Problem 61 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The commonly used pain reliever, aspirin, has the molecular formula \(\mathrm{C}_{2} \mathrm{H}_{8} \mathrm{O}_{4}\). If a sample of aspirin contains \(0.968 \mathrm{~g}\) of carbon, what is the mass of hydrogen in the sample? (a) \(0.717 \mathrm{~g}\) (b) \(0.0717 \mathrm{~g}\) (c) \(8.000 \mathrm{~g}\) (d) \(0.645 \mathrm{~g}\)

Step-by-Step Solution

Answer

0.0717 g

Step 1: Determine the molar mass of carbon and hydrogen

Use the periodic table to find the atomic mass of carbon and hydrogen. Carbon has an atomic mass of about 12.01 g/mol, and hydrogen has an atomic mass of about 1.01 g/mol.

Step 2: Calculate moles of carbon in the sample

Use the mass of carbon and its molar mass to find the number of moles of carbon present in the sample. Number of moles of carbon (n) can be calculated by the formula: \( n = \frac{mass}{molar\;mass} \).

Step 3: Determine the mole ratio of carbon to hydrogen in aspirin

From the molecular formula \(\mathrm{C}_{2}\mathrm{H}_{8}\mathrm{O}_{4}\), we know that the mole ratio of carbon to hydrogen is 2:8, which reduces to 1:4.

Step 4: Calculate the number of moles of hydrogen in the sample

Multiply the number of moles of carbon by the mole ratio of hydrogen to carbon to find the number of moles of hydrogen in the sample.

Step 5: Calculate the mass of hydrogen in the sample

Multiply the number of moles of hydrogen by the molar mass of hydrogen to find the mass of hydrogen in the sample.

Key Concepts

Molar Mass

Understanding the concept of molar mass is crucial when solving stoichiometry problems. Molar mass is defined as the mass of one mole of a substance and is expressed in grams per mole (g/mol). It can be determined by summing up the atomic masses of all the atoms present in a molecule, as found on the periodic table. For example, in the case of carbon (C), which has an atomic mass of approximately 12.01 g/mol, and hydrogen (H), with an atomic mass of about 1.01 g/mol, determining the molar mass of each element allows us to convert between grams and moles — a fundamental step in stoichiometric calculations.

To solidify this concept, consider the molar mass of water (H₂O): with two hydrogen atoms and one oxygen atom, the molar mass is \(2 \times 1.01 g/mol + 16.00 g/mol = 18.02 g/mol\). This is essential knowledge when analyzing chemical reactions, as the molar mass serves as a conversion factor between the mass of a substance and the number of particles (moles) it contains.

Mole Ratio

When dealing with chemical reactions, the mole ratio is the bridge that allows us to translate quantities of one substance into quantities of another. It is derived from the coefficients of a balanced chemical equation or the subscripts in the molecular formula, indicating the relative amounts of each substance involved. In the example of aspirin (C₂H₈O₄), the mole ratio between carbon and hydrogen atoms is derived from the subscripts: for every 2 moles of carbon, there are 8 moles of hydrogen, giving us a ratio of 1:4 after simplification.This proportional relationship is the heartbeat of stoichiometry. Once we know the number of moles of one substance, we can use the mole ratio to calculate the number of moles of any other substance in the chemical formula. It's like a recipe that tells us how much of each ingredient is needed to bake a cake, ensuring that we always have the right proportions for a perfect outcome.

Chemical Formula Calculations

Chemical formula calculations are the math behind the science. They involve using the chemical formula of a compound to calculate various quantitative aspects of its composition. By mastering chemical formula calculations, students can determine the number of moles, mass, or number of atoms of each element within a compound.If we revisit our aspirin sample (C₂H₈O₄), knowing that we have 0.968 grams of carbon enables us to calculate its moles using the molar mass of carbon. Then, with the mole ratio, we can find out how many moles of hydrogen correspond to the amount of carbon. Finally, by multiplying the moles of hydrogen by its molar mass, we determine the mass of hydrogen in the sample.

Here's a simplified approach to these calculations:

• Find the molar masses of the elements in the compound.
• Determine the moles of a given element from the mass presented.
• Apply the mole ratio to find the moles of the desired element.
• Convert the moles of the desired element back to mass using its molar mass.

With this methodology, students can analyze and predict the outcomes of chemical reactions with greater accuracy and understand the composition of compounds better.