# Question e7817

Jan 9, 2018

$x = 2$

#### Explanation:

Your unknown compound has ${\text{C"_7"H"_8"O}}_{x}$ as the molecular formula, so you know for a fact that $1$ mole of this compound contains

• $7$ moles of carbon, $7 \times \text{C}$
• $8$ moles of hydrogen, $8 \times \text{H}$
• $x$ moles of oxygen, $x \times \text{O}$

This means that the molar mass of the compound can be calculated like this

M_"M" = overbrace(7 xx "12.011 g mol"^(-1))^(color(blue)("7 moles C")) + overbrace(8 xx "1.00794 g mol"^(-1))^(color(blue)("8 moles H")) + overbrace(x xx "15.9994 g mol"^(-1))^(color(blue)(x quad "moles O"))

${M}_{\text{M" = (94.14052 + 15.9994 * x)quad "g mol}}^{- 1}$

Now, you know that this compound contains 26.2% oxygen by mass. This means that if you take the mass of $1$ mole of this compound, you can say that you have

( 15.9994 * x color(red)(cancel(color(black)("g mol"^(-1)))))/((94.14052 + 15.9994 * x)color(red)(cancel(color(black)("g mol"^(-1))))) xx 100% = 26.2%

This basically means that if you take the mass of oxygen present in $1$ mole of this compound, divide it by the total mass of $1$ mole of the compound, and multiply the result by 100%, you will end up with 26.2%, the percent concentration by mass of oxygen.

You will thus have

 (15.9994 * x)/((94.14052 + 15.9994 * x)) xx 100 color(red)(cancel(color(black)(%))) = 26.2color(red)(cancel(color(black)(%)))#

Rearrange to solve for $x$

$15.9994 \cdot 100 \cdot x = 26.2 \cdot 94.14052 + 26.2 \cdot 15.9994 \cdot x$

$x \cdot 15.9994 \cdot \left(100 - 26.2\right) = 26.2 \cdot 94.14052$

You will end up with

$x = \frac{26.2 \cdot 94.14052}{15.9994 \cdot \left(100 - 26.2\right)} = 2.089 \approx 2$

You can thus say that you have $x = 2$, which implies that the molecular formula of the compound is ${\text{C"_7"H"_8"O}}_{2}$.