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Understanding chemical reactions is crucial for graspng the intricacies of stoichiometry, which involves calculations based on the quantitative relationships of the reactants and products involved in a chemical reaction. In our example, the reactions provided are between ammonia (\r\( \mathrm{NH}_{3}(g) \)) and hydrazine (\r\( \mathrm{N}_{2}\mathrm{H}_{4}(g) \)), with oxygen (\r\( \mathrm{O}_{2}(g) \)). Each reaction was balanced according to the law of conservation of mass, which states that matter cannot be created or destroyed in a chemical reaction.

When writing a balanced chemical equation, it is essential to ensure that the number of atoms of each element on the reactant side is equal to the number on the product side. The exercise showcased two balanced equations, with specific mole ratios between the reactants and the products pivotal in later calculations. It is these balanced equations that form the backbone of stoichiometry and lead us to further explore the mole concept.

The mole concept is a method of quantifying particles, like atoms, molecules, or in this case, the amount of gas. One mole corresponds to Avogadro's number (\r\( 6.022 \times 10^{23} \)) particles. In our exercise, we are dealing with the initial and remaining amount of oxygen gas in moles. Knowing the amount of moles involved in a reaction allows us to calculate the amounts of other reactants or products using the balanced equation as a ratio. This relationship is key for solving complex stoichiometry problems.

For instance, in the given problem, calculating the moles of \r\( \mathrm{O}_{2} \) that reacted was critical. By subtracting the remaining \r\( \mathrm{O}_{2} \) moles from the initial moles, we used the mole concept to move towards determining the composition of the gas mixture. From the mole concept, we can infer the proportions of reactants and products, leading to a deeper understanding of the reaction mechanics and the subsequent calculations involving mass percent composition.

The mass percent calculation is often used in chemistry to express the concentration of an element or compound within a mixture. Mass percent is calculated by the formula: \r
\[ \text{Mass Percent} = \left( \frac{\text{Mass of Solute}}{\text{Total Mass of Solution}} \right) \times 100 \]

This concept allows us to determine how much of a certain substance is present compared to the total mass. In the context of the exercise, it was initially thought that we would need to calculate the mass percent of \r\( \mathrm{N}_{2}\mathrm{H}_{4}(g) \) in the gaseous mixture. However, through the process of elimination using the equations derived from the chemical reactions and mole concept, we deduced that only \r\( \mathrm{NH}_{3} \) was present and thus the mass percent of \r\( \mathrm{N}_{2}\mathrm{H}_{4}(g) \) is 0%.

This part of the exercise offers a prime learning moment, emphasizing the importance of systematically following stoichiometric principles to reach a logical conclusion, highlighting the synergy between chemical reactions, the mole concept, and mass percent calculations in the realm of stoichiometric analysis.