Difficult molecular formula question? (See below)

The following appeared on my practise test: You discover the formula of a new compound to be ${C}_{4 y} {H}_{7 y} {O}_{y}$ , where y is a unit positive integer between 1 and 9; for example, if y = 2, then the molecular formula would be ${C}_{42} {H}_{72} {O}_{2}$. Determine the molecular formula if the molar mass ofhe compound is 725.22 g/mol. Show all work – maximum marks will be rewarded for solving using a single equation.

Feb 1, 2017

The molecular formula of the given compound is ${C}_{4 y} {H}_{7 y} {O}_{y}$.Here y represents the unit place of a number as evident from the given example. 4 and 7 represent the tens place of the number of Carbon atoms and the number of H atoms in the molecule.

Hence molecular formula of the compound should actually be represnted as follows ${C}_{\left(4 \cdot 10 + y\right)} {H}_{\left(7 \cdot 10 + y\right)} {O}_{y}$

Using atomic masses of constituent elements the caculated molar mass becomes
$12 \left(40 + y\right) + 1 \left(70 + y\right) + 16 y$
$= 550 + 29 y g \text{/mol}$

Equating this with the given molar mass we get

$550 + 29 y = 725.22$
$\implies y = \frac{175.22}{29} \approx 6$

So the molecular formula of the compound is

${C}_{\left(4 \cdot 10 + y\right)} {H}_{\left(7 \cdot 10 + y\right)} {O}_{y}$

$= {C}_{\left(4 \cdot 10 + 6\right)} {H}_{\left(7 \cdot 10 + 6\right)} {O}_{6}$

$= {C}_{46} {H}_{76} {O}_{6}$