Chemical Box Models
Markus Rex
Alfred Wegener Institute
Potsdam
Germany
(1) Basic concepts, simplified systems (Saturday)
(2) The Ox, NOy/NOx, HOx, Cly/ClOx systems (Monday)
(3) Application for polar ozone loss studies (Thursday)
Chemical Models
Goal: Calculate the time evolution of the concentration of chemical species.
E.g. for the a species A this can be done ...
• ... at a fixed location: Eulerian formulation -> grid point model (e.g. 1d, 2d, 3d)
production
loss
advection
fixed grid points
wind

divergence

chemical changes
dynamical changes
• ... for an individual air mass: Lagrangian formulation -> box model
air mass with trajectory
wind
=> In our box model we focus on chemistry alone !
disappears if vmr is
used instead of conc.
subscale mixing and
molecular diffusion
are neglected
What governs chemical reactions in a system of species ?
To build a chemical box model we need to understand what governs chemical reactions in
the atmosphere and how to interpret published thermodynamical and kinetic data
Thermodynamics of chemical processes
Chemical thermodynamics describe the energetic balance of
chemical processes. This tells us which processes can principally
occur and whether they consume or or release energy.
Kinetics of chemical processes
Reaction kinetics describe at which rate chemical reactions
occur.
Reaction thermodynamics
The energetic balance of reactions
Goal: calculate the energy consumed or released by a chemical reaction
In general reactions can only occur if they have a positive energy balance
e.g. the photolysis of ozone
O2 + hn -> O + O
Reaction thermodynamics tell how much energy the photon need to supply for this
reaction.
Process at constant volume:
Things are simple !
energy supplied to the system (DQ) by the photons
is just the change of 'internal energy' (DU).
DU = DQ
Process at constant pressure:
(constant volume)
In the atmosphere reactions occur at constant
presure => Things are less simple !
If number of molecules is changed by the reaction,
the volume changes and produces mechanical work
(in case of positive volume change a negative
contribution to energy balance of the system):
DW = - p.DV
DU = DQ + DW
Enthalpy: H = U + pV
With H, things again become simple:
DH = DQ
(constant pressure)
Process at constant volume:
Relevant forms of internal energy for atmospheric
chemical processes
chemical
reactions
chemical
reactions
Collisions
chemical
reactions
Process at constant pressure:
permamnent exchange
by collision
Things are simple !
energy
energykinetic
supplied
to the system (DQ) by the photons
is just the change of 'internal energy' (DU).
DU = DQ
(constant volume)
energy of rotation
In the atmosphere reactions occur at constant
presure => Things are less simple !
energy of vibration
If number of molecules is changed by the reaction,
the volume changes and produces mechanical work
(in case of positive volume change a negative
contribution
to energy
of the system):
potential
energybalance
in the molecular
bonds
DW = - p.DV
DU = DQ + DW
potential
Enthalpy:
H = U energy
+ pV of the electron shell
With H, things again become simple:
DH = DQ
(constant pressure)
Process at constant volume:
Things are simple !
energy supplied to the system (DQ) by the photons
is just the change of 'internal energy' (DU).
DU = DQ
Process at constant pressure:
(constant volume)
In the atmosphere reactions occur at constant
presure => Things are less simple !
If number of molecules is changed by the reaction,
the volume changes and produces mechanical work
(in case of positive volume change a negative
contribution to energy balance of the system):
DW = - p.DV
DU = DQ + DW
Enthalpy: H = U + pV
With H, things again become simple:
DH = DQ
(constant pressure)
Standard enthalpy of formation (DHf0)
"enthalpy of formation" (DHf)
DHf for all relevant chemical species is listed in the literature.
E.g. JPL2003
DHA-B
AC
BC
products C
A
Enthalpy is a state function I.e. it does not matter how we get
from a state A to a state B, the net amount of enthalpy needed
(or released) by this transition is the same
DHB-C
=>
The enthalpy needed to produce a chemical species from the
elements is an individual fixed property of the species. We call it:
DHA-C
reactants A
DHA-C = DHA-B + DHB-C
• By definition DHf of the elements in their most stable form (N2, O2, H2, etc) is zero
• Obviously the the enthalpy of formation varies with pressure and temperature. But the
variation is small and in practice DHf for standard conditions (DHf0) can be used even
at stratospheric conditions.
Standard enthalpy
of formation (DHf0)
carries a huge amount of
enthalpie
=> enough to break
most bonds
examples
Can the following reactions occur ?
(1)
O + H2O ->
HO + HO
(59.6 - 57.8)
(9.3 + 9.3)
1.8
<
18.3
NO !
(2)
O(1D) + H2O ->
Lies in a deep enthalpie
valley. Difficult to get out
there
=> very passive molecule
HO + HO
(104.9 - 57.8)
(9.3 + 9.3)
47.1
>
18.3
YES !
JPL, 2002
Reaction enthalpy (Hr)
The enthalpy consumed (or released) by a chemical reaction (DHr) is the
difference between the enthalpy of formation for all products and all reactants:
DHr = SDHf(products) - SDHf(reactants)
In general the reaction can only occur if Hr is negative (enthalpy is released)
or the missing enthalpy is supplied by photons.
Example: photolysis of O3
(1) O3 + hv -> O + O2
DHr = DHf(O) + DHf(O2) - DHf(O3)
= 25.5 kcal/mol
= 107 kJ/mol
= 1.78 10-19 J/molecule
=> photon lmax = 1120 nm
(2) O3 + hv -> O(1D) + O2
=> photon lmax = 340 nm
O1D
production
Reaction kinetics
Goal:
Describe in a quantitative way how concentrations of species change
with time due to chemical reactions
1st order reactions
Reactions with only one reactant. E.g. photolysis:
The number of molecules of A lost in a volume per unit of time is proportional to the
concentration of A. Hence, changes of [A] with time are described by:
...... is a differential equation for [A](t).
Changes of concentrations in chemical systems are described by a
system of differential equations. A "chemical box model" is nothing else
than the numerical solver for this system.
1st order reactions ...more
This "one species" / "one reaction"
system has a simple analytical solution:
[A](t) = A0
.
e-kt
Reaction constant:
k
(for photolysis often called J-value or
"photolysis frequency")
Reaction rate:
exponential decay
R = k.[A]
(or "photolysis rate")
Lifetime:
t = 1/k
• is the e-folding time of the exp. decay
t
Line with constant initial slope
(constant loss rate)
• or, as more general definition:
t = [A]/R
=> after t, [A] would have been
consumed if R would remain constant
1st order reactions ...still more
The key is to determine J:
Dissociation
Quenching
Fluorescence
The rate of photolysis processes in our volume is determined by: For the Box model we get:
• the rate of absorption processes
 the actinic flux I(l) ..........................................................from radiation transfer calc.
 the absorption cross section s(l) ....................................from lab. studies
• the fraction of dissociation vs. other processes (quenching, ...)
 quantum yield F(l) ...........................................................from lab. studies
Actinic flux
In the UV:
strongly dependent on altitude
In near UV and vis:
weakly dependent on altitude
Variations of photolysis frequencies with altitude
Which of these species photolyses at longer wavelengths than the others ?
l < 300 nm
l < 410 nm
Sunrise as seen by different molecules
UV: near complete absorption
in the ozone layer
In the visible:
At 20 km altitude sun climbs above the
horizont at ~95 deg sza
=> abrupt sunrise at ~95 deg sza
In the UV:
at 90-95 deg sza the sun is still hidden
behind the ozone layer !
=> slow sunrise between ~90-85 deg sza
visible light: little attenuation
What does that mean for the chemistry ?
l < 410 nm
NO2 + hn -> NO + O
l < 300 nm
ClONO2 + hn -> ClO + NO2
Which photolysis occurs at
longer wavelengths ?
Wennberg et al., 1994
2nd order reactions
Reactions with two reactants:
2nd order reactions
Reactions with two reactants:
The rate of the reaction is determined by:
• collision frequency: proportional to [A].[B]
(...and proportional to sqrt(T), usually neglected)
• fraction of collisions that result in a reaction:
- steric factor (slightly negative temperature dependence)
- activation energy needed to form AB* (if high => strong positive T dependence)
Ea: Activation energy
(R: gas constant)
Ea and A are determined in the lab by plotting
ln(k) vs. 1/T "Arrhenius plot" (=> slope is Ea/R)
Sometimes this leads to negative Ea. For these reactions:
=> Ea is very small
=> T dependency dominated by steric factor
Reaction systems
Example: Three species, four reactions
The system O / O2 / O3 and the Chapman reactions
The evolution of the concentrations in the system is described by the
continuity equation (one for each species):
This is a set of coupled differential equations !
Here:
Simplified systems
In general numerical models are needed to solve the set of differential equations that describe a
system of interest. Here we look at two simplified cases first, that can be solved analytically:
(1) All production and loss terms are constant
, Pi and Li all constant
with
transient solution
transient solution
dissappears with
e-folding time 1/SLi
'steady state'
e
=[A]e steady state solution
Lifetime:
controls how long the system needs to reach steady state,
is dominated by the shortest individual lifetime in the system
Simplified systems, continued
(2) All production and loss terms are periodic (same frequency) or constant
e.g. diurnal cycle or seasonal cycle
solution has the general form:
stationary solution
("diurnal steady state")
W(t) is a periodic function
with the same frequency as
the forcing
transient solution
dissappears with 1/S
Example
Solution (a): Lifetime (1/L) << period of forcing
lifetime = 0.05 days
• diurnal steady
state rapidly
reached
Forcing
• One constant loss process L
• Production: Harmonic diurnal cycle,
i.e. period = 1 day
• strong diurnal
variation of
concentration
• no lag to forcing
Production term P
midday
midnight
P [mol day-1 m-3]
t [days]
Solution (b): Lifetime (1/L) >> period of forcing
lifetime = 20 days
• slow decay of
transient
solution
t [days]
• virtual no diurnal
variation of
concentration
t [days]
Example, continued
Solution (c): Lifetime (1/L) ~ period of forcing
lifetime = 2 days
• weaker diurnal
variation of
concentration
Forcing
• One constant loss process L
• Production: Harmonic diurnal cycle,
i.e. period = 1 day
• dirunal variation
lags the forcing
(we will come
back to that)
Production term P
midday
midnight
P [mol day-1 m-3]
t [days]
t [days]
Solution (d): Lifetime (1/L) ~ period of forcing
lifetime = 2 days, 10 times faster production
• Shape of curve
unchanged
• Absolute
concentrations
ten times larger
Still same example, the stationary solutions
short lifetime (0.05 days)
• strong diurnal variation
• maximum concentration ~midday
long lifetime (20 days)
• weak diurnal variation
• maximum concentration ~sunset !
intermediate lifetime (2 days)
midnight
midday
midnight
• some diurnal variation
• maximum concentration ~afternoon !
Atmospheric measurements of ClO
Dominating reactions:
(1) production: ClONO2 + hv -> ClO + NO2
(2) loss:
ClO + NO2 -> ClONO2
From these measurements alone, what do
we learn about the rate of (1) ?
=> not much !
What can we say about the rate of (2) ?
=> The lifetime of ClO with respect to (2)
is much shorter than one day
=> The reaction consumes much more
than 25 pptv ClO per day.
You should now be able to set up the system of
differential equations that describes the chemistry for a
given set of reactions and use lab data to calculate the
relevant kinetic parameters.
You should also know the fundamental behaviour of
such systems under simplified conditions.
Tomorrow:
... how to make a box model out of this
... real systems that actually exist in the atmosphere