Question

Consider \(100.0-\mathrm{g}\) samples of two different compounds consisting only of carbon and oxygen. One compound contains \(27.2 \mathrm{~g}\) of carbon and the other has \(42.9 \mathrm{~g}\) of carbon. How can these data support the law of multiple proportions if \(42.9\) is not a multiple of \(27.2\) ? Show that these data support the law of multiple proportions.

Short answer

The mass ratios of carbon to oxygen in the two compounds are \( \frac{27.2 \mathrm{~g}}{72.8 \mathrm{~g}} \) and \( \frac{42.9 \mathrm{~g}}{57.1 \mathrm{~g}} \). When comparing these ratios, \( \frac{\frac{42.9}{57.1}}{\frac{27.2}{72.8}} \approx 2 \), which is a whole number. Therefore, these data support the law of multiple proportions.

Step-by-step solution

Calculate the mass of oxygen in each compound

To find the mass of oxygen in each compound, subtract the mass of carbon from the total sample mass (100 g). For the first compound: Mass of oxygen = Total mass - Mass of carbon = \( 100 - 27.2 = 72.8 \mathrm{~g} \) For the second compound: Mass of oxygen = Total mass - Mass of carbon = \( 100 - 42.9 = 57.1 \mathrm{~g} \)

Find the mass ratios of carbon to oxygen in both compounds

Calculate the mass ratios of carbon to oxygen for both compounds: For the first compound: Mass ratio (C : O) = \( \frac{27.2 \mathrm{~g}}{72.8 \mathrm{~g}} \) For the second compound: Mass ratio (C : O) = \( \frac{42.9 \mathrm{~g}}{57.1 \mathrm{~g}} \)

Compare the mass ratios to support the law of multiple proportions

According to the law of multiple proportions, when two elements combine to form different compounds, the masses of one element that combine with a fixed mass of another element are in simple whole-number ratios. To check if these mass ratios are in a simple whole-number ratio, divide the higher mass ratio by the lower mass ratio: \( \frac{\frac{42.9}{57.1}}{\frac{27.2}{72.8}} = \frac{\frac{42.9 \times 72.8}{57.1 \times 72.8}}{\frac{27.2 \times 57.1}{72.8 \times 57.1}} = \frac{42.9 \times 72.8}{27.2 \times 57.1} \) We can now simplify the expression: \( = \frac{3130.92}{1552.12} \approx 2 \) The ratio is very close to a whole number (2), indicating that these mass ratios support the law of multiple proportions.

Key concepts

Stoichiometry

Stoichiometry is at the heart of chemistry, serving as the quantitative bridge between reactants and products in a chemical reaction. It allows chemists to predict the amount of substances consumed and produced in a reaction, guiding the creation of compounds and the understanding of reaction dynamics.

With stoichiometry, scientists calculate relationships within a reaction using balanced chemical equations, which reflect the conservation of mass. Each substance involved in the reaction is represented with a chemical formula, and coefficients in the equation represent the ratio in which these substances react or are produced—also known as the stoichiometric coefficients. These coefficients are central to stoichiometry, as they allow chemists to determine the proportions of reactants needed to form a specific amount of product.

For instance, when calculating the mass of oxygen needed to react with a known mass of carbon to form a compound, stoichiometry provides a framework to ensure that the atomic economy—no atom is wasted—is adhered to. This is particularly useful in industrial settings, where the efficiency and cost-effectiveness of a chemical process are paramount.

Chemical Composition

Chemical composition refers to the identity and quantity of the elements that make up a substance. Each substance has a unique chemical composition characterized by a specific set of elements in a definite proportion by mass.

To comprehend chemical composition, one needs to understand elements, compounds, and the law of definite proportions. This law states that a chemical compound always contains its component elements in a fixed ratio and does not depend on its source or the method of preparation. For example, water (H₂O) is always made up of two parts hydrogen and one part oxygen by mole ratio, regardless of how or where it is produced.

Understanding the chemical composition of compounds is crucial for various applications, including material synthesis, pharmaceutical formulations, and nutritional analysis. It allows scientists and engineers to design materials with specific properties, pharmacists to concoct medications with desired therapeutic effects, and nutritionists to determine the caloric and nutritional content of food.

Mass Ratios in Compounds

Mass ratios in compounds describe how the masses of different elements are related to each other within a compound. This concept is a practical application of the law of multiple proportions. It states that when elements form more than one compound, the masses of one element that combine with a fixed mass of the other are in small whole number ratios.

In context, consider compounds formed by carbon and oxygen. If we take 100-gram samples of two different carbon-oxygen compounds, the mass of carbon will be different for each, yet when we calculate the ratio of the carbon masses to those of oxygen, we often find a simple whole number ratio. This is evidence of the law of multiple proportions in action, highlighting nature's preference for simplicity and order in the composition of compounds.

The precise determination of these mass ratios is essential for identifying compounds and analyzing their chemical structure—a fundamental requirement in disciplines such as materials science, geology, and forensic analysis, where the identification of substances comes into play.

Empirical Formula

The empirical formula of a compound provides the simplest whole-number ratio of the elements within it. While the molecular formula details the exact number of atoms of each element in one molecule of a compound, the empirical formula distills this down to the most reduced ratio.

Finding the empirical formula involves analyzing a compound's composition either by mass or by percent composition and converting these values to a molar ratio. This ratio is then simplified to the smallest whole numbers, leading to the empirical formula. For example, glucose has a molecular formula of C₆H₁₂O₆, which can be simplified to the empirical formula CH₂O, representing the same ratio of elements.

The calculation of the empirical formula is a critical step in the chemical analysis and synthesis of compounds. It is the starting point for determining molecular structure in organic chemistry, as well as for calculating nutritional content in food chemistry—since empirical formulas allow for a simplified view into the basic building blocks of compounds.