(b)Determine an integrated form for [B] over time. Do this
by finding an integrated expression for [A]t(Hint:refer to the
chapter!), substitute for [A]tin the second expression, re-
arrange the infinitesimals, and integrate one side over [B] as
the variable and the other side over time, t, as the variable.
The integrations are actually simpler than for the consecutive
first-order reaction example in the chapter itself.
(c)Determine an integrated form for [C] over time.
(d)Rough values of k 1 and k 2 for people are 3.00 10 ^3 s^1
and 4.44 10 ^5 mol/s, respectively. Use your expressions for
the amounts over time to plot a graph of [A]t, [B]t, and [C]t
versus time. (A graphing calculator or a computer with a
graphing program would be useful.) Use 1.00 mol of C 2 H 5 OH
as [A] 0. Vary this value and see how it affects the graphs of
[A]t, [B]t, and [C]t.
20.39.Find limiting forms of equations 20.47 for (a)k 1
k 2 , and (b)k 1 k 2.
20.6 Temperature Dependence
20.40.Express equation 20.48 as if it were referring to a
graph. What is being plotted, and what does the expression
(rxnH)/Rrepresent?
20.41.Nitrous oxide, N 2 O, can be decomposed thermally.
The following values for the rate constant kwere determined
(S. K. Ross et al., J. Phys. Chem. A, 1997, 101: 1104) at the fol-
lowing temperatures:
k[cm^3 /(moleculesecond)] T(K)
6.79 10 ^162056
8.38 10 ^162095
1.03 10 ^152132
1.39 10 ^152173Does this reaction follow the Arrhenius equation? What is the
estimated pre-exponential factor?
20.42.Derive equation 20.54 from equation 20.53. Beware
of the 1/Tpart of the denominator in the derivative.
20.43.One rule of thumb has been that a chemical reaction
doubles in speed for every 10° increase in absolute tempera-
ture from room temperature (nominally 295 K). If this is strictly
true, what is the activation energy of the chemical reaction?
20.44.At room temperature (22°C), the rate constant for
proton transfer between water molecules in solution is about
1 1011 s^1 , which is one of the fastest known reactions. If
the activation energy for this process equals one-half of the
O–H bond strength of 498 kJ/mol, what is the pre-exponen-
tial factor Afor this reaction?
20.45.Recently, researchers studying the kinetics of metal
atom reactions with small gas molecules measured a rate con-
stant for the gas-phase reaction of
Co NO →productsas 9.9 10 ^12 cm^3 /s at 1153 K. If the activation energy of
this reaction is 1.9 kJ/mol, what is the value of the pre-expo-
nential factor?
20.46.A reaction has k1.77 10 ^6 1/(Ms) at 25.0°C and
an activation energy of 20.0 kJ/mol. (a)What order is the re-
action? (b)What is the value of the rate constant at 100°C?
20.47.A reaction has k1.77 10 ^6 1/(Ms) at 25.0°C and
an activation energy of 20.0 kJ/mol. What is the value of the
rate constant at 100° C? Compare this answer to the answer
in part b of the previous exercise and comment on the effect
of the activation energy on rate law constants.
20.48.Scientists use very focused “molecular beams” of gas-
phase atoms and molecules in high-vacuum systems to study
one-step chemical processes. For the gas-phase molecular
beam reaction between sodium atoms and methyl chloride to
make sodium chloride and the methyl radical,
Na CH 3 Cl →NaCl CH 3
the probability of the reaction depends on the orientation of
the methyl chloride molecule with respect to the sodium atom.
Suggest a possible interaction geometry that would account
for this result. Why would the probability for reaction notde-
pend on the orientation of the other reactant?
20.49.Chemical processes that are triggered by photons are
also understood using the kinetics concepts mentioned in this
chapter. The chemistry of vision is one example. The currently
accepted mechanism for vision involves a compound called
rhodopsin, which is composed of a protein molecule (opsin)
attached to a colored polyene molecule called cis-retinal. (Cis-
retinal is chemically related to a class of molecules called
carotenes, which are highly colored compounds responsible
for the colors of carrots and tomatoes. Eating carrots does help
vision, specifically in low light.) The vision process begins when
the cis-retinal absorbs a photon and is isomerized about one
of its double bonds to make trans-retinal:
h
cis-retinal →trans-retinal
The reaction is energetically favorable; the photon’s energy is
apparently necessary only to overcome an activation energy
barrier.
(a)The least energetic photon that is considered visible light
has a wavelength of 750 nm. What is the activation energy for
the isomerization reaction?
(b)If the rate constant at 37°C is 3 1011 s^1 , what is the
pre-exponential factor Afor this reaction?
(c)The Antarctic icefish lives in Southern Hemisphere waters
that can reach 2°C. Assuming that this chemical process is
the same in an icefish eye, what is kat this temperature?
20.50.Nitric oxide, NO, is known to break down ozone, O 3 ,
by the following bimolecular reaction:
NO (g) O 3 (g) →NO 2 (g) O 2 (g)
If the activation energy of this reaction is 10.5 kJ/mol and the
pre-exponential factor is 7.9 1011 cm^3 /(mols), (a)what is
the rate constant of this reaction at 298 K? (b)If the ozone
concentration were 5.4 10 ^12 mol/cm^3 and the NO con-
centration were 2.0 10 ^12 mol/cm^3 (conditions that might
exist in a high-pollution area), what is the rate of this reaction?728 Exercises for Chapter 20