Question

Ascorbic acid (vitamin C) contains \(\mathrm{C}, \mathrm{H},\) and \(\mathrm{O} . \mathrm{In}\) one combustion analysis, \(5.24 \mathrm{~g}\) of ascorbic acid yields \(7.86 \mathrm{~g} \mathrm{CO}_{2}\) and \(2.14 \mathrm{~g} \mathrm{H}_{2} \mathrm{O} .\) Calculate the empirical formula and molecular formula of ascorbic acid given that its molar mass is about \(176 \mathrm{~g}\).

Short answer

Empirical formula: \(\mathrm{C}_3\mathrm{H}_4\mathrm{O}_3\); Molecular formula: \(\mathrm{C}_6\mathrm{H}_8\mathrm{O}_6\).

Step-by-step solution

Convert CO2 and H2O to Moles of C and H

First, find the moles of carbon using the mass of \(\mathrm{CO}_2\):\[\text{Molar mass of CO}_2 = 44.01 \mathrm{g/mol}.\]The moles of carbon is equal to the moles of \(\mathrm{CO}_2\):\[\text{Moles of } \mathrm{CO}_2 = \frac{7.86 \, \mathrm{g}}{44.01 \, \mathrm{g/mol}} = 0.1786 \, \text{mol} \]\[ \text{Moles of } \mathrm{C} = 0.1786 \, \text{mol}\]Next, calculate the hydrogen moles using \(\mathrm{H}_2\mathrm{O}\):\[\text{Molar mass of } \mathrm{H}_2\mathrm{O} = 18.02 \mathrm{g/mol}.\]The moles of hydrogen is twice the moles of \(\mathrm{H}_2\mathrm{O}\):\[\text{Moles of } \mathrm{H}_2\mathrm{O} = \frac{2.14 \, \mathrm{g}}{18.02 \, \mathrm{g/mol}} = 0.1188 \, \text{mol} \] \[ \text{Moles of } \mathrm{H} = 2 \times 0.1188 \, \text{mol} = 0.2376 \, \text{mol}\]

Determine Mass of Oxygen and Convert to Moles

Calculate the mass of oxygen in ascorbic acid by subtracting the mass of carbon and hydrogen obtained from the combustion from the original ascorbic acid sample: \[\text{Mass of C from } \mathrm{CO}_2 = 0.1786 \, \text{mol} \times 12.01 \, \mathrm{g/mol} = 2.144 \, \mathrm{g} \] \[\text{Mass of H from } \mathrm{H}_2\mathrm{O} = 0.2376 \, \text{mol} \times 1.01 \, \mathrm{g/mol} = 0.24 \, \mathrm{g} \]\[\text{Mass of O} = 5.24 \, \mathrm{g} - (2.144 \, \mathrm{g} + 0.24 \, \mathrm{g}) = 2.856 \, \mathrm{g} \] \[\text{Moles of O} = \frac{2.856 \, \mathrm{g}}{16.00 \, \mathrm{g/mol}} = 0.1785 \, \text{mol}\]

Find the Empirical Formula

Find the smallest number of moles among C, H, and O. Divide each element's moles by the smallest number to find the simplest ratio.\[\text{Ratio of C} = \frac{0.1786}{0.1785} \approx 1\] \[\text{Ratio of H} = \frac{0.2376}{0.1785} \approx 1.33\] \[\text{Ratio of O} = \frac{0.1785}{0.1785} = 1\]\(1.33 \approx \frac{4}{3}\), so multiply all ratios by 3 to get whole numbers\[\text{C: } 3, \text{ H: } 4, \text{ O: } 3 \]Thus, the empirical formula is \(\mathrm{C}_3\mathrm{H}_4\mathrm{O}_3\).

Determine the Molecular Formula

Calculate the empirical formula mass \[(12.01 \times 3) + (1.01 \times 4) + (16.00 \times 3) = 88.06 \, \mathrm{g/mol}\]Determine the multiplier \(n\) for the molecular formula: \[n = \frac{\text{Molar mass}}{\text{Empirical formula mass}} = \frac{176.00}{88.06} \approx 2\]Multiply the subscripts of the empirical formula by \(n\):\[\text{Molecular formula is } \mathrm{C}_{3 \times 2}\mathrm{H}_{4 \times 2}\mathrm{O}_{3 \times 2} = \mathrm{C}_6\mathrm{H}_8\mathrm{O}_6\]

Key concepts

Combustion Analysis

Combustion analysis is a powerful technique used to determine the elemental composition of a compound. This approach can be especially helpful when analyzing organic substances, which typically contain carbon, hydrogen, and sometimes oxygen. During combustion analysis, the compound is completely burned in the presence of excess oxygen, resulting in carbon dioxide (CO extsubscript{2}) and water (H extsubscript{2}O) as primary products.
To determine the amount of carbon in the compound, we measure the amount of CO extsubscript{2} produced. Similarly, the hydrogen content is found by measuring the water produced. If the compound contains oxygen, it is often deduced by the difference in mass from the initial sample weight and the weight of carbon and hydrogen found.
This understanding forms the backbone of learning how to calculate empirical and molecular formulas for unknown compounds using combustion data.

Stoichiometry

Stoichiometry involves calculating the relative quantities of reactants and products in chemical reactions. It relies on balanced chemical equations to ensure that the law of conservation of mass is followed—meaning that the number of atoms of each element remains the same before and after a reaction.
In the context of combustion analysis, stoichiometry is essential to convert the masses of CO extsubscript{2} and H extsubscript{2}O obtained from the reaction into moles of elements within the original compound. This conversion is crucial for unveiling the mole ratios which help in identifying the empirical formula.
For example, by knowing the molar masses of CO extsubscript{2} (44.01 g/mol) and H extsubscript{2}O (18.02 g/mol), we can calculate the moles of carbon and hydrogen originating from a combustion experiment, providing the basis for finding the empirical formula.

Molar Mass Calculation

The molar mass of a compound—expressed in grams per mole—is critical in converting between mass and amount in moles. Molar mass is determined by summing the atomic masses of all atoms in a chemical formula. For example, the molar mass of carbon dioxide (CO extsubscript{2}) is the sum of the mass of one carbon atom and two oxygen atoms.
When calculating empirical and molecular formulas, knowing the molar mass of a compound helps verify and convert the simplest mole ratios obtained from empirical formulas into actual numbers that define the precise molecular formula. The given molar mass can verify whether the empirical formula needs to be multiplied to reach the appropriate size that matches the compound's known molar mass.
For ascorbic acid, a known molar mass of about 176 g/mol was used to determine the molecular formula, showing why calculating molar masses correctly can alter the understanding of a compound's full structure.

Chemical Composition Determination

Determining chemical composition involves finding the relative amounts of each element present in a compound. It is a cornerstone of analytical chemistry and crucial for understanding the substance's properties and behaviors.
In steps involve finding empirical formulas first, which represent the simplest whole-number ratio of atoms within the compound. This requires analysis of the combustion data, converting the mass of combustion products into elemental mass, and deducing the composition based on mass and mole computations.
With the empirical formula in hand, the molecular formula can be determined if the compound's molar mass is known. The molecular formula is essentially a multiple of the empirical formula, revealing the actual number of atoms within a molecule of the compound.
This level of determination is vital for understanding compounds like ascorbic acid, revealing its true chemical identity as C extsubscript{6}H extsubscript{8}O extsubscript{6}.