A compound of \(\mathrm{P}\) and \(\mathrm{F}\) was analyzed as follows: Heating \(0.2324 \mathrm{~g}\) of the compound in a \(378-\mathrm{cm}^{3}\) container turned all of it to gas, which had a pressure of \(97.3 \mathrm{mmHg}\) at \(77^{\circ} \mathrm{C}\). Then the gas was mixed with calcium chloride solution, which turned all of the \(\mathrm{F}\) to \(0.2631 \mathrm{~g}\) of \(\mathrm{CaF}_{2}\). Determine the molecular formula of the compound.

When embarking on the journey to determine the molecular formula of a compound, one invaluable tool is the Ideal Gas Law, represented by the formula: \( PV = nRT \). This equation relates the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas to the ideal gas constant (R).

To comprehend this law is to unlock the ability to solve a multitude of gas-related problems. In the context of our exercise, we use it to calculate the number of moles of gas. Before plugging values into the equation, we need to ensure they are in standard units: pressure in atm, volume in liters, and temperature in Kelvin. For temperature conversion, remember \( T(K) = T(^{\rc}C) + 273.15 \).

Once the units are correctly converted, we substitute the known values into the equation, solving for n, which stands for the moles of our unknown compound.

Stoichiometry, the backbone of chemical reactions, details the quantitative relationships between reactants and products. It's akin to a recipe: knowing the amount of each ingredient allows you to predict the final dish.

In our problem, stoichiometry reveals itself when fluorine from the unknown compound reacts with calcium chloride to form calcium fluoride (\( CaF_2 \)). The balanced chemical equation provides the ratio of reactants to products. For every mole of \( CaF_2 \), there is one mole of fluorine atoms involved, unveiling a 1:1 mole ratio.

Using the mass of \( CaF_2 \) and its molar mass (78.07 g/mol), we find the moles of fluorine that reacted. This stoichiometric conversion is essential to establish the relationship between the known substance (\( CaF_2 \)) and the fluorine in the unknown compound.

Identifying a compound's molecular formula demands an understanding of molar mass calculation. Every element has an atomic weight, and the molar mass of a compound is the sum of the atomic weights of its elements, proportionate to their occurrence in the formula.

In order to decrypt the compound's molecular formula, we compare the molar mass to the measured mass of the sample. This process can seem cumbersome, but it's straightforward with some practice. By knowing the mass of the substance and the number of moles calculated through the ideal gas law and stoichiometry, we can determine the molar mass.

With the molar mass of fluorine and the mole ratio between phosphorus and fluorine in the compound, we can deduce the subscripts for each element, simplifying to the smallest whole numbers, which finally reveals the molecular formula to us.