Download Chemical Synthesis (sat6)

Chemical Synthesis (sat6)
Problem
: The following chemical reactions are given:
MgO + H2 --> Mg + H2O
C + O2 --> CO2
CO2 + H2O --> H2CO3
Show that from a blend of
(H2O p.e. means water)
MgO, H2, O2,
and
C, the molecule H2CO3 will be produced.
Modeling Steps
To be able to model this with Boolean logic, some suppositions and simplications are
needed:
1. A Boolean variable for each molecule is introduced with the meaning that the
molecule takes part or does not take part of a chemical reaction.
2. The molecules which are the inputs in a reaction are linked by a
∧-connector meaning
that they must all be present in a chemical reaction at the same time.
3. Also the molecules which are output of a chemical reaction are connected by a
∧
operator meaning that they are all produced at the same time.
4. Input and outputs of a reactions are linked by implication. This expresses the fact
that if the inputs are present then the reaction takes place and the outputs must
result.
With these assumptions, the reaction is modeled as a simple Boolean (satisbility)
problem. It must be shown that:
E→F
with
E = (M gO ∧ H2 → M g ∧ H2 O) ∧ (C ∧ O2 → CO2 )∧
(CO2 ∧ H2 O → H2 CO3 ) ∧ M gO ∧ H2 ∧ O2 ∧ C
F = H2 CO3
The complete model code in LPL for this model is as follows (see [1]):
Listing 1: The Model
model Make_H2CO3 "Chemical Synthesis";
binary variable MgO; H2; O2; C; H2CO3; Mg; CO2; H2O;
constraint
A1: MgO and H2 -> Mg and H2O;
A2: C and O2 -> CO2;
A3: CO2 and H2O -> H2CO3;
A4: MgO and H2 and O2 and C;
minimize obj: H2CO3;
Write(’Yes, H2CO3 is produced’);
Write(’No, H2CO3 is not produced’);
end
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The constraints
A1
to
A3
form the reaction.
are present. In the implementation,
E
A4
means that these four molecules
is the set of constraints (which must hold). The
F . It is minimized, If the objective function F is zero (false)
E , that is E → F is false. On the other side, if F is one
then F is the consequence of E and E → F is true.
: In this model, H2 CO3 is produced from the reaction.
objective function represents
then
F
(true),
does not follow from
Solution
References
[1] T. Hürlimann. Reference Manual for the LPL Modelling Language, most recent version.
www.virtual-optima.com.
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