Question

A compound with the molar mass of approximately \(142 \mathrm{~g} / \mathrm{mol}\) has the composition \(50.7 \% \mathrm{C}, 9.9 \% \mathrm{H}\), and \(39.4 \% \mathrm{~N}\). What is its molecular formula?

Short answer

The molecular formula of the compound is C9H21N6

Step-by-step solution

Convert percentage composition to moles

Convert the percent composition of each element (Carbon - C, Hydrogen - H, and Nitrogen - N) to moles, assuming a 100g sample. In such a sample, the mass of each element is equal to its percentage. using the formula \( n = m/M \), where m is the mass and M is the molar mass, \n\nFor carbon: \(n_C = \frac{50.7\ g}{12.01\ g/mol} \approx 4.22 \ mol \)\n\nFor hydrogen: \(n_H = \frac{9.9\ g}{1.01\ g/mol} \approx 9.80\ mol\) \n\nAnd for nitrogen: \(n_N = \frac{39.4\ g}{14.01\ g/mol} \approx 2.81\ mol\)

Find empirical formula

To find the empirical formula, divide all the calculated moles by the smallest number of moles, in this case, 2.81 mol for nitrogen: \n\n For Carbon: \(4.22\ div\ 2.81 = 1.5\) , for Hydrogen: \(9.80\ div\ 2.81 = 3.5\) , for Nitrogen: \(2.81\ div\ 2.81 = 1 \) \n\nThe empirical formula thus far would be C1.5H3.5N, but fractional values are not accepted in formulas, so multipling through by 2 for all to get integer values, we obtain the empirical formula as: C3H7N2

Determine the Molecular Formula

To find the molecular formula, calculate the empirical formula weight first. Given the atomic masses: Carbon (12.01 g/mol), Hydrogen (1.01 g/mol), Nitrogen (14.01 g/mol), the empirical formula weight is: \(3*12.01 + 7*1.01 + 2*14.01 \approx 43.06 g/mol \) \n\nThen divide the molar mass of the compound (142 g/mol) by the empirical formula weight and round to the nearest whole number. \n\n \(n = 142/43.06 \approx 3\),\n\n Multiply the empirical formula by this number to get the molecular formula as C3H7N2*3 = C9H21N6

Key concepts

Percent Composition

Understanding percent composition is crucial for chemists in determining the formula of a compound. It represents the percent by mass of each element within a compound. To calculate the percent composition, you divide the mass of a specific element in one mole of the compound by the compound’s molar mass and multiply by 100 to get a percentage.

For example, if a compound has a molar mass of 180 grams per mole and contains 60 grams of carbon, the percent composition of carbon in the compound would be \(\frac{60}{180} \times 100 = 33.3\%\). When given the percent composition, as in our exercise, it allows students to work backwards to deduce the amount of each element in grams when assuming a sample size, typically 100 grams, which simplifies the calculation.

Molar Mass

The molar mass of an element or compound is the mass in grams of one mole of that substance. A mole is simply a unit in chemistry that represents Avogadro's number, which is approximately \(6.022 \times 10^{23}\) entities (atoms, molecules, ions, etc.).

The molar mass is used to convert between the number of moles of a substance and its mass in grams. Using the periodic table, we can determine the molar mass by adding the atomic masses of each element in the compound. For instance, the molar mass of water (H₂O) would be the sum of the molar masses of hydrogen (1.01 g/mol, multiplied by 2 because there are 2 hydrogens) and oxygen (16.00 g/mol), resulting in approximately 18.02 g/mol.

Empirical Formula

The empirical formula represents the simplest whole-number ratio of elements in a compound. It does not necessarily represent the actual numbers of atoms in a molecule, which the molecular formula indicates, but it provides the proportional comparison. To derive the empirical formula, divide the number of moles of each element by the smallest number of moles from the elements in the compound.

Then, if necessary, we multiply these numbers by a common factor to achieve whole numbers, which is a critical step because chemical formulas cannot have fractions. As seen in the exercise, multiplying the ratios of the elements by 2 resulted in the empirical formula C₃H₇N₂, providing a foundational ratio upon which the actual molecular formula can be constructed.

Moles to Grams Conversion

The conversion from moles to grams is a fundamental skill in chemistry and it is conducted using the molar mass. It's particularly relevant when dealing with chemical reactions, where you need to know how much of a substance you'll produce or require.

The formula to convert moles to grams is simple: \( \text{mass (g)} = \text{moles} \times \text{molar mass (g/mol)} \). Conversely, to find the number of moles given the mass, you would divide the mass by the molar mass. For example, if you have 9.8 moles of hydrogen and want to find the mass, you would multiply 9.8 moles by 1.01 g/mol (the molar mass of hydrogen), resulting in about 9.9 grams.