Problem 89 - Chemistry The Science in Context

Question

Asbestosis is a lung disease caused by inhaling asbestos fibers. In addition, fiber from a form of asbestos called chrysotile is considered to be a human carcinogen by the U.S. Department of Health and Human Services. Chrysotile's composition is \(26.31 \%\) magnesium, \(20.20 \%\) silicon, and \(1.45 \%\) hydrogen with the remainder of the mass as oxygen. Determine the empirical formula of chrysotile.

Step-by-Step Solution

Answer

Answer: The empirical formula for chrysotile is Mg₃Si₂H₄O₉.

Step 1: Convert the given percentage composition to grams

Assume a 100 gram sample of chrysotile. Therefore, the mass of each element present in the sample will be: - Magnesium: \(26.31g\) - Silicon: \(20.20g\) - Hydrogen: \(1.45g\) - Oxygen: remainder of the mass, which is \(100g - 26.31g - 20.20g - 1.45g = 52.04g\)

Step 2: Convert the mass of each element to moles

To convert the mass of each element to moles, we divide the mass by the element's atomic mass. We have: - Magnesium: \(\frac{26.31}{24.31} = 1.08\) moles - Silicon: \(\frac{20.20}{28.09} = 0.72\) moles - Hydrogen: \(\frac{1.45}{1.01} = 1.44\) moles - Oxygen: \(\frac{52.04}{16.00} = 3.25\) moles

Step 3: Find the ratio of moles of each element

To find the simplest whole-number ratio of the moles of each element, we divide the moles of each element by the smallest number of moles calculated in the previous step (which is 0.72 for Silicon). The results are: - Magnesium: \(\frac{1.08}{0.72} = 1.5\) - Silicon: \(\frac{0.72}{0.72} = 1\) - Hydrogen: \(\frac{1.44}{0.72} = 2\) - Oxygen: \(\frac{3.25}{0.72} = 4.5\) Notice that the ratio for Magnesium and Oxygen contains decimals (1.5 and 4.5). To get whole numbers, we can multiply all ratios by 2: - Magnesium: \(1.5 * 2 = 3\) - Silicon: \(1 * 2 = 2\) - Hydrogen: \(2 * 2 = 4\) - Oxygen: \(4.5 * 2 = 9\)

Step 4: Write the empirical formula

The empirical formula of chrysotile is derived from the whole-number mole ratios we calculated: Empirical Formula: \(\text{Mg}_3\text{Si}_2\text{H}_4\text{O}_9\)

Key Concepts

Percentage Composition

Percentage composition refers to the percent by mass of each element present in a compound. It’s a critical starting point for determining the empirical formula, which shows the simplest whole-number ratio of elements in a compound.

Take chrysotile, for example, where the analysis starts with the known percentage composition of magnesium, silicon, hydrogen, and oxygen. Imagining a 100 gram sample makes the math intuitive: the given percentages translate directly into grams of each element. From there, the empirical formula is just a few conversion steps away.

Understanding and being able to calculate percentage composition is essential for anyone working with chemical substances. It gives insight into the relative abundance of each element in a sample and is foundational for deeper stoichiometric calculations.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and the macroscopic universe we experience. It represents Avogadro's number (approximately \(6.022 \times 10^{23}\)) of items, whether they're atoms, molecules or other particles.

By converting the masses of elements from the example of chrysotile to moles, we are essentially counting how many atoms of each element we have in a given sample. This step is vital since chemical reactions occur atom by atom, mole by mole, and not on a mass basis.

The use of the mole concept standardizes measurements across chemistry, allowing scientists and students alike to compare and predict how substances will react in differing amounts, all based on this fundamental unit of chemical quantity.

Chemical Stoichiometry

Chemical stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. It’s the measure of these ratios and how they inform creation of new substances.

In the context of the chrysotile example, after converting the masses to moles, we investigate the mole ratio of each element to reveal the simplest whole-number mole ratio. Stoichiometry isn't just about balance; it's about proportions - the 'recipe' for a compound, if you will. This ratio translates the quantitative chemical ‘sentence’ that is the empirical formula.

It's a fundamental concept compelling students to consider not just the qualitative aspects of chemistry but the precise quantitative relationships that sustain it.

Stoichiometric Calculations

Stoichiometric calculations are the mathematical workhorse of chemistry. They convert the qualitative knowledge of what reacts and what's produced into categorical numerical predictions.

For chrysotile, stoichiometric calculations reveal how the moles of each element compare. However, since our goal is an empirical formula, we need integer ratios. Multiplying by the smallest number necessary to achieve this makes the math align with the physical reality: you can't have a fraction of an atom in a molecule.

These calculations are about more than just solving a problem; they're about understanding the relationships that govern the behavior of the elemental building blocks of our world. There's a beauty in the numbers that correspond with the exactness of natural laws.