Consider the reaction: $$ \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(g) $$ A reaction flask initially contains 112 torr of \(\mathrm{CO}\) and 282 torr of \(\mathrm{H}_{2}\). The reaction is allowed to proceed until the pressure stops changing, at which point the total pressure is 196 torr. Determine the percent yield for the reaction. Assume that temperature is constant and that no other reactions occur other than the one indicated.

Understanding the percent yield of a reaction is crucial to evaluating the efficiency of a chemical process. The percent yield denotes the ratio of the actual yield to the theoretical yield, multiplied by 100. In simpler terms, it tells us what percentage of the theoretical amount was actually produced when the reaction was carried out. To calculate the percent yield, one must first determine the theoretical yield based on the limiting reactant's stoichiometry. Then, the actual yield, which is often determined experimentally or, in the case of our problem, inferred from pressure changes, is compared to the theoretical yield. The formula used is:
\[\begin{equation}\text{Percent Yield} = \left(\frac{\text{Actual Yield}}{\text{Theoretical Yield}}\right) \times 100\%\end{equation}\]
In this exercise's context, the total pressure after the reaction indicates the actual yield, while the theoretical yield is derived from the stoichiometric assumptions assuming complete reaction of the limiting reactant.

The concept of the limiting reactant is pivotal in stoichiometry as it determines which reactant will be completely consumed first, thus limiting the extent of the chemical reaction. When determining the limiting reactant, we must compare the mole ratios of the reactants to those required by the balanced chemical equation. The reactant that is present in the lesser amount, in terms of its stoichiometric necessity, is the limiting reactant. For gas reactions held at constant temperature and volume, partial pressures can act as a proxy for mole ratios, simplifying calculations. In the exercise, the pressures of the gases, CO and H2, are given, and through stoichiometric analysis, the limiting reactant is identified, guiding the subsequent yield calculations.

Stoichiometry, the heart of chemical reaction calculations, involves quantifying the relationship between reactants and products. It relies on the balanced chemical equation to provide the mole ratio of the substances involved. For the gas-phase reaction in the exercise, stoichiometry is applied to determine how moles—or in the context of gases under constant temperature and volume, pressures—of reactants relate to each other and to the resulting product. A deeper understanding of this relationship is essential for interpreting the pressure changes in the reaction flask and for calculating the theoretical yield of the product. The correct use of stoichiometry not only predicts how much product will form under ideal conditions but also is fundamental to the concept of limiting reactants and percent yield.

The ideal gas law provides a link between the physical properties of ideal gases. Expressed as PV=nRT, it shows the relationship between pressure (P), volume (V), amount in moles (n), temperature (T), and the ideal gas constant (R). This law comes in handy when working with reactions involving gases, as seen in the task. Here, the ideal gas law is used to interpret changes in pressure as changes in the number of moles of gas, assuming constant temperature and volume. By simplifying the ideal gas law under these conditions, we can use the initial and final pressures to deduce the stoichiometry of the reaction and to solve for the theoretical and actual yields of the product. It is this application that allows us to understand the entirety of the experiment and complete the calculation for the percent yield. The proper use of the ideal gas law is integral to accurately comprehending and solving problems involving gas-phase chemical reactions.