Problem 92 - Chemistry The Central Science

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?

Step-by-Step Solution

Answer

The empirical formula of eucalyptol is \(C_{10}H_{18}O\), and its molecular formula is also \(C_{10}H_{18}O\), as the empirical formula mass (154.26 g/mol) is almost equal to the given peak mass (154 amu).

Step 1: Calculate moles of each element

To determine the moles of each element, we will first assume that we have 100 grams of eucalyptol. This allows us to interpret the percentages as the mass of each element in grams. For Carbon (\(C\)): 77.87 g For Hydrogen (\(H\)): 11.76 g For Oxygen (\(O\)): (100 - (77.87 + 11.76)) g = 10.37 g Using the molar masses of each element, we can calculate their respective moles. Moles of \(C\) = \(\frac{77.87 \ \mathrm{g}}{12.01 \ \mathrm{g/mol}}\) = 6.485 moles Moles of \(H\) = \(\frac{11.76 \ \mathrm{g}}{1.008 \ \mathrm{g/mol}}\) = 11.667 moles Moles of \(O\) = \(\frac{10.37 \ \mathrm{g}}{16.00 \ \mathrm{g/mol}}\) = 0.648 moles

Step 2: Determine empirical formula

To find the empirical formula, we will divide the moles of each element by the smallest number of moles calculated (in this case, it is moles of \(O\)) to get the smallest whole number ratio. Ratio of \(C: H: O\) = \(\frac{6.485}{0.648} : \frac{11.667}{0.648} : \frac{0.648}{0.648}\) ≈ 10 : 18 : 1 Thus, the empirical formula for eucalyptol is \(C_{10}H_{18}O\).

Step 3: Determine molecular formula

We are given that the peak in the mass spectrum for eucalyptol is 154 amu. First, let's calculate the mass of the empirical formula: Empirical formula mass = \(10 \times 12.01\ \mathrm{g/mol}\ (C) + 18 \times 1.008\ \mathrm{g/mol}\ (H) + 1 \times 16.00\ \mathrm{g/mol}\ (O) = 154.26\ \mathrm{g/mol}\) The empirical formula mass is almost equal to the given peak mass (154 amu). This indicates that the empirical and molecular formulas are the same, since there is no whole number multiple necessary to match the given mass. Therefore, the molecular formula of eucalyptol is also \(C_{10}H_{18}O\).