Consider the reaction: $$ \mathrm{P}_{4}(s)+6 \mathrm{H}_{2}(g) \longrightarrow 4 \mathrm{PH}_{3}(g) $$ (a) If \(88.6 \mathrm{~L}\) of \(\mathrm{H}_{2}(g)\), measured at STP, is allowed to react with \(158.3 \mathrm{~g}\) of \(\mathrm{P}_{4}\), what is the limiting reactant? (b) If \(48.3 \mathrm{~L}\) of \(\mathrm{PH}_{3}\), measured at \(S T P\), forms, what is the percent yield?

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. Understanding stoichiometry is crucial for determining the amount of substances needed or produced in a chemical reaction. To perform stoichiometric calculations, we must first balance the chemical equation and then use the mole ratio to convert between amounts of reactants and products.

For example, in the balanced equation provided in the exercise: $$\mathrm{P}_{4}(s) + 6 \mathrm{H}_{2}(g) \longrightarrow 4 \mathrm{PH}_{3}(g)$$we see that one mole of phosphorus tetrahydride (\(\mathrm{P}_{4}\)) reacts with six moles of hydrogen gas (\(\mathrm{H}_{2}\)) to produce four moles of phosphine (\(\mathrm{PH}_{3}\)). The mole ratio of hydrogen to phosphine is therefore 6:4, or 3:2. This stoichiometric ratio allows us to calculate the required or expected amount of reactants and products when given one quantity in a reaction.

The theoretical yield is the amount of product that would be formed if every reactant was completely converted into products without any loss or side reactions. Calculating the theoretical yield involves using the stoichiometry of the balanced equation and the quantities of the reactants. In the stoichiometric calculation, the limiting reactant determines the theoretical yield because it is the substance that is completely consumed first.

To calculate the theoretical yield, we must:

• Balance the chemical equation to obtain the mole ratio between reactants and products.
• Convert all reactant quantities to moles.
• Use the mole ratio to determine the amount of product formed from the limiting reactant.
• Convert the moles of product to the desired units (grams, liters, etc.), using the appropriate conversion factors, like molar mass or molar volume at STP.

The percent yield is a measure of the efficiency of a chemical reaction. It compares the actual yield, the amount of product actually obtained from a reaction, to the theoretical yield, the maximum amount of product that could be formed as predicted by stoichiometry. The percent yield can be impacted by factors like incomplete reactions, side reactions, or purifying losses.

The formula to calculate the percent yield is:
$$\text{Percent Yield} = \left(\frac{\text{Actual Yield}}{\text{Theoretical Yield}}\right) \times 100\%$$For instance, if we have an actual yield of 48.3 liters of \(\mathrm{PH}_{3}\) gas, and our theoretical yield was calculated to be higher, using this formula will give us the efficiency of the reaction. Percent yield values above 100% are not possible theoretically and usually indicate measurement error or incorrect calculation.

The Ideal Gas Law is a fundamental equation in chemistry that establishes a relationship between pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas, along with the gas constant (R). The equation for the Ideal Gas Law is:$$PV = nRT$$At standard temperature and pressure (STP), which is 0°C (273.15 K) and 1 atm, the volume occupied by one mole of any ideal gas is 22.4 liters. This value is often used to convert between volume and moles at STP without knowing the pressure or temperature explicitly. The Ideal Gas Law is particularly useful in stoichiometry when dealing with gaseous reactants or products.

The molar volume of a gas at STP (0°C and 1 atm) is the volume one mole of the gas occupies under these conditions. For any ideal gas, this volume is 22.4 liters per mole. It is a convenient figure for chemists to use in calculations as it simplifies the conversion between the volume of a gas at STP and the amount in moles.

In the context of the textbook exercise, knowing that 22.4 liters is the molar volume at STP allows us to directly calculate the number of moles of hydrogen gas \(\mathrm{H}_{2}\) from the given volume (88.6 liters), by simply dividing by 22.4. Likewise, the theoretical and actual volumes of phosphine gas (\(\mathrm{PH}_{3}\)) can be converted to moles when evaluating the theoretical yield and percent yield of the reaction.