For reference, the “big numbers” are usually referred to as “stoichiometric coefficients” by chemists and so that is how I will refer to them in any subsequent posts.
Let’s run through some examples all based on the following reaction:
XeF6(s) —> Xe(g) + 3F2(g)
(I don’t know why this was the first reaction to come to mind - maybe I’ve been doing an unhealthy amount of main group chemistry lately LOL).
Let’s suppose in each example our desired product is fluorine, F2(g).
For atom economy, the big numbers do matter. Atom economy is the total Mr of all the desired product divided by the total Mr of reactants.
The total Mr of reactants in this case is just the Mr of XeF6(s) (Ar of Xe = 131.3, Ar of F = 19.0), which is 245.3.
The total Mr of desired product is 3 x the Mr of F2(g), or 3 x 38.0 = 114.0.
The atom economy is therefore 100% x 114.0/245.3 = 46.5% (3 sf)
For percentage yield, it is important to factor in via the molar ratio.
Supposing I made 24.0 dm^3 of F2(g) under RTP from 98.12 g of XeF6(s), I would first work out the moles of each:
n(XeF6) = (98.12 g)/(245.3 g mol^-1) = 0.4 mol
n(F2) = (24.0 dm^3)/(24.0 dm^3 mol^-1) = 1.0 mol
The theoretical yield of F2(g) should be 3 x the number of moles of XeF6(s) as per the equation and so this is how we factor in the stoichiometric coefficients.
Theoretical n(F2) = 3 x 0.4 mol = 1.2 mol
% yield = 100% x (1.0 mol)/(1.2 mol) = 83.3% (3 sf)
For any other stoichiometry calculations, the equation gives the molar ratios in which your reactants react and the molar ratios in which your products form.
Under no circumstances, however, must you multiply an Mr by a stoichiometric coefficient when attempting to use the formula moles = mass/Mr.