Chapter 3, Problem 3.49P

Interpretation Introduction

Interpretation:

The molecular formula of dimethyl phthalate $\left(\text{M}=\text{194}\text{.2 g/mol}\right)$ that contains $\text{C}$, $\text{H}$, and $\text{O}$ and form $\text{0}\text{.583 g}$ of carbon dioxide $\left({\text{CO}}_2\right)$ and $0.119\text{ g}$ of water $\left({\text{H}}_2\text{O}\right)$ on combustion of $0.2572\text{ g}$ of dimethyl phthalate is to be determined.

Concept introduction:

An empirical formula gives the simplest whole number ratio of atoms of each element present in a molecule. The molecular formula tells the exact number of atoms of each element present in a molecule.

Following are the steps to determine the empirical formula of a compound when the masses of ${\text{CO}}_2$ and ${\text{H}}_2\text{O}$ are given,

Step 1: One mole of ${\text{CO}}_2$ gives one mole of carbon $\left(\text{C}\right)$. Calculate the mass and moles of carbon in the original sample by using the mass of ${\text{CO}}_2$ and molar masses of $\text{C}$ and ${\text{CO}}_2$. One mole of water $\left({\text{H}}_2\text{O}\right)$ gives two moles of hydrogen $\left(\text{H}\right)$. Calculate the moles and mass of hydrogen in the original sample by using the mass of ${\text{H}}_2\text{O}$ and molar masses of $\text{H}$ and ${\text{H}}_2\text{O}$. The formula to calculate the moles of carbon in the sample when moles of ${\text{CO}}_2$ is given is as follows:

$\text{Amount of C}\left(\text{mol}\right)=\left({\text{mass of CO}}_2\left(\text{g}\right)\right)\left(\frac{{\text{1 mol CO}}_2}{{\text{Mass of 1 mol CO}}_2\left(\text{g}\right)}\right)\left(\frac{1\text{ mol C}}{1{\text{ mol CO}}_2}\right)$ (1)

The formula to calculate the mass of carbon in the sample is as follows:

$\text{Mass of C}\left(\text{g}\right)=\left(\text{Amount of C}\left(\text{mol}\right)\right)\left(\begin{array}{l}\text{Molar mass of }\\ \text{C}\end{array}\right)$ (2)

The formula to calculate the moles of hydrogen in the sample when the mass of ${\text{H}}_2\text{O}$ is given is as follows:

$\text{Amount of H}\left(\text{mol}\right)=\left({\text{mass of H}}_2\text{O}\left(\text{g}\right)\right)\left(\frac{{\text{1 mol H}}_2\text{O }}{\begin{array}{l}\text{Mass of 1 mol}\\ {\text{H}}_2\text{O}\left(\text{g}\right)\end{array}}\right)\left(\frac{2\text{ mol H}}{1{\text{ mol H}}_2\text{O}}\right)$ (3)

The formula to calculate the mass of hydrogen in the sample is as follows:

$\text{Mass of H}\left(\text{g}\right)=\left(\text{Amount of H}\left(\text{mol}\right)\right)\left(\begin{array}{l}\text{Molar mass of }\\ \text{H}\end{array}\right)$ (4)

Step 2: If the sample contains any other element X, then calculate the mass of element X as follows:

$\text{Mass of element X}\left(\text{g}\right)=\text{Mass of sample}\left(\text{g}\right)-\left(\text{mass of C}+\text{mass of H}\right)$ (5)

Step 3: Divide mass of element X by its molar mass to convert the mass to moles. The formula to calculate moles from the mass is as follows:

$\text{Amount}\left(\text{mol}\right)=\left(\text{Given mass}\left(\text{g}\right)\right)\left(\frac{1\text{ mol}}{\text{No}\text{. of grams}}\right)$ (6)

Step 4: The number of moles of the elements is the fractional amounts, thus, write the calculated amount $\left(\text{mol}\right)$ of each element as the subscript of the element’s symbol to obtain a preliminary formula for the compound.

Step 5: Convert the moles of each element to the whole number subscripts. The steps for this math conversion are as follows:

(a) Each subscript is divided by the smallest subscript.

(b) If the whole number is not obtained after division, multiply the obtained subscripts by the smallest integer. This gives the empirical formula of the compound.

Step 6: Add the molar mass of each element multiplied by its number of atoms present in the empirical formula to obtain the empirical formula mass for the compound.

Step 7: Divide the molar mass of the compound by its empirical formula mass to obtain the whole number. The formula to calculate the whole number multiple is as follows:

$\text{Whole}-\text{number multiple}=\frac{\text{Molar mass of compound}}{\text{Empirical formula mass}}$ (7)

Step 8: Multiply the whole number with the subscript of each element present in the empirical formula. This gives the molecular formula of the compound.