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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 1 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. 2