Chemical Kinetics in the Earth’s Atmosphere

This individual assignment comprises three compulsory sections:
Section A: Chemical Kinetics in the Earth’s Atmosphere
Section B: Problems in Chemical Kinetics
Section C: Problems in Basic Reactor Engineering
You are required to complete the following:
• all questions in Section A
• one question from Section B
• one question from Section C.
You must submit your solutions to the problems, evidencing your calculations, by the deadline:
Tuesday, November 22, 2022 at 2 p.m. GMT
Submission is through a .pdf file upload to Canvas.
This assessment represents 60% of the total grade for this module.
If you will need to undertake a reassessment, you will be required to “make good” this
assignment.
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Section A: Chemical Kinetics in the Earth’s Atmosphere
Background
The Earth’s atmosphere can be considered as a giant photochemical reactor.
There has been an explosion of knowledge of Earth’s atmosphere in the last few decades, primarily due to
advances in technology (materials and remote sensing): investigations of the composition of atmospheres
(especially of the trace gases present) has been greatly facilitated by the use of space probes, satellites,
rockets, balloons, drones and other aircraft, to take instruments to remote regions of our atmosphere.
Alongside this, laboratory investigations of the mechanisms and rate of chemical transformations have
enabled quantitative interpretations of atmospheric chemistry.
Linking the atmospheric measurements and the laboratory work are computational (numerical) simulation
models of the physics and chemistry of the real atmosphere, which can be used diagnostically to understand
atmospheric observations, and prognostically to predict future changes of the atmosphere to a perturbation
that is chemical or physical, natural or anthropogenic (“human-made”).
Our current understanding is that Nature and humans have both affected the atmosphere and its chemistry.
Discoveries such as the “Antarctic Ozone Hole”, the heightened awareness of the importance of surface
phenomena in the Earth’s atmosphere, and the growing emphasis on the way in which the composition of
the Earth’s atmosphere is changing (climate change) allow atmospheric science and chemistry to be the
focus of much media attention, which sometimes hides the true reasons of, say, CO2 being a greenhouse gas.
Human activity, in particular, appears to be leading to an increase in the concentration of trace gases in the
atmosphere. The changes occur not on a geological timescale, but at rates of ca. 1% annually. This has led
to the naming of our current era as the “Anthropocene”. Important gases in this respect include methane
(CH4), carbon monoxide (CO), nitrous oxide (N2O), carbon dioxide (CO2), chlorofluorocarbons (CFCs), sulphur
dioxide (SO2), sulphur hexafluoride (SF6) and hydrogen (H2). Some of these species are as a result of
industrial production or release including fugitively (e.g. CFCs, SF6, CH4, H2), or the burning of fossil fuels
(CO2, SO2). However, changes in land use and agricultural practice are also important causes of the increase
in gases such as CH4 and N2O. Biomass burning is another very important contributor to atmosphere of a
variety of trace species.
The effects of a changing composition on atmospheric behaviour encompass a considerable range. The
oxidising capacity of the atmosphere may itself be changing, and, accordingly, the capacity of the lower
atmosphere to process materials released even by natural biological and geological phenomena. [For a
discussion on the effect of changing oxygen composition and the effect this has had on changing the size of
living organisms, such as large dinosaurs, see N. Lane, Oxygen – the molecule that changed the world, Oxford
University Press, Oxford, 2003.] Many of the compounds of interest are greenhouse gases, meaning they trap solar infra-red radiation
together with the Earth’s albedo, in the lower atmosphere, and so increase temperatures. CO2 is particularly
important here, together with water vapour. But other trace gases act synergistically with CO2 in trapping IR
radiation. The risk that the global climate alters significantly is very real, probably with adverse socioeconomic consequences: a change of 32 K of the Earth’s surface temperature (up or down) would make the
Earth large uninhabitable.
A further complication arises in that gases that trap radiation near the ground and lead to a warming, may
radiate at greater altitudes and produce a cooling. Reduced temperatures in the ozone layer may modify
the chemistry and the extent of ozone destruction.
So, when dealing with atmospheres and their chemistries, it is important to consider multiple feedbacks
between chemical compositions, temperature and the rates of chemical change, particularly along the chain
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of source to transformation to sink, with the chemical changes that occur in the atmosphere being driven by
sunlight (photochemical transformation).
The questions in this section of the assignment concern the kinetics of reactions in the Earth’s atmosphere,
and drawing appropriate conclusions, and presenting them. You will need to answer all of the questions.
You will have the opportunity to (1) develop rate laws and use your skills in kinetics to solve them, (2)
interpret your findings in a relevant, real-world context, and (3) develop your ICT and written presentation
skills through reporting your findings.
How good is the steady-state approximation for reactive intermediates in
the Earth’s atmosphere?
Chemical reactions in the Earth’s atmosphere typically involve a number of consecutive and parallel steps
involving reactive intermediates, which include atoms, radicals, ions and excited species (one where the
electronic configuration does not correspond to the ground state – that with lowest energy configuration).
Multi-step reaction schemes typically involve a number of coupled differential equations; solution of these
enables the prediction of the concentration-time variation of each species. Unfortunately, analytical
solution of many reaction mechanisms is not possible, and computational solutions tend to be used. These
solutions sometimes do not readily enable the insight into the underlying chemistry of the system, so often,
particularly for reactive intermediates, the use of the steady-state approximation (or stationary state
hypothesis) allows for a simplification that permits the algebraic solution to the kinetic equations.
But how good is this approximation for reactive intermediates in the Earth’s atmosphere?
Let us consider the Earth’s atmosphere at an altitude of 80 km above the Earth’s surface, where the
temperature is ~200 K.
Here the combined concentration, [M], of nitrogen gas (N2) and molecular oxygen (O2) is
3 x 1014 molecules cm-3
, with [M] = [N2] + [O2]; molecular oxygen comprises around 20% of this.
Molecular oxygen (O2) at these altitudes can be photochemically decomposed to excited oxygen atoms, O*
(often reported as O(
1
D) and ground-state oxygen atoms,O (often reported as O(
3
P):
�� + ℎ� → �∗ + � (A.1)
In this reaction hv represents a photon of light of sufficient energy to cause the decomposition of O2. The
rate of this reaction can be represented as Iabs.
The major loss processes for these atoms are:
�∗ + �
!!
� + � (A.2)
and
� + �� + �
!!
�� + � (A.3)
In reactions (A.2) and (A.3), M = N2 or O2, with kq being the second-order rate constant for reaction (A.2),
and kt being the third-order rate constant for equation (A.3). At the altitude and temperature considered,
these composite rate constants have the following values.
�! = 3×10!!! ��! ��������!! �!!
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�! = 1.4×10!!! ��! ��������!! �!!
In the following questions you will need to use the reaction mechanism given by the sequence of reactions
(A.1), (A.2) and (A.3), to assess the validity of the steady-state approximation for the reactive intermediates
O* and O.
Question A1: The excited oxygen atom
Write down a kinetic equation (net rate law) for the concentration change of excited oxygen atoms with
time, ! �∗
!” , where t represents time.
(2 marks)
Integrate your rate law, to afford an analytical expression showing the actual variation of the excited oxygen
atom concentration with time, given that no O* exists prior to t = 0.
HINT: you may find that making assumptions that Iabs and [M] are independent of time to be helpful!
(6 marks)
Use the steady-state approximation and your kinetic equation to determine the steady-state concentration
of excited oxygen atoms., and identify under what conditions the actual variation of the excited oxygen
atom concentration corresponds to the steady-state variation.
(2 marks)
Then, estimate the minimum illumination time required for the non-steady state and steady-state
concentrations to be identical to within 1%.
(4 marks)
Comment on whether the steady-state hypothesis is a good approximation for the atmospheric behaviour of
excited oxygen atoms where the solar intensity changes over periods of hours.
(2 marks)
Question A2: The ground state oxygen atom
Write down a kinetic equation (net rate law) for the concentration change of ground-state oxygen atoms
with time, ! �
!” , where t represents time, neglecting equation (A.2).
(2 marks)
Integrate your rate law, to afford an analytical expression showing the actual variation of the ground-state
oxygen atom concentration with time, under conditions of constant irradiation, and that all species M are
present in excess. Make sure you identify appropriate boundary conditions.
(4 marks)
Use the steady-state approximation and your kinetic equation to determine the steady-state concentration
of ground-state oxygen atoms.
(2 marks)
Then, identify whether the steady-state hypothesis is a suitable approximation to apply to ground-state
oxygen atoms in atmospheric modelling.
(4 marks)
Comment on any similarities or differences you have found in the application of the steady-state
approximation for the reactive intermediates (O and O*
) considered in Questions A1 and A2.
(2 marks)
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Section B: Numerical Problems in Chemical Kinetics
The questions in this section of the assignment concern the kinetics of chemical reactions. You will need to
answer only one of the three questions provided. You will have the opportunity to (1) develop rate laws and
use your skills in kinetics to solve them, and (2) develop your ICT and written presentation skills through
reporting your findings.
Question B1: Temperature-dependence of a first-order reaction
The thermal decomposition of gaseous dimethylether, (CH3)2O, to form methane (CH4), hydrogen (H2) and
carbon monoxide (CO), follows the single-step reaction given in equation (B.1):
��! !� � → ��! � + �! � + �� � (B.1)
In experiments undertaken in studying the kinetics of this reaction, for initial pressures, P0 = 41.6 kPa at a
temperature of 777 K, and P0 = 56.0 kPa at 825 K, the pressure increases, x, given in Tables B.1 and B.2 were
measured as a function of time (t).
t/s x/kPa
770 23.5
1195 33.3
3155 62.3
∞ 83.2
Table B.1: Experimental data corresponding to the dimethylether decomposition a temperature of 777 K and initial
pressure of 41.6 kPa.
t/s x/kPa
114 43.1
219 71.2
405 98.6
∞ 112.0
Table B.2: Experimental data corresponding to the decomposition of dimethylether a temperature of 825 K and
initial pressure of 56.0 kPa.
Provide an algebraic expression indicating how the pressure increase during the course of the thermal
decomposition of dimethylether is related to the fraction of dimethylether that has decomposed.
(4 marks)
Use the experimental data given in Tables B.1 and B.2 to demonstrate that the thermal decomposition of
dimethylether is first-order at both temperatures considered in the experiment.
(8 marks)
Then, estimate the rate constants for the first-order decomposition reaction at 777 K and at 825 K, and
calculate the activation energy of the reaction.
(3 marks)
Question B2: Kinetics of a reversible, first-order reaction
The acid-catalysed conversion of γ-hydroxybutyric acid (GHBA) into its lactone, γ-butyrolactone (GBL) is a
reversible reaction. The GHBA to GBL forward reaction is first-order with respect to the GHBA
concentration; the GBL to GHBA reverse reaction is first-order with respect to the GBL concentration.
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An experimental study of the kinetics of this reaction was undertaken in 0.2 mol L
-1 hydrochloric acid (HCl) at
298 K. The initial concentration of GHBA was 18.23 × 10-3 mol L
-1
. The concentration of GBL in solution was
followed as a function of time (t), as indicated in Table B.3.
Time/min 0 21 36 50 65 80 100 ∞
GBL
concentration
/10-3 mol L
-1
0 2.41 3.73 4.96 6.10 7.08 8.11 13.28
Table B.3: Experimental data corresponding to the GHBA to GBL conversion.
Use the data in Table B.3 to determine the equilibrium constant and the first-order rate constants for both
forward and reverse reactions.
(15 marks)
Question B3: Heterogeneous kinetics
Carbon monoxide (CO) is found to adsorb onto a specific crystallographic surface of a particular transition
metal single crystal. In an experiment, the number of CO molecules adsorbed per square metre of the
surface, N, was measured as a function of CO gas pressure, �!”, at 400 K, and gave rise to the results
reported in Table B.4.
���/10-4 Pa 0.44 1.7 4.0 9.3
N/1018 m-2 0.80 2.4 4.0 5.6
Table B.4: Experimental data corresponding to the adsorption of CO at 400 K.
Demonstrate that the data in Table B.4 conform to the Langmuir isotherm equation, and calculate the
values of both the Langmuir adsorption coefficient, K, and the maximum surface concentration of CO which
can be chemisorbed, N∞.
(8 marks)
Then, given that at the higher temperature of 420 K, a pressure of 5.6 x 10-3 Pa is required to achieve the
highest fractional coverage given in Table B.4, calculate the enthalpy of adsorption of CO at this particular
surface concentration.
HINT: you may assume that the enthalpy of adsorption of CO is independent of temperature!
(7 marks)
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Section C: Problems in Basic Reactor Engineering
The questions in this section of the assignment concern ideal chemical reactors. You will need answer only
one of the three questions provided. You will have the opportunity to (1) apply your knowledge to solve
them, and (2) develop your ICT and written presentation skills through reporting your findings.
Question C1: Batch reactor
The following gas phase, first-order, irreversible reaction is undertaken in a batch reactor, of volume V.
A à B + C (C.1)
At the temperature of interest, the rate constant for the reaction is 0.23 s
-1
.
How long should the reaction proceed in the reactor for the rate of conversion of A to be 90% with the initial
pressure being (i) 1.0 atm, and (ii) 50 atm?
(15 marks)
Question C2: Plug-flow reactor
The following irreversible, first-order, gas phase reaction occurs at a constant temperature T, and a constant
pressure P, in a plug-flow reactor, of volume 2.0 m3
.
A à 2B (C.2)
Under these conditions, the rate constant is 0.69 min-1
.
What volume flow rate of A is required for its conversion to be 99%?
(15 marks)
Question C3: Continuous stirred tank reactor (CSTR)
For the second-order, liquid phase, irreversible reaction,
2A à B (C.3)
a CSTR is used. Under operating conditions, the rate constant for the reaction is 3.0 mol-1 m3 s
-1
.
What is the volume of the reactor required to achieve an 80% conversion of the reactant in a single-pass, if
the flow rate of A is 0.2 L s
-1 and its concentration is 1.0 mol L
-1
?
(15 marks)
END
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