It is easiest to approach the effect of pressure on reaction rates through the formalism of transition-state theory, whereby we can write
k = — e x p ( - — ) or In* = 111-5 ^ .
It follows from Eq. (6-47) that
(15-103) Strictly speaking, the effect of pressure on activity coefficients is being neglected, but this contribution should be small compared to the Δ F°* term if the system is dilute, so that departures from ideality are not large.
Partial molal volume changes for a reaction are generally small—of the order of 20 c m3 m o l e- 1, and pressures of thousands of atmospheres are needed to make an appreciable change in k. Experiments therefore require special high-pressure equipment. Some representative data are given in Fig. 15-18. It appears that Δ Vot values tend to lie between zero and Δ V°, the value for the overall reaction.
The results also tend to make qualitative sense in terms of structural considerations.
If a reaction involves bond breaking, it seems reasonable that J F ° * should be positive, corresponding to an expanded or loose activated complex structure relative to that of the reactants, as in the case of the decomposition of benzoyl peroxide (curve 1 in the figure). Conversely, J F ° * should be negative for an association reaction since the transition state should now involve incipient bond formation and hence a compaction. This case is illustrated by curve 2 in the figure.
An ionization reaction also tends to give a negative Δ Vot, presumably as a
conse-1.5 Γ
ΙΟ"3 P , a t m
FIG. 15-18. Variation of some rate constants with pressure. Curve 1: rate of decomposition of benzoyl peroxide at 70° C. curve 2: rate of dimerization of cyclopentadiene at 50° C (in monomer as solvent): curve 3: rate for C2H5I + C2H50 - C2H5O C2H5 + I " in ethanol at 25°C. (From
"The Foundations of Chemical Kinetics" by S. W. Benson. Copyright 1960, McGraw-Hill, New York. Used with permission of McGraw-Hill Book Company.)
EXERCISES 647
quence of the electrostatic compaction of solvent around ions; that is, one assumes the transition state to correspond to incipient ionization.
Example. W e read from curve 2 o f Fig. 15-18 that log(fcP/fc) at 50°C is a b o u t 1.0 at Ρ =
BENSON, S. W. (1960). "Foundations o f Chemical Kinetics." McGraw-Hill, N e w York.
BRESLOW, R. (1969). "Organic Reaction Mechanisms," 2 n d ed. Benjamin, N e w York.
CALDIN, E. F. (1964). "Fast Reactions in Solution." Wiley, N e w York.
C O R N I S H - B O W D E N , A . (1976). "Principles o f E n z y m e Kinetics." Butterworths, L o n d o n .
BENSON, S. W . (1960). "Foundations o f Chemical Kinetics." McGraw-Hill, N e w York.
B U R N S , R. C , A N D H A R D Y , R. W . (1975). " M o l e c u l a r B i o l o g y , Biochemistry a n d Physics,"
Vol. 21 ( A . Kleinzeller, G . F . Springer, and Η . G . W i t t m a n n , eds.). Springer-Verlag, Berlin and N e w Y o r k .
EIGEN, M . (1963). Angew. Chem. 7 5 , 498.
HAMMETT, L. P. (1940). "Physical Organic Chemistry." McGraw-Hill, N e w York.
M I C H A E L I S , L . , A N D M E N T E N , M . L . ( 1 9 1 3 ) . Biochem. Z. 4 9 , 3 3 3 . time observed in a temperature-jump experiment using a 2 χ 10~a Μ solution is 6.5 /xsec.
Calculate the rate constants for dissociation a n d association, kx and .
Ans. 2.98 x 1 0s s e c- 1, 9.93 x 1 08 liter m o l e "1 s e c "1. 15-3 Consider the reaction C H3B r -f I- occurring in aqueous solution (see Table 15-1). Assume
both species to be 4 A in diameter and that Eq. (10-42) applies. Taking 25°C and the solution to be 1 Μ in each species, calculate (a) the reaction rate, (b) the frequency factor A, (c) the encounter rate, (d) the encounter frequency factor Ae , (e) the collision frequency from collision theory, and (f) the value of Ke assuming that AHe° is zero. (both values for 25°C). Calculate the rate constant for the forward or anation reaction (a) in a low-ionic-strength medium and (b) in 0.1 Μ N a C 1 04. for the forward and the reverse reactions, and AS0 and AH0 for the overall reaction.
15-3 A mixed ethanol-water solvent is 0.0677 Μ in formic acid (plus some added HC1 as catalyst) and the esterification reaction is followed by periodic titration of 5 c m3 aliquots by 0.010 m base. The following data (for 25°C) are obtained:
Time (min) 0 50 100 160 290 oo A m o u n t of base (cm3) 43.52 40.40 37.75 35.10 31.09 24.29 Calculate the rate constants kx and k-x for the formation and decomposition of the ester and the equilibrium constant K. If the rate law is written to include ( H+) (as catalyst), what are the values of kx and k_x ?
15-4 Calculate kx and k2 for the curve labeled Κ = 2 of Fig. 15-1. A t a different temperature, kx is doubled and Κ increases by 20 %. Calculate the new half-time.
15-5 The first-order catalyzed decomposition of H202 in aqueous solution is followed by titration of the undecomposed H2Oz with K M n 04 solution. By plotting the proper func
tion, ascertain from the following data the value of the rate constant.
/ ( m i n ) 0 5 10 20 30 50
c m3K M n 04 46.1 37.1 29.8 19.6 12.3 5.0
per given amount of H2Oa solution
15-6 A solution at 25°C initially contains 0.063 Μ F e C l3 and 0.0315 Μ S n C l2. After the elapsed
PROBLEMS 649 times given, the concentration of the ferrous chloride produced is determined by a titration procedure:
/ ( m i n ) 1 3 7 17 4 0 F e2 +( M ) 0.0143 0.0259 0.0361 0.0450 0.0506 Determine the reaction order and the rate constant. (This o n e is hard!)
15-7 Calculate Ae at 25°C for the dimerization of cyclopentadiene, assuming that the molecular diameter is 4.5 A and that Eq. (10-42) applies. Compare Ae with the experimental A factor;
alternatively, calculate Ke . Assume the solvent t o be benzene, obtain necessary data from a handbook and from Table 15-2.
15-8 The kinetics of the reaction S2Og" + 2I~ 2SOJ" + I2 is being studied. A solution was made up which was 0.1 Μ in KI and 0.001 Μ in K2S208, and the concentration of iodine was measured at 3 min intervals with the following results:
/ ( m i n ) 0 3 6 9 12 (I2) (mole liter-1) 0 0.00010 0.00019 0.00027 0.00034 The rate of the reaction will be s o m e function of the concentrations of the species appearing in the overall equations, such as
dQJIdt =
« S208 2- )e( I - )b( S 02- )c( I / .(a) Derive from the data as much information as y o u can about the values of the ex
ponents a, b, c, and d. (b) The following mechanism is proposed for the reaction:
2°8~ +
S I_ = S°4
_I + S O! ~ ( r a p i d e £lu i l ib r i u m ) , I - + S O J - i I2 + S O2" (slow).S h o w what the values of a, b, c, and d are that would be predicted by this proposed mecha
nism. Discuss whether the rate data are compatible with the mechanism.
15-9 The following data are obtained o n the kinetics of the lactonization of hydroxyvaleric acid. The overall reaction is
C H3C H O H C H2C H2C O O H C H3- C H - C H2- C H2- C = Q + H20 .
25°C 50°C
(H+) = 0.025 Μ (H+) = 0.050 Μ (H+) = 0.025 Μ
t Concentration of / Concentration of / Concentration of (min) acid ( M ) (min) acid (M) (min) acid ( M )
0 0.080 0 0.100 0 0.080
2 0.0570 2 0.0510 1 0.0408
4 0.0408 4 0.0260 2 0.0210
6 0.0295 6 0.0133 3 0.0107
8 0.0210
The rate is to be expressed by the equation i/(hydroxyvaleric acid)/<# = fc(H+)a (hydroxy-valeric acid)b. Calculate the values of a and b and the value of the rate constant k at 25°C and 50°C, and the activation energy. Estimate the entropy of activation.
15-11 The reaction 2Fe2+ + 2 H g2 + = H g2 + + 2 F ea + has been studied by measurement of the optical density of the solution at various times. The initial solutions contained only ferrous and mercuric ions, and at the wavelength employed the optical density D increased owing to the increased absorption of the products. The following t w o runs are given: Run 1, initially ( F e2 +) = 0.1, (Hg2+) = 0.1; run 2, initially (Fe2+) = 0.1, ( H g2 +) = 0.001.
Run 1 R u n 2
t (sec) D (Hg2+)/0.1 / (sec) (Hg2+)/0.001
0 0.100 — 0 1.000
1 χ 105 0.400 — 0.5 x 105 0.585
2 χ 105 0.500 — 1.0 χ 105 0.348
3 χ 105 0.550 — 1.5 χ 105 0.205
0 0 0.700 — 2.0 χ 105 0.122
0 0 0
(a) Calculate the ratios ( H g2 +) / 0 . 1 for run 1, that is the fraction of H g2 + remaining at the various times.
(b) S h o w what the order of the reaction is in run 1 and in run 2.
(c) If the rate equation is written in the form: Rate = ^ ( F e2 +)p( H g2 +)< /, what are the values of ρ and ql
(d) Write a possible mechanism which would give this rate law.
15-12 The reaction C H3I + C2H50 - = C H3O C2H5 + I" is second order and k varies with temperature as follows:
/ ( ° C ) 0 12 18 24 k (liter mole"1 sec"1) 5.60 χ 10~5 24.4 χ 10"5 48 χ 10"5 100 χ 10"5 Calculate A, E*9 AS0*, and AH0*.
15-13 Find the rate law for the reaction 3 H N 02 = H20 + 2 N O + H+ + N Os" if the actual mechanism is the following, where the first t w o steps rapidly attain equilibrium and the third step is slow:
(1) 2 H N 02 = N O + N 02 + H20 (2) 2 N 02 = N204
(3) N204 + H20 H N 02 + H+ + N 03~
15-10 The aquation reaction C o ( N H3)5B r2+ + HaO = C o ( N H3)5( H20 )3 + + Br~ is followed spectrophotometrically at 25°C by monitoring of the optical density of a 0.001 Μ solution at 370 n m . Added B r- is present, and also other absorbing but inert species, so that the measured optical density does not directly give the concentration of the reactant, nor is the value at infinite time known. Obtain the aquation rate constant.
/ ( m i n ) 0 10 20 30 4 0 50 Z ) ( a t 3 7 0 n m ) 1.03 0.91 0.81 0.730 0.665 0.615 H i n t : Since equal t i m e intervals are involved, it turns out that ( A ) , / ( A )0 — (A)t + i / ( A )0 oc exp(/& At) where (A), is the concentration at the e n d o f the ith time interval and At is the time interval.
PROBLEMS 651
15-15 Cobaltic trioxalate undergoes a thermal decomposition into cobaltous dioxalate and carbon dioxide. The following mechanism is assumed;
0 ) ( Ο χ)Γ ^ Co(Ox)2 2"+ C204" ,
C204~ + C o ( O x )3 3" - ^ Co(Ox)2 2" + 2 C 02 + C2o\~.
Derive the corresponding rate expression if (a) the first reaction is assumed t o be a rapid reversible equilibrium, and the second reaction is rate determining, (b) the forward direction of the first reaction is rate determining, and (c) a stationary state is assumed. Derive likewise the three cases if instead of the second reaction, the assumed reaction is
2 C204- - 2 C 02 + C2O J - .
15-16 Find the constants b and <* of the Bronsted relation given the following data for the general base-catalyzed reaction of nitramide, H2N N 02 H20 + N20 :
Base Pyridine Acetate ion Formate ion Dichloracetate ion k (liter m o l e- 1 m i n- 1) 4.6 0.50 0.082 7 χ 1 0 -4
2.3 x 1 0 -9 5.5 χ 1 0 -1 0 4.8 χ ΙΟ"1 1 2.0 χ 1 0 -1 3
Estimate k in water ( ^w must be in liter m o l e- 1 for consistency).
15-17 The exchange reaction
* C e4 + + C e3 + = * C e8 + + C e4 +
is a simple o n e , w i t h AS0* = — 4 0 cal K "1 m o l e "1 a n d ΔΗ0* = 7 . 7 k c a l m o l e "1. Calculate k at (a) 25°C, at negligible ionic strength, (b) 25°C and / = 0.1, and'(c) 35°C and / = 0.1.
15-14 A proposed mechanism for the reaction
2Cr(VI) + 3As(III) = 2Cr(III) + 3As(V) is the following:
Cr(VI) + As(III) % Cr(IV) + As(V),
Cr(IV) + Cr(VI) % 2Cr(V),
Cr(V) + As(III) Ϊ Cr(III) + As(V).
T h e observed forward rate law is R = fcapp[Cr(Vi)][As(III)]. What are possible rate laws for the reverse reaction?
15-1 S h o w that t h e equations
15-4 A type of enzyme mechanism is the following:
Ε + S ^ ES, ES + S ^ E S2,
ES -i- products + E .
The first t w o equilibria are rapid; only ES reacts to give products, regenerating E. If (S) ^> (E) and the total concentration of S is varied at constant (E), find an expression for (S) when the rate of decomposition is at a maximum.
15-5 A series of solutions is made u p which have the same concentration o f the enzyme saccha-rase (E) but different initial concentrations of substrate saccharose (S). The enzyme Ε is present in some small (unknown) concentration and catalyzes the hydrolysis of the saccharose, and the rate is measured from the change in optical rotation of the solutions.
With increasing (S) the rate reaches a maximum Rao; it is half of this value when (S) is 1.1% by weight. A s s u m i n g the M i c h a e l i s - M e n t e n m e c h a n i s m , calculate as m u c h infor
m a t i o n as y o u c a n about kx, k- x, a n d k2.
15-6 Referring to the reaction scheme of Eq. (15-82), suppose that kx = 0.01 m i n- 1, k_x = 2 χ 1 0- 4 m i n- 1, k3 = 1 χ 1 0- 7 m i n- 1, L3 = 5 χ 1 0- 7 m i n- 1, and k2 and fc_2 are very small. Calculate the composition of a system consisting initially of A only after 10 min, 100 min, 1 03 min, 1 05 min, and 107 min. Also, find the ratio k2\k_2.
15-7 A sequential reaction, such as given by Eq. (15-84) is being studied. It is found that (B) reaches a maximum at 200 min and eventually disappears with a half time of 345 min, which is also the half-life with which ( A ) decreases with time. Calculate kx and k2 and plot ( A ) / ( A )0 and (B)/(A)0 as a function o f time u p to 1000 min.
15-8 Calculate kx, k2, and Κ from the plot o f Fig. 15-16.
15-9 The treatment for the sequence of Eq. (15-84) yields an indeterminate result if kx = k2. R e d o the derivation s o as to obtain (B) and (C) as functions of time.
15-10 Find the integrated expression for (C) as a function of time for the case A + Β # C . *1
*2
15-11 A study of the kinetics of the exchange of radiocyanide ion with the hexacyanomanganate (III) ion, M n ( C N )3 -, gave the following values for the slope c of the exchange plot.
SPECIAL T O P I C S P R O B L E M S
SPECIAL TOPICS PROBLEMS 653
Concentration (A/)
C N - Mn(CN)3 6" />H c ( m i n- 1)
0.0596 0.0199 10 1.31 χ ΙΟ"2
0.0571 0.0102 10 0.883 χ 1 0 -2
0.104 0.0199 10 0.943 χ ΙΟ"2
0.0596 0.0199 9 1.33 χ ΙΟ"2
Determine the form of the kinetic equation for the reaction leading to exchange and calculate the rate constant. On the basis of these results suggest a possible mechanism for the exchange.
15-12 The rate of exchange between I2* and I 03" is studied with 0.00050 ΜI2 and 0.00100 Μ H I 03. The radioactivity of the I 03~ at various subsequent times is given (corrected for radioactive decay). The total radioactivity of the system (or that initially present as I2) is 1650.
/ ( h r ) 19.1 47.3 92.8 169.2 oo
Radioactivity of I O r 107 246 438 610 819 Calculate c of Eq. (15-101) and the rate R of the exchange reaction.
15-13 T h e rate constant for the exchange reaction
C r ( HaO ) 2+ + H2 1 80 - > C r ( H20 )6( H2 1 80 )8 + + H20
is 5.00 x 1 0 "5 M_1 s e c- 1 at 2 5 ° C and 1 a t m pressure, and JF°* is —9.3 c m8 m o l e "1. Calculate the rate constant for 2 kbar pressure.