Development of the Rate Law for a Simple System
4.5 PROBLEMS FOR CHAPTER 4
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4-1 The kinetics of the pyrolysis of mixtures of 2-butyne (A, C4H6) and vinylacetylene (B, Cab) have been investigated by Harper and Heicklen (1988). Pyrolysis is a factor in soot formation, which involves polymerization at one stage. Although the major product in this case was a polymer, o-xylene (C, CsHro) was also produced, and this was chosen as the species of interest.Reaction was carried out in a constant-volume BR, and analysis was by mass spectrometry.
4.5 Problems for Chapter 4 81 Initial rates of formation of C for various initial concentrations of A and B at 400°C are as follows:
I@ CAo 104 CBO 109 rcO mol L-l mol L-l mol L-l s-l
9.41 9.58 12.5
4.72 4.79 3.63
2.38 2.45 0.763
1.45 1.47 0.242
4.69 14.3 12.6
2.28 6.96 3.34
1.18 3.60 0.546
0.622 1.90 0.343
13.9 4.91 6.62
6.98 2.48 1.67
3.55 1.25 0.570
1.90 0.67 0.0796
(a) Test the hypothesis that the initial rate of formation of o-xylene is first-order with respect toeachofAandB.
(b) For a rate law of the form rc = kAcAcB ,a s determine values of kA, (Y, and /3 by nonlinear regression.
(c) From the following values of the rate constant, given by the authors, at five temperatures, determine the values of the Arrhenius parameters A and EA , and specify their units.
T/“C 350 375 400 425 450
lo3 k/L mol-i s-l 4.66 6.23 14.5 20.0 37.9
4-2 The rate of decomposition of dimethyl ether (CHsOCHs) in the gas phase has been determined by Hinshelwood and Askey (1927) by measuring the increase in pressure (AP) accompany- ing decomposition in a constant-volume batch reactor at a given temperature. The reaction is complicated somewhat by the appearance of formaldehyde as an intermediate product at the conditions studied, but we assume here that the reaction goes to completion according to CH30CH3 -+ CHq + Hz + CO, or A + M + H + C. In one experiment at 504°C in which the initial pressure (P, = PAo, pure ether being present initially) was 41.6 kPa, the following values of AP were obtained:
0 207 390 481 665 777
AP = (P - pAo)kPa 0 7.5 12.8 15.5 20.8 23.5
916 1195 1587 2240 2660 3155 to
AP = (P - PAo)lkPa 26.7 33.3 41.6 53.6 58.3 62.3 82.5 Test the hypothesis that the reaction is first-order with respect to ether.
4-3 The hydrolysis of methyl bromide (CHsBr) in dilute aqueous solution may be followed by titrating samples with AgNOs. The volumes of AgNOs solution (V) required for 10 cm3 sam- ples at 330 K in a particular experiment in a constant-volume batch reactor were as follows
(Millard 1953, p. 453):
t/ruin 0 88 300 412 reaction complete
V/cm3 0 5.9 17.3 22.1 49.5
(a) Write the equations for the reactions occurring during hydrolysis and analysis.
(b) If the reaction is first-order with respect to CHsBr(A), show that the rate constant may be calculated from k~ = (l/t) ln[VJ(V, - V)], where t is time, V, is the volume of AgN03 required for titration when the reaction is complete, and V is the volume required at any time during the course of the reaction.
(c) Calculate values of kA to show whether the reaction is first-order with respect to CHsBr.
4-4 Ethyl acetate reacts with sodium hydroxide in aqueous solution to produce sodium acetate and ethyl alcohol:
CHsCOOC2Hs(A) + NaOH + CHsCOONa + CzHsOH
This saponification reaction can be followed by withdrawing samples from a BR at various times, adding excess standard acid to “quench” the reaction by neutralizing the unreacted hydroxide, and titrating the excess acid with base. In a particular experiment at 16”C, samples of 100 cm3 were withdrawn at various times; the concentration of acid used (HCl) was 0.0416 mol L-l. The following results were obtained (V, is the volume of acid solution required to neutralize umeacted NaOH at time t) (Glasstone, 1946, p. 1058).
tls 0 224 377 629 816 00
V,/cm3 62.09 54.33 50.60 46.28 43.87 33.06 Using this information, obtain the rate law for the reaction.
4-5 The rate of decomposition of gaseous ethylene oxide (Cz&O), to C& and CO, has been stud- ied by Mueller and Walters (1951) by determination of the fraction (f~) of oxide (A) reacted after a definite time interval (t) in a constant-volume batch reactor. In a series of experiments, the initial pressure of the oxide (PAo) was varied. Some of the results are as follows:
PA&h 27.1 37.2 40.4 55.3 58.6
tls 2664 606 2664 2664 1206
fA 0.268 0.084 0.274 0.286 0.139
From these results, determine the order of reaction and the value of the rate constant (specify its units).
4-6 The rate of reaction between hydrocyanic acid (HCN) and acetaldehyde (CHsCHO) to give acetaldehyde cyanohydrin has been studied in a constant-volume batch reactor at 25°C in dilute aqueous solution, buffered to keep the pH constant (Svirbely and Roth, 1953). The reaction is
HCN + CH3CH0 -+ CH3CH(OH)CN
A typical set of results is given below, where the concentrations are in mol L-l
tlmin 3.28 11.12 24.43 40.35 67.22 00
CHCN X 10’ 6.57 6.19 5.69 5.15 4.63 2.73
CCH$HO x lo2 3.84 3.46 2.96 2.42 1.90 0.00
Determine the rate law for this reaction at 25”C, and calculate the rate constant, and the initial concentrations of HCN(CA,) and CHsCHO(ca,).
4-7 The rate of acetylation of benzyl chloride in dilute aqueous solution at 102°C has been studied by Huang and Dauerman (1969). The reaction is
4.5 Problems for Chapter 4 83 CHsCOONa + CsHsCH&I -+ CHsCOOC6HsCH2 + Na’ + Cl-
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v0yap:
or A + B + products
Some of the data they obtained for a solution equimolar in reactants (CA0 = 0.757 mol L-l) in a constant-volume batch reactor are as follows (fi; is the fraction of B unconverted at time t):
10-3tls 24.5 54.7 88.6 126.7 fr; 0.912 0.809 0.730 0.638
Determine the form of the rate law and the value of the rate constant at 102°C based on these data.
4-8 The rate of decomposition of nitrogen pentoxide (NzOs) in the inert solvent CC14 can be fol- lowed by measuring the volume of oxygen evolved at a given temperature and pressure, since the unreacted NzOs and the other products of decomposition remain in solution. Some results at 45°C from a BR are as follows (Eyring and Daniels, 1930):
tls 162 409 1721 3400 00
02 evolved/cm3 3.41 7.78 23.00 29.33 32.60
What is the order of the decomposition reaction (which for this purpose can be written as N20s + Nz04 + ~OZ)? Assume the reaction goes to completion.
4-9 Rate constants for the first-order decomposition of nitrogen pentoxide (N205) at various tem- peratures are as follows (Alberty and Silbey, 1992, p. 635):
T/K 273 298 308 318 328 338
lo5 k/s-’ 0.0787 3.46 13.5 49.8 150 487
Show that the data obey the Arrhenius relationship, and determine the values of the Arrhenius parameters.
4-10 Rate constants for the liquid-phase, second-order, aromatic substitution reaction of 2- chloroquinoxaline (2CQ) with aniline in ethanol (inert solvent) were determined at sev- eral temperatures by Pate1 (1992). The reaction rate was followed by means of a conductance cell (as a BR). Results are as follows:
TI”C 20 25 30 35 40
105k/dm3 mol-t s-t 2.7 4.0 5.8 8.6 13.0
Calculate the Arrhenius parameters A and EA for this reaction, and state the units of each.
4-11 Suppose the liquid-phase reaction A --z B + C was studied in a 3-L CSTR at steady-state, and the following results were obtained:
Assuming that the rate law is of the form (-rA) = kAct = A exp(-E,JRT)ci, determine A, EA, and n, and hence kc at 25°C and at 35°C. CAM in all three runs was 0.250 mol L-‘.
4-12 The oxidation of nitric oxide, NO(A) + :O, -+ NOz, is a third-order gas-phase reaction (second-order with respect to NO). Data of Ashmore et al. (1962) for values of the rate constant at various temperatures are as follows:
T/K 377 473 633 633 692 799
lo-3 kA/L’ mOl-2 S-l 9.91 7.07 5.83 5.73 5.93 5.71
(a) Calculate the corresponding values of & in kPaa2s-‘.
(b) Determine the values of the Arrhenius parameters based on the values of ka given above.
(c) Repeat (b) using the values calculated in (a) to obtain EAT and A,,.
(d) Compare the difference EA - E.+, as calculated in (b) and (c) with the expected result.
(e) Which is the better representation, (b) or (c), of the experimental data in this case?
(See also data of Bodenstein et al. (1918,1922), and of Greig and Hall (1967) for additional data for the range 273 to 622 K).
4-13 The chlorination of dichlorotetramethylbenzene (A) in acetic acid at 30°C has been studied by Baciocchi et al. (1965). The reaction may be represented by
A + B + products,
where B is chlorine. In one experiment in a batch reactor, the initial concentrations were CA0 = 0.0347 mol L-l, and caO = 0.0192 mol L-‘, and the fraction of chlorine reacted (fa) at various times was as follows:
tlmin 0 807 1418 2255 2855 3715 4290
fB 0 0.2133 0.3225 0.4426 0.5195 0.5955 0.6365
Investigate whether the rate law is of the form (-7~) = (-ra) = kcAcB, and state your con- clusion, including, if appropriate, the value of k and its units.
4-14 The reaction 2N0 + 2Hz + N2 + 2HzO was studied in a constant-volume BR with equimolar quantities of NO and HZ at various initial pressures:
P,lkPa 47.2 45.5 50.0 38.4 33.5 32.4 26.9
t112ls 81 102 95 140 180 176 224
Calculate the overall order of the reaction (Moore, 1972, p. 416).
4-15 The hydrolysis of ethylnitrobenzoate by hydroxyl ions
N02C6H4COOC2Hs + OH- + NO&J-LCOO~ + CzHsOH
proceeds as follows at 15°C when the initial concentrations of both reactants are 0.05 mol L-’
(constant-volume batch reactor):
tls 120 180 240 330 530 600
% hydrolyzed 32.95 41.75 48.8 58.05 69.0 70.4
Use (a) the differential method and (b) the integral method to determine the reaction order, and the value of the rate constant. Comment on the results obtained by the two methods.
4-16 The kinetics of the gas-phase reaction between nitrogen dioxide (A) and trichloroethene (B) have been investigated by Czarnowski (1992) over the range 303-362.2 K. The reaction ex- tent, with the reaction carried out in a constant-volume BR, was determined from measure- ments of infrared absorption intensities, which were converted into corresponding pressures by calibration. The products of the reaction are nitrosyl chloride, NOCl (C), and glyoxyloxyl chloride, HC(O)C(O)Cl.
In a series of seven experiments at 323.1 K, the initial pressures, PA0 and Pno, were varied, and the partial pressure of NOCl, PC, was measured after a certain length of time, t. Results are as follows:
t/mm 182.2 360.4 360.8 435.3 332.8 120.0 182.1
pAofl<Pa 3.97 5.55 3.99 2.13 3.97 2.49 2.08
pBoma 7.16 7.66 6.89 6.77 3.03 8.57 9.26
p&Pa 0.053 0.147 0.107 0.067 0.040 0.027 0.040
4.5 Problems for Chapter 4 85
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(a) Write the chemical equation representing the stoichiometry of the reaction.
(b) Can the course of the reaction be followed by measuring (total) pressure rather than by the method described above? Explain.
(c) Determine the form of the rate law and the value of the rate constant (in units of L, mol, s) at 323.1 K, with respect to NO*.
(d) From the following values of the rate constant, with respect to NO2 (units of kPa, min), given by Czarnowski, determine values of the Arrhenius parameters, and specify the units of each:
T/K 303.0 323.1 343.1 362.2
lo6 kp (units of kPa, min) 4.4 10.6 20.7 39.8
4-17 A La(Cr, Ni) 0, catalyst was tested for the cleanup of residual hydrocarbons in combustion streams by measuring the rate of methane oxidation in a differential laboratory flow reactor containing a sample of the catalyst. The following conversions were measured as a function of temperature with a fixed initial molar flow rate of methane. The inlet pressure was 1 bar and the methane mole fraction was 0.25. (Note that the conversions are small, so that the data approximately represent initial rates.) The rate law for methane oxidation is first-order with respect to methane concentration.
TPC 250 300 350 400 450
% conversion 0.11 0.26 0.58 1.13 2.3
(a) Explain why initial methane molar concentrations are not constant for the different runs.
(b) Calculate k (s-l) and kb (mol s-l L-’ bar-‘) for each temperature, given that the void volume in the bed was 0.5 cm3 and the methane molar flow rate into the reactor was 1 mm01 min- l.
(c) Show whether these data obey the Arrhenius rate expression for both k and kb data. What are the values of EA and Eip? (Indicate the units.)
(d) Explain why, if one of the Arrhenius plots of either k or kb is linear, the other deviates from linearity. Is this effect significant for these data? Explain.
(e) Calculate the pre-exponential factors A and A6,. Comment on the relative magnitudes of A and A; as temperature approaches infinity.
(f) How would you determine if factors involving the reaction products (CO2 and H20) should be included in the rate expression?
4-18 The Ontario dairy board posted the following times for keeping milk without spoilage.
T/Y 1 Safe storage time before spoilage
0 30 days
3 14 days
15 2 days
22 16 hours
30 3 hours
0
7O-vv(a) Does the spoilage of milk follow the Arrhenius relation? Assume spoilage represents a given “fractional conversion” of the milk. Construct an Arrhenius plot of the data.
(b) What value of activation energy (EA) characterizes this process? (State the units.) 4-19 The reactions of the ground-state oxygen atom O(3P) with symmetric aliphatic ethers in the gas
phase were investigated by Liu et al. (1990) using the flash photolysis resonance fluorescence technique. These reactions were found to be first-order with respect to each reactant. The rate constants for three ethers at several temperatures are as follows:
1014 k/cm3 molecule-’ s-l Ether
diethyl di-n-propyl di-n-butyl
2 4 0 K 2 9 8 K 3 3 0 K 3 5 0 K 4 0 0 K
17.0 38.1 55.8 66.1 98.6
25.8 58.2 75.3 90.0 130
36.0 68.9 89.7 114 153
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Determine the Arrhenius parameters A and EA for each diether and specify the units of each.4-20 Nowak and Skrzypek (1989) have measured the rates of decomposition separately of (1) NbHCOs (A) (to (N&)zCOs), and (2) (NH&C03 (B) in aqueous solution. They used an open, isothermal BR with continuous removal of gaseous products (CO2 in case (1) and NH3 in (2)) so that each reaction was irreversible. They measured CA in case (1) and cB in case (2) at predetermined times, and obtained the following results at 323 K for (1) and 353 K for (2).
lo-Q/s lOc,Jmol L-l locB/mol L-’
0 8.197 11.489
1.8 6.568 6.946
3.6 5.480 4.977
5.4 4.701 3.878
7.2 4.116 3.177
9.0 3.660 2.690
10.8 3.295 2.332
12.6 2.996 2.059
14.4 2.748 1.843
16.2 2.537 1.668
18.0 2.356 1.523
(a) Write the chemical equations for the two cases (H20 is also a product in each case).
(b) Determine the best form of the rate law in each case, including the numerical value of the rate constant.