The reaction \(2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{O}_{2}(g)\) is first order in \(\mathrm{H}_{2} \mathrm{O}_{2}\) and under certain conditions has a rate constant of \(0.00752 \mathrm{~s}^{-1}\) at \(20.0^{\circ} \mathrm{C}\). A reaction vessel initially contains \(150.0 \mathrm{~mL}\) of \(30.0 \% \mathrm{H}_{2} \mathrm{O}_{2}\) by mass solution (the density of the solution is \(1.11 \mathrm{~g} / \mathrm{mL}\) ). The gaseous oxygen is collected over water at \(20.0^{\circ} \mathrm{C}\) as it forms. What volume of \(\mathrm{O}_{2}\) forms in \(\begin{array}{lllll}85.0 & \text { seconds at a barometric pressure of } & 742.5 & \mathrm{mmHg} ?\end{array}\) (The vapor pressure of water at this temperature is \(17.5 \mathrm{mmHg}\).)

Understanding a first-order reaction is fundamental in the study of chemical kinetics. This type of reaction has a rate that is directly proportional to the concentration of one reactant. In mathematical terms, if we represent the reactant concentration as [A], the rate of the reaction can be expressed as Rate = k[A], where 'k' is the rate constant. A key characteristic of first-order reactions is that their half-life, the time it takes for half of the reactant to be consumed, is constant and independent of its initial concentration.

When solving problems involving first-order reactions, one often uses the integrated rate law, given by \( \text{ln} \frac{[A]_0}{[A]} = kt \), where \( [A]_0 \) is the initial concentration, [A] is the concentration at time 't', and 'k' is the rate constant. In practical terms, this equation allows us to determine how the concentration of a reactant changes over time, given the rate constant and the period of time.

The rate constant, denoted as 'k', is a crucial parameter in chemical kinetics, providing a quantitative measure of how quickly a reaction proceeds. It is unique for each reaction and can be affected by factors like temperature and the presence of a catalyst. For a first-order reaction, the units of the rate constant are \( s^{-1} \), indicating the reaction rate per second.

Knowing the rate constant allows us to predict the speed of a reaction under specific conditions. This enables chemists to control and optimize reactions, whether it's speeding up a desired chemical process or slowing down a potentially hazardous one. In our original exercise, the rate constant is given, allowing us to calculate the concentration of remaining reactants at any given time.

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It's the 'recipe' for a reaction, telling us how much of each substance is needed to react completely with other substances, and what amount of product will be formed. The coefficients in a balanced chemical equation provide the molar ratio of reactants and products.

In our example reaction \(2 \text{H}_2\text{O}_2 \rightarrow 2 \text{H}_2\text{O} + \text{O}_2\), the stoichiometry tells us that 2 moles of hydrogen peroxide produce 1 mole of oxygen gas. Hence, by determining the amount of hydrogen peroxide that has reacted, we can directly calculate the moles of oxygen gas produced. Accurate stoichiometric calculations are essential for predicting the outcomes of reactions and are the basis for the design of chemical processes.

Dalton's Law of Partial Pressures is a principle in chemistry that states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. A gas's partial pressure is the pressure it would exert if it occupied the entire volume of the mixture alone.

This law becomes extremely relevant when collecting gases over water, as in our exercise. Since water vapor exerts its own pressure, we must account for it to determine the pressure of the gas of interest—oxygen, in this case. Subtracting the water vapor pressure from the total pressure gives us the dry oxygen gas's partial pressure. Applying Dalton's Law correctly is vital to accurately determine the amount of gaseous product in gas collection experiments.

The Ideal Gas Law is a well-established principle that describes the behavior of an ideal gas. Represented by the equation \(PV = nRT\), it relates the pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) of a gas. For the calculation, 'R' is a known constant (0.0821 L·atm/mol·K), and the temperature must be converted to Kelvin (K).

In the scenario provided in our exercise, we use the Ideal Gas Law to find the volume of oxygen gas produced from the known moles and temperature. Correct application of this law is crucial to determining the volume of gases in a variety of conditions, especially in laboratory settings where gas properties are to be measured or predicted.