Concentrated hydrogen peroxide solutions are explosively decomposed by traces of transition metal ions (such as Mn or Fe): $$ 2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{O}_{2}(g) $$ What volume of pure \(\mathrm{O}_{2}(\mathrm{~g})\), collected at \(27^{\circ} \mathrm{C}\) and 746 torr, would be generated by decomposition of \(125 \mathrm{~g}\) of a \(50.0 \%\) by mass hydrogen peroxide solution? Ignore any water vapor that may be present.

Stoichiometry is a key concept in chemistry that involves the quantitative relationships between the reactants and products in a chemical reaction. It is based on the principle of conservation of mass, where the number of atoms of each element must be the same before and after the reaction.

For example, in the decomposition of hydrogen peroxide, the balanced chemical equation demonstrates stoichiometry:
\[2 \text{H}_2\text{O}_2 \longrightarrow 2 \text{H}_2\text{O} + \text{O}_2\]
This equation tells us that two moles of hydrogen peroxide decompose to form two moles of water and one mole of oxygen gas. Using this stoichiometric relationship, we can determine the amount of oxygen produced if we know the quantity of hydrogen peroxide we started with.

Understanding stoichiometry is crucial when aiming to predict the outcomes of reactions. For instance, in our exercise, we utilize stoichiometry to establish that the decomposition of 1.837 moles of hydrogen peroxide yields 0.9185 moles of oxygen. Remember, the stoichiometric coefficient of hydrogen peroxide is twice that of oxygen, reflecting the 2:1 ratio.

The Ideal Gas Law is a fundamental equation in chemistry, relating the pressure, volume, temperature, and number of moles of an ideal gas. The law is commonly expressed as:\[PV = nRT\]
In this formula, P stands for pressure in atmospheres (atm), V is the volume in liters (L), n is the number of moles, R is the ideal gas constant with a value of 0.0821 L·atm/mol·K, and T is the temperature in Kelvin (K).

In the exercise, we are asked to determine the volume of oxygen gas produced, so we rearrange the Ideal Gas Law to solve for V:\[V = \frac{nRT}{P}\]
Performing the appropriate conversions for temperature and pressure to their standard units (Kelvin and atmospheres, respectively), we can calculate the volume of gas produced from the reaction. The Ideal Gas Law assumes that the gas behaves 'ideally', meaning intermolecular forces and molecular volume are negligible. While real gases do not perfectly follow these assumptions, the Ideal Gas Law provides a good approximation for gases under many conditions.

Calculating molar mass is fundamental in chemistry as it ties together the microscopic world of atoms and molecules with the macroscopic world we can measure. The molar mass is the weight of one mole of a substance, usually expressed in grams per mole (g/mol).

For hydrogen peroxide (H2O2), the molar mass calculation is straightforward: each molecule consists of two hydrogen atoms and two oxygen atoms. The atomic mass of hydrogen is approximately 1.01 g/mol, and the atomic mass of oxygen is approximately 16.00 g/mol. Therefore, the molar mass of hydrogen peroxide is:\[2(1.01\, \text{g/mol}) + 2(16.00\, \text{g/mol}) = 34.02\, \text{g/mol}\]
In our sample problem, we leveraged this calculation to determine how many moles of hydrogen peroxide were present in a 125 g sample that is 50% by mass H2O2. By dividing the mass of H2O2 (62.5 g) by its molar mass, we found the number of moles available to participate in the decomposition reaction.