Serotonin \((\mathscr{M}=176 \mathrm{~g} / \mathrm{mol})\) transmits nerve impulses between neurons. It contains \(68.2 \%\) C, \(6.86 \%\) H, \(15.9 \%\) N, and \(9.08 \%\) O by mass. What is its molecular formula?

The empirical formula is the simplest ratio of elements in a compound. It gives the smallest whole number ratio of the atoms of each element in the compound.
To find the empirical formula, we first assume a sample mass (usually 100 grams) and convert the percentage of each element directly into grams.
Then, we convert the mass of each element to moles using their respective molar masses.
The key step is to find the smallest ratio by dividing all mole values by the smallest number of moles calculated.
Finally, the resulting whole numbers give us the empirical formula, which ensures that the proportions of atoms are expressed in their simplest form.

Mole calculation is essential in finding the empirical formula. By converting the mass of an element to moles, we can compare quantities of different elements accurately.
To perform mole calculations, use the molar mass of each element as a conversion factor:
1. Divide the mass of each element by its molar mass.
2. This gives us the number of moles of each element.
For example, if we have 68.2 grams of Carbon (C) with a molar mass of 12.01 g/mol, the moles of Carbon would be calculated as follows:
\(\frac{68.2 \text{ g}}{12.01 \text{ g/mol}} = 5.68 \text{ moles}\).
Repeating this for all elements allows us to determine their mole ratios, which are crucial for the resultant empirical formula.

The mass percentage of an element in a compound tells us how much of the total mass of the compound comprises that specific element. This is usually expressed as a percentage.
Determining mass percentage is the first step in solving problems related to empirical and molecular formulas.
With mass percentage, you directly translate the percentages into the grams when assuming a 100-gram sample. For example, if a compound is made up of 68.2% Carbon, in 100 grams of the compound, there would be 68.2 grams of Carbon.
This simplification is fundamental as it sets up the next steps for converting these masses into moles and eventually finding the empirical formula.

The molecular mass is the total mass of a molecule, calculated as the sum of the atomic masses of all the atoms in the molecule.
This value is crucial when determining the molecular formula of a compound from its empirical formula.
The formula for molecular mass uses the molecular weights (molar masses) of each constituent element:
1. Multiply the number of atoms of each element by their molar mass.
2. Sum these values.
For example, for the empirical formula \(C_{10}H_{12}N_{2}O\):
\(10 \times 12.01 \text{ g/mol} + 12 \times 1.01 \text{ g/mol} + 2 \times 14.01 \text{ g/mol} + 1 \times 16.00 \text{ g/mol}\).
This gives us the empirical formula mass.
When this empirical formula mass closely matches the given molecular mass (e.g., 176 g/mol), the empirical formula is also the molecular formula.