At \(\mathrm{STP}, 1.0 \mathrm{L}\) \(\mathrm{Br}_{2}\) reacts completely with \(3.0 \mathrm{L}\) \(\mathrm{F}_{2},\) producing \(2.0 \mathrm{L}\) of a product. What is the formula of the product? (All substances are gases.)

At STP (Standard Temperature and Pressure), 1 L of any gas contains the same number of moles. The volume ratio of the reactants, \(\mathrm{Br}_2\) and \(\mathrm{F}_2\), is 1:3, which means for every liter of \(\mathrm{Br}_2\), 3 liters of \(\mathrm{F}_2\) react completely with it. Let's denote the volume of the produced product as \(\mathrm{X}\)L.

Assume that there are y moles of \(\mathrm{Br}_2\) that react completely with 3y moles of \(\mathrm{F}_2\) to yield 2y moles of the product. Now, apply Avogadro's law: at the same temperature and pressure, equal volumes of all gases contain the same number of moles. So if the moles of \(\mathrm{Br}_2\) is y, and the moles of \(\mathrm{F}_2\) needed to react completely is 3y, we can write the proportion: \( \frac{moles \space of \space Br_2}{moles \space of \space F_2} = \frac{1}{3} \) That is: \(\frac{y}{3y}= \frac{1}{3}\)

Since the reaction is balanced, 1 mole of \(\mathrm{Br}_2\) will react completely with 3 moles of \(\mathrm{F}_2\) (as we found from the proportion above), producing 2 moles of the product. This indicates that 1 \(\mathrm{Br}\) atom combines with 1.5 \(\mathrm{F}\) atoms to form the product. However, we cannot have a fraction of an atom in a chemical formula. To balance this, we can multiply both the reactants and the product by 2: \(2 \mathrm{Br}_2 + 3 \cdot 2 \mathrm{F}_2 \rightarrow 4 \mathrm{X}\) This means that 2 \(\mathrm{Br}\) atoms will react with 3 \(\mathrm{F}\) atoms, yielding 4 \(\mathrm{X}\) atoms. Therefore, the formula of the product, where \(\mathrm{Br}\) and \(\mathrm{F}\) are combined, is \(\mathrm{BrF}_3\).