Stoichiometric number for methanol synthesis

[chemengisfun]

I realised that the stoichiometric number of the syngas required for methanol synthesis is often defined as
(H2-CO2)/(CO+CO2), and preferably at 2. Please may I know how to arrive to this definition?

Thank you.

 

[25362]

I assume it is related to the stoichiometric molar balance in two parallel reactions of hydrogen with carbon oxides:

CO + 2 H[sub]2[/sub] => CH[sub]3[/sub]OH​

Stoichiometrically 2 moles of hydrogen react with 1 mole of CO to produce 1 mole of methanol.

The xs hydrogen from syngas (as obtained from methane reforming) is reacted with CO[sub]2[/sub]. In this reaction 3 moles of hydrogen react with 1 mole of carbon dioxide to produce 1 mole of methanol and 1 mole of water.

CO[sub]2[/sub] + 3 H[sub]2[/sub] => CH[sub]3[/sub]OH + H[sub]2[/sub]O​

Combining both reactions we see that 5 moles of hydrogen react with 1 mole of CO and 1 mole of CO[sub]2[/sub].

Therefore, moles of hydrogen less moles of CO, divided by the sum of the moles of both carbon oxides is 2. In practice this number is adjusted by addition of hydrogen or CO[sub]2[/sub] as needed.