Question

Vitamin A has a molar mass of 286.4 g and has a general molecular formula of C_xH_yE, where E is an unknown element. If vitamin A is 83.86% C and 10.56% H by mass, what is the molecular formula of vitamin A?

Short answer

Theunknown element E is Oandthemolecular formula of vitamin Ais C₂₀H₃₀O

Step-by-step solution

Definition

The molar mass of a materialis defined as the mass of the substance contained in one moleof the substance, and it is equal to the total massof the component atoms. It may be stated mathematically as follows:
Molar mass =$\frac{Weighting}{numberofmoles}$

Determine the moles of C, H, and E

From the given,

Mass of vitamin A is 100g, the mass of C is 83.86g, vitamin H is 10.56g, and the mass of E = 100 - 83.86 - 10.56

E = 5.58g

Moles of C =$\frac{83.86}{12}$

= 6.988

Moles of H =$\frac{10.56}{1.0078}$

= 10.478

Moles of E =$\frac{mass}{Atomicmass}$

$=\frac{5.58}{Atomicmass}$

From the formula C_xH_yE, E = 1i.e. Number of atoms = 1

Determine of the empirical formula

By obtaining an empirical formula, divide each number of moles by the number of moles of E.

$$\begin{gathered} \frac{{\displaystyle \frac{6.988}{5.58}}}{AtomicmassofE}=\left(1.252\right)\left(AtomicmassofE\right)-C \\ \frac{{\displaystyle \frac{10.478}{5.58}}}{AtomicmassofE}=\left(1.877\right)\left(AtomicmassofE\right)-H \\ \frac{{\displaystyle \frac{5.58}{AtomicmassofE}}}{{\displaystyle \frac{5.58}{AtomicmassofE}}}=\left(1AtomofE\right) \end{gathered}$$

Here the molar mass of vitamin A

$$\begin{gathered} \left(1.252\right)\left(AtomicmassofE\right)(AtomicmassofE)+(1.877)\left(AtomicmassofE\right)(AtomicmassofH)+\left(AtomicmassofE\right)=286.64 \\ =⌊\left(1.252\right)\left(12\right)+(1.877)(1.0078)+1⌋\left(AtomicmassofE\right) \end{gathered}$$

Determine the molecular formula of vitamin A

$$\begin{gathered} AtomicmassofE=\frac{286.4}{17.915} \\ =15.986\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$mol$}\right. \end{gathered}$$

Number of C atoms = (1.252)(15.986)

= (1.252)(15.986)

=20

Number of H atoms = (1.877) (Atomic mass of E)

= (1.877)(15.986)

= 30

Thus the molecular formula of Vitamin Ais C₂₀H₃₀O.