Problem Set 2: Chemical Kinetics
1. The second-order rate constant for the reaction:
CH3COOC2H5(aq) + OH-(aq)  CH3COO-(aq) + C2H5OH(aq)
is 0.11 L mol-1 s-1. The reaction is first order in the concentrations of both ethyl
acetate and OH-(aq). What is the concentration of the ester (ethyl acetate,
CH3COOC2H5) after: (i) 10 seconds, (ii) 10 minutes, when ethyl acetate is added
to an aqueous solution of sodium hydroxide such that the initial concentrations
are: [NaOH] = 0.05 mol L-1, and [CH3COOC2H5] = 0.1 mol L-1 ?
2. The reaction mechanism:
A2  A + A, Fast
A + B  P, Slow
involves an intermediate, denoted A. (Each of the above steps represents an
elementary reaction). What is the overall rate law for this reaction (i.e., for the
formation of P from the reactants A2 and B) ?
3. The enzyme-catalyzed conversion of a substrate at 25o C has a Michaelis constant
(KM) of 0.035 mol L-1. The rate of the reaction is 1.15 x 10-3 mol L-1 s-1 when the
substrate concentration is 0.11 mol L-1. What is the maximum possible velocity
for this enzyme-catalyzed reaction, at the same temperature and with double the
concentration of enzyme ?
4. The gas phase reaction 2 NO2(g) + O3(g)  N2O5(g) + O2(g) is described by the rate
constant k = 2 x 104 L mol-1 s-1 at T = 300 K. What is the order of this reaction ?
5. The first-order reaction: 2A  2B + C , with only reactant A and none of the
products B and C present initially, is 35% complete after 325 seconds. Determine
(i) the rate constant, k, and (ii) the time it will take for the reaction to reach 90 %
completion (i.e., for 90 % of the quantity of A present initially to have undergone
reaction).
6. What is the activation energy of an elementary reaction for which the rate
constant increases by a factor of 6.5 when the temperature is raised from 300 K to
310 K ?
7. Consider the following suggested mechanisms for the gas-phase reaction:
2 NO(g) + O2(g) 2 NO2(g)
(a) 2 NO  N2O2 ,
Equilibrium
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N2O2 + O2  2 NO2 , Slow
(b) NO + O2  NO3 , Equilibrium
NO3 + NO  2 NO2 , Slow
(c) 2 NO + O2  2 NO2
Verify that each of these proposed mechanisms predicts the same overall reaction
rate, namely, rate = k [NO]2 [O2].
8. The radioactive decay of radium (half-life  1600 years) leads to formation of the
inert gas radon. Radon itself undergoes further decay with a half-life of
approximately 3.8 days.
Derive an expression for the number of radon atoms present as a function of time
(for time intervals of 1 year or less) if a 1 gram sample of radium is sealed inside
a vial that is impermeable to radon. Assume that the vial contains no radon
initially. (Atomic weight of radium: 226 gram mol-1). Why does the number of
radon atoms in the vial approach a constant, plateau-like, value for timescales
short compared to the half-life of radium ?
9. Consider the reaction mechanism:
k1
A M
k 1
k2
M A B
(a) Obtain expressions for d [A] / dt, d [M] / dt, and d [B] / dt.
(b) Apply the steady-state approximation to the intermediate species M to obtain an
expression for [M] in terms of the concentrations of the major reagents, [A] and
[B].
(c) Calculate the rate of formation of the product B, i.e., d [B] / dt, within the steadystate approximation as applied to the intermediate M.
(d) Show that the rate of formation of B can appear to be either first-order, or secondorder, with respect to the concentration of A, depending upon the relative values
of k1, k-1, k2, and [A].
10. For the overall, multi-step, gas-phase reaction: 2 A  B + C, suppose that the rate
law for the forward reaction is:
2
d A
  k A .
dt
Assuming that A, B, and C are all ideal gases, suggest two possible rate laws for
the reverse reaction that are consistent with equilibrium thermodynamics.
(In other words, suggest two possible ways to complete the following equation:
d A
  (Forward Rate)  (Reverse Rate) , where: Forward Rate  k A ).
dt
11. The radioactive isotope of carbon, C14
6 , is generated in the atmosphere by cosmic
ray bombardment. Atmospheric CO2 thus contains a trace proportion of this
isotope, and it becomes incorporated within the tissues of living plants. On
average, approximately one out of every 1012 carbon atoms present in the
atmosphere is the radioactive C14
6 isotope. (This ratio is referred to as the “natural
abundance” of the isotope). Upon the death of a tree, the radioactivity of the wood
gradually decays, as the C14
6 nuclide has a half-life of 5730 years. The nuclide
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C14
6 undergoes first order decay to nitrogen ( N 7 ) by emission of a beta particle.
(i)
(ii)
(iii)
Consider a sample of 1 gram of oxalic acid (C2O4H2), extracted from a
present-day plant. How many radioactive disintegrations of C14
6 atoms
would you expect to see if this sample is observed over a period of one
day ? (Assume that one out of every 1012 carbon atoms in the sample is
C14
6 ).
For an archeological sample of wood, it is found that one out of every
2.08 x 1012 carbon atoms is the C14
6 isotope. Estimate the age of the
archeological sample, assuming that the natural abundance of the
14
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C14
carbon
6 isotope has remained steady at one atom of C 6 per 10
atoms.
The relative atmospheric abundance of the radioactive isotope C14
6
(relative to the stable isotope C12
6 ) is known to have varied
historically, owing to fluctuations in cosmic ray intensity, volcanic
eruptions, release of emissions from the combustion of fossil fuels,
atmospheric testing of nuclear devices, variations in the strength of the
magnetosphere of the earth, etc etc etc. Assume that the variability in
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the ratio of C14
6 / C 6 has not exceeded 5 % of the present-day value for
this quantity (one atom of carbon-14 per 1012 atoms of carbon).
Estimate the potential error in the calculated age of the sample from
Part (ii) due to the uncertainty in the ambient level of C14
6 .
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(Dendrochronology, i.e., the use of tree ring samples, provides one
method for the calibration of radiocarbon dating methods).
12. Consider the following mechanism for the thermal decomposition of the
hydrocarbon R2:
(i)
(ii)
(iii)
(iv)
R2  R + R
R + R2  PB + R/
R/  PA + R
R + R  PA + PB
The symbols R2, PA, and PB represent stable hydrocarbons, while R and R/ denote
reactive free radicals. Use the steady-state approximation to estimate the
dependence of the rate of decomposition of R2 upon the concentration of R2.
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