Question

The koala dines exclusively on eucalyptus leaves. Its digestive system detoxifies the eucalyptus oil, a poison to other animals. The chief constituent in eucalyptus oil is a substance called eucalyptol, which contains \(77.87 \% \mathrm{C}\), \(11.76 \% \mathrm{H}\), and the remainder O. (a) What is the empirical formula for this substance? (b) A mass spectrum of eucalyptol shows a peak at about 154 amu. What is the molecular formula of the substance?

Short answer

The empirical formula for eucalyptol is C10H18O, and its molecular formula is also C10H18O since the multiplier factor is approximately 1.

Step-by-step solution

Assume 100g of eucalyptol

We can assume we have 100g of eucalyptol, so the percentages correspond directly to the mass of each element in the sample.

Calculate moles of each element

Divide the mass of each element by its molar mass to find the moles: Carbon (C): \(\frac{77.87g}{12.01g/mol} = 6.49mol\) Hydrogen (H): \(\frac{11.76g}{1.01g/mol} = 11.64mol\) Oxygen (O): Since only Carbon, Hydrogen, and Oxygen are present and the total must be 100%, the mass of Oxygen is \(100 - 77.87 - 11.76 = 10.37g\). Therefore, the moles of Oxygen are: \(\frac{10.37g}{16.00g/mol} = 0.648mol\)

Find the lowest whole number ratio

Divide all the moles by the smallest value: Carbon: \(\frac{6.49}{0.648} \approx 10\) Hydrogen: \(\frac{11.64}{0.648} \approx 18\) Oxygen: \(\frac{0.648}{0.648} = 1\) The empirical formula for eucalyptol is C10H18O. #b) Find the molecular formula of eucalyptol#

Calculate the molar mass of empirical formula

Calculate the molar mass of the empirical formula: \(10(12.01) + 18(1.01) + 16.00 = 154.24g/mol\)

Find the molecular formula

Divide the given molecular weight (154 amu) by the empirical formula's weight (154.24 g/mol) to find the multiplier factor: \(\frac{154}{154.24} \approx 1\) Since the factor is 1, the molecular formula is the same as the empirical formula. The molecular formula of eucalyptol is C10H18O.

Key concepts

Moles and Molar Mass

Understanding moles and molar mass is essential in the calculation of chemical formulas. Moles provide a bridge between atomic scales and observable, macroscopic quantities of materials. One mole of any substance contains Avogadro's number of atoms or molecules, which is approximately \(6.022 \times 10^{23}\). Molar mass, on the other hand, is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It is calculated based on each element's atomic mass found on the periodic table.

In the given problem, assuming 100 grams of eucalyptol simplifies the calculation, as the percentages directly translate to grams. We use molar mass to convert the mass of each element to moles. For example, the molar mass of carbon is \(12.01 \text{ g/mol}\), so with \(77.87\, \text{g}\) of carbon, we have \( \frac{77.87}{12.01} \approx 6.49 \text{ moles}\). Similarly, hydrogen’s and oxygen’s moles are calculated using their respective molar masses (\(1.01 \text{ g/mol}\) for hydrogen and \(16.00 \text{ g/mol}\) for oxygen).

These calculations allow us to determine the amount of each element and eventually find the stoichiometric ratio, leading to the empirical formula. This process is foundational in stoichiometry and stoichiometric calculations.

Mass Spectrometry

Mass spectrometry is a powerful analytical tool used to identify the composition of a sample by measuring the masses of its components. It operates by ionizing chemical compounds and sorting these ions based on their mass-to-charge ratio. The result is a mass spectrum that displays the mass and abundance of different ions.

In the problem, the mass spectrum of eucalyptol displays a peak at about \(154\, \text{amu}\). This peak represents the mass of the molecule as a whole, providing crucial information for determining the molecular formula. Knowing that the mass of the empirical formula \(\text{C}_{10}\text{H}_{18}\text{O}\) is approximately the same as the peak, we confirm that the molecular formula is identical to the empirical formula.

This illustrates the use of mass spectrometry in verifying molecular weights and ensuring the empirical formula is accurately reflected by the molecular formula of the substance.

Elemental Composition Analysis

Elemental composition analysis is used to determine the percentage of each element within a compound. This process provides the foundational data required to establish empirical formulas. By assessing the mass percentage of components within a sample, we gain insight into its molecular makeup.

For eucalyptol, the given elemental composition is \(77.87\%\) carbon, \(11.76\%\) hydrogen, with the remainder being oxygen, which we calculated as \(10.37\%\) oxygen. By assuming a sample mass of 100 grams, these percentages directly translate into mass values. Each element’s mass can be converted to moles using their molar masses.

Dividing the calculated moles by the smallest value among them yields the simplest whole number ratio of atoms, giving the empirical formula. Such analyses are critical in providing a life's blueprint for each compound, determining how elements are combined at the molecular level.