
A M = %w[MIJ (6) This dependence is not linear, since c.ff depends on [MI,. By dividing termwise eq 5 by eq 6 we obtain [M]/[Id], = %ff/fM = a.By substituting the expression obtained for a into eq 4 we obtain
A plot of tff versus (~.ff)~[M], gives a straight line with an
(1)
where [MI and [Dl are concentrations of the monomer and dimer under equilihrium conditions. These are related by the equation [Dl, - [Dl = '1, [MI
(2)
where [Dl, and [MI, are the concentrations if all of the
'
Stowell. J. C. J. Org. Chem. 1971, 36. 3055; this substance is widely used as a free radical scavenger in spin-trapping experiments. Sergeev, G. B.; Leenson, I. A. Zhurn. Fiz. Khim. 1978, 53, 546. "The Chemistly of the Nitro and Nitroso Groups;" Feuer, H., Ed.; interscience: 1969: Pt 1. Chap 3. Keussler. V.; Liinke, W. Z. Elektrochem. 1959, 63.614.
A,nm. Figure 1. Absorption spectra of t-BuNO ([MI, = 4.4 X M) in heptane: 1) 40°C. 2) 59'C. (All temperature changes are completely reversible if measures are taken to prevent evapwation of the goivent.) Volume 63 Number 5 May 1986
437
intercept CM while the equilibrium constant K can be calculated from the slope of a plot using the determined value of EM.
To determine the enthalpy of the reaction, equilihrium constants a t several temperatures can he calculated by the above method and the value of AHcan he derived from van't Hoff's equation
However, in practice it is convenient to use another method. As can be seen from Figure 1, a temperature rise does not appreciably change the AMvalue (a slightly lowered intensity in the maximum is due to the broadening of the band with beating), whereas the intensity of the dimer absorption hand drops sharply. This means that, with the concentrations and temperatures given in Figure 1, the equilihrium is almost completelv shifted towards the monomer: indeed. a change in a f r o m 0.98 to 0.99 results in a change in [MI b; appro;imatelv 1%. while the value of ID1 chanees hv 100%. ~ h u its is more convenient ;se thehim& band to determine the enthalpy of the reaction a.I t is not necessary even to know the value of CD;indeed from equation AD = ~ ~ l D 1where 1, ADis the absorbance a t X = 294 nm.. hv .comhination with eqs 1 and 8, provided [MI = const, we obtain dlogApldT = AH/2.303RP or
tb
Thus the dependence of IogAn versus 1/T is linear. and from its slope the enthalpy of t h e dissociation reaction can he calculated. From the values obtained for AH and K it is possible to calculate the entropy change in this reaction according to the equations
Dissociation-DimerizationKinetics in Solution The high sensitivity of the dimer absorption hand to the dissociation degree makes i t useful for the study of kinetic processes. These are carried out by the relaxation method by following the system approach to equilibrium, with the initial dimer concentrations either higher or lower than the equilihrium ones. According to the first method a small amount of t-BuNO solution is poured into a thermostatted cell with solvent. As h,
a result of such dilution, equilihrium D s 2M shifts to the k,
right and AD decreases from a certain initial value (Ainit)to the equilibrium value (AD). The rate of the reaction is described by the equation
where kl[D] > kz[MI2. Since the experimental conditions assume that the equilibrium is almost completely shifted towards the monomer and its concentration remains virtually unchanged with time, the rate of the reverse reaction is constant (k2[MI2 k2[MIo2).By integrating eq 11under this condition, provided [Dl = [Dlsit a t t = 0, kl/k2 = K and [MIo2/K = [Dl (where [Dl is the dimer equilihrium concentration), we obtain In ([Dl - [Dl) = klt + I n ([Dl - [Dlinit). Substituting concentrations by absorbance8 gives where Ai,;, and AD > AD. I t follows from this equation that the reversible reaction is governed by first-order kinetics,
Benson. S. W. "Thermochemical Kinetics. Methods for the Estimation of Thermochemical Data and Rate Parameters"; Wiley: New York, 1968; Chaps 1.3. 438
Journal of Chemical Education
i.e., the plot of l o g ( A ~- AD) versus time is linear and from its slope it is easy to calculate kl. By carrying out this experiment a t several temperatures, we obtain from a n Arrhenius plot (logkl versus 1/T) kinetic parameters for the reaction of dimer dissociation, i.e., activation energy El and pre-euponential factor Al in
T h e pre-exponential factor for the monomolecular reaction of decomposition in two fragments can he written as5 A1 = (ekeT/h)exp(AS'/R), whence AS' = 2.303Rlog(A,hlekBT)
(14)
where k s is Boltzmann constant, h is Planck's constant, AS$ is activation entropy for the dissociation reaction. According to the second method a heated solution of tBuNO is quickly cooled. The equilihrium shifts to the left and AD increases with time. The reaction rate is also described by eq 11, with the exception that in this case kl[D] < k2[MI2.Finally we obtain a n equation similar to eq 12 where A D > Ai,it and AD. If the values of El and m a r e known, i t is easy to calculate activation energy for the reverse dimerization reaction
For complete thermodynamic and kinetic description of the system it is necessary t o determine the pre-exponential factor A2 in the equation k2 = A2exp(-EZIRT). I t follows from eqs 13 and 16 that kllk2 = (A1/Az)exp[-(El - EZ)/RT]. On the other hand kllkz = K = exp(AS/R)exp(-AHIRT), and, since El - EZ= AH, Al/A2 = exp(ASIR) or
where AS is calculated from the equilibrium data (eq 10). The Experiment Reagents
Dimer (t-BuN0)s was obtained by oxidation of t-BuNH2 with Hz02 in the presence of NazW01 catalyst.' The dimer is a colorless crystallinesolid that can bestored at O°C in tightly stoppered flasks for years. Solutions with known concentrations [MI,or [Dl, were prepared by weighing the solid dimer. Spectroscopically pure nheptane was used as a rather high-hailing solvent, which made it possible to record the spectra safely up to 70°C. To avoid solvent evaporation, Teflon-stoppered cells were used. All solutions were kept away from direct sunlight since the monomer is suhject to a slow photolysis. Absorption spectra were recorded with dual-beam instruments Unicam SP 800 and Cary 15 with thermostatted cell compartments. Constant temperatures were maintained with a standard thermostat connected with the cell holder. To ohtain temperatures helow ambient the water in the thermostat was ice cooled. Determination of a Molar Absorptivity of the Monomer and Equilibrium Constant at Room Temperature
Stock solution is obtained hy placing 0.355 g of crystalline dimer intoa 5-ml flask or apicnometer, which is then filled with heptane to the mark ([MI, = 0.816 M). This solution is diluted with pure heptanein six flasks with tight stoppers (see Table I),and the flasks are kept in thedarkuntilequilibrium is established (ahout40min at 20°C and significantly more quickly at higher temperatures); then spectra are registered in 0.1-cm cells. The reference cell is filled with pure heptane. The use of a thin cell makes it possible to expand greatly the range of concentrations to be measured. The AMvalues = 678 nm should he determined as accurately as possible. at A., The plot afAMversus [MIo(Fig. 2) deviates considerablyfrom the straight line due to increased monomer association at higher concentrations of the solution. The "effective" molar absorptivityof the monomer drops from 20.2 M-'em-' to 15.4 M-'cm-' (see Tahle 1). The results of calculations by the least-squares method are also given in Table 1, and the correspondingplot is presented in Figure 3.
The intercept gives r M = 22.5 f 0.3M-'cm-' and K = 2.4 f 0.2M-I (in CC4 solution K = 1.92 M-I at 2 0 T and 2.77 M-' at 26.5-C a s
Table 1.
determined by NMR technique'). With the value of K known, it is possible to calculate by eq 4 the concentration dependence of ol a t 20°C:
Data for Calculation of r. and Ktor CBuNO Solution at 2D0C*
Stock Flask number 1 2 3 4 5 6
solution (mi)
Heplane (ml)
0.2 1.0 0.4 0.8 0.6 0.6 0.8 0.4 1.0 0.2 nondlluted
ru(intercept)= 22.5 M-'. r = 0.993.
[MI.
(4
AM
fat
rev2
dlM10
0.136 0.272 0.408 0.544 0.680 0.816
0.275 0.52 0.73 0.92 1.08 1.26
20.2 19.1 17.9 16.9 15.9 15.4
408 365 320 286 253 237
55.5 99 130 156 172 193
+ 0.3 W' cm-', tga = -0.0365
i 0.0022. K = 2.4 i 0.2
Determination of Reaction Enthalpy A 31.8-mg portion of crystalline dimer is dissolved in 10 ml of heptane ([MI, = 3.65 X 10W M ) and kept a t 20°C in the dark for 40 min. A 0.5-em cell witha stopper is filled with this solution, and the = 294 nm is measured. Then the temperature is absorhance at A,, increased by 10% and the spectrum is recorded again (constant absorbance AD at 294 nm implies that equilibrium in the solution has been reached). The spectra at several temperatures are presented in Figure 4 and corresponding calculations are given in Table 2. In calculations according to eq 9 the AD values are corrected for the "tail" of the short-wave absorption of nitroso compound (dotted line in Fig. 4). From the values obtained for K (2.4 M-1 a t 293 K) and AH (50.5 kdlmol) it is easy to calculate by eq 10 A S = 179.3J molFIK-'. This value is typicalfor reactions of a molecule decomposition in two large fragments; for example, for reaction N104t 2 NO2 AS = 177 J
Table 2. t('C)
etgn
Dependence 01 A. at 294 nm on Temperature'
T(K)
10slKl
Figure 2. Dependence of monomer absorbance at 678 nm on concentration.
AoD
log AoB+ I
+
2.0 kJlmol, r = 0.998. absaption by ~ub*anion of 0.08 from ho sylaimental
= 2640 i 100. A n = 50.5
bcarected IN --wave abswbance at 294 nm.
Ao
Me solution
Figure 4. Absorption specba of the dimer ([MI. = 3.65 X 1) 20%. 2) 30%. 3139%. 4)4a°C, 5)65'C.
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m~l-'K-'.~ I t is noteworthy that the AHvalue obtained is also close to the dissociation enthalpy of nitrogen tetroxide (58 kJ/molS), but the energetic profiles of these reactions differ considerably: while the recombination of NO? molecules proceeds virtually without activation energy and consequently the dissociation enthalpy of NzOd is equal to activation energy, in the ease of t-BuNO a large barrier hamperingdimerization (see below) Leads to the fact that AH and El for dimer dissociation differ greatly.
Kinetics of Dissociation-DimeriationReaction "Cone~nrrolion-JumpWMerhod. A 81-mg portion of solid dimer is dissolved in 10 mlufheptsne ([MI, = 9.31 X 10VMl and kept in the dark for 40 mi". A l-cmstoppered cell is filled with 2.2 mlof heptane and placed in a cell holder maintained at a desired temperature. In about 10 min, when the cell with the solution aasumcs the tempera-
0
4
8
12
t, rnin
lureofthecell holder,theformer isquickly filled with0.35mlofthe prepared r.BuNO wdution, shaken vigorously once (or stirred witha Tellonagitator without removing thecell from therell holder) a n d a time-dependence trace of absorbance (0-2 scale) is run a t 294 nm. The experimental conditions are chosen so that the initial concentration of the dimer right after diluting corresponds t o a n ADvalue slightly exceeding 2.0, so the pen of the recorder remains off-scale for a while. However, a gradual shift of equilibrium toward the monomer caused by more than seven-fold dilution results in a continuous decrease of dimer concentration with time. It is convenient to switch scanning on when the absorbance is passing the 2.0 mark (Fig. 5). Owing to first-order kinetics, partial progress of the reaction before recording does not affect the calculations. Moreover, during the "dead time" period the solution temperature, somewhat altered by the addition of 0.35 ml of t-BuNO solution, regains the pre-set value. The recording of dissociation kinetics is continued with ADvalue is obtained (in all experiments Ahit = 2). From the plots of log(Ao - AD)versus time a t each temperature, we obtain by eq 12 the rate constants kl, and, from the plot of logk, versus 1/T, Arrhenius parameters in eq 13 are determined (see Fig. 6 and Table 3). The kinetic equation thus ohtained has the form kl = 1O1"Seexp(-90400lRT) = 10165-m10 where 8 = 2.303RT,R = 8.31 5 ~ O I - ~ K - 'The . value logA1 = 16.5 0.3 agrees well with a mean value for the monomolecular decomposition reactions in two bulky groupss: A = 1016+'s-I. The A S value obtained by eq 14 (63.2
-
Figure 5. Kinetic C U N S obtained by the "concernration-lump" melhod: 1)13.3OC,2)20.8'C. 3)25.0°C. 4)30.4*C. Equilibrium absorbances /T. at each temperature are shown at the right of t h e figure.
4
0
8
12
.
,
-
t, rnin Figure 7. Kinetic curves obtained by lhe "temperature-jump" method: 1) 15.6Y 2) 20.7'C. 3) 26.0°C. Equilibrium absorbances & T,a tech temperature are shown at the right of the figure.
;.
Table 3.
t('C)
Data lor Calculation of Klnetlc Parameters 01 the Dlmer Dlssaciatlon Reaction T(K)
l o g A? (IMerceptI =
Flgure 6. Plots of logkl (6-') versus 11T: 1)lhe "concentration-jump" memod. 2) the "temperaturejump" method.
440
Journal of
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0.999. 'log 0.999.
1O9Il7K-'1
k, (mi"-')
k, Is-')
log k,
+3
18.5 f 0.3, t g u = -4725 f 76. El = 90.4 + 1.5 Wlmal. r =
AT (IMacoPI = 16.8
* 0.4. $a = -4795 =k 123. El = 91.8 i 2.3 kJ1mol.
r=
J mol-'K-') means that the transition state has a considerably looser structure than does the initial "Tenperature-Jump" Method. Previously prepared solution ([MI. = 9.31 X 10WM) is poured into a thin cell (0.1 em). The cell is placed in the hot water (65-70°C) for about 40 s, which results in almost complete dissociation of the dimer. After that, the cell is transferred to cold water with temperature equal to that of the cell holder, kept there for 50 s, quickly wiped with a soft tissue moistened in ethanol and placed in the cell holder. Then, immediately, an increase of AD with time is recorded. Working properly, the "dead time" period can he reduced to about 1min. I t is seen from Figure 7 that during this period the reaction has proceeded to a certain depth, and again first-order kinetics ensures that this fact does not affect the calculated kinetic parameters. The corresponding data are presented in Table 3 and Figure 6.
As can he seen, both methods are in satisfactory agreement with respect to the final results. However, the "concentration-jump" method yields more accurate results, which were used in calculation of A, and El. In conclusion Let us calculate kinetic parameters for the reverse dimerization reaction. In accordance with eqs 16 and 17 Ez = El AH = 39.9 kJ/mol, and Ap = 1073M-1s-1. Thus, the Arrhenius equation for the dimerization reaction has the form kz = 107.3-s99W/8 (M-'s-'). The lower value of the pre-exponential factor in comparison with a "normal" value5 (1010-1011 M-'s-I) fol. the himolecular association reactions may be explained by steric hindrances in the course of this reaction: two molecules t-BuNO must be rigidly oriented to he transformed into reaction produet-trans-azodioxy compound.
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