Question

Suppose that a large volume of 3\(\%\) hydrogen peroxide decomposes to produce 12 \(\mathrm{mL}\) of oxygen gas in 100 \(\mathrm{s}\) at 298 \(\mathrm{K}\) . Estimate how much oxygen gas would be produced by an identical solution in 100 s at 308 \(\mathrm{K}\)

Short answer

In conclusion, without additional information about the activation energy or the rate constants at both temperatures, it is not possible to estimate the amount of oxygen gas produced by an identical solution in 100 s at 308 K.

Step-by-step solution

1. Determine the initial rate of the reaction

Given that 12 mL of oxygen gas is produced in 100 sec at 298 K, we can calculate the initial rate of the reaction as follows: \(Initial\: rate = \frac{Volume\: of\: O_2\: produced}{Time} = \frac{12\: mL}{100\: s} = 0.12\: \frac{mL}{s}\)

2. Determine the rate constant using Arrhenius equation

The Arrhenius equation is given by: \(k = A e^{-\frac{E_a}{RT}}\) Where: - \(k\) is the rate constant, - \(A\) is the pre-exponential factor, - \(E_a\) is the activation energy, - \(R\) is the gas constant (\(8.314\: \frac{J}{mol \cdot K}\)), - \(T\) is the temperature in Kelvin. To estimate the rate constant at the new temperature of 308 K, we will first need to determine the activation energy, \(E_a\). Given the initial rate of reaction, we will assume that the pre-exponential factor, \(A\), and the activation energy, \(E_a\), are constant for the reaction in both cases. Since we don't have the value of the activation energy, we cannot directly find the value of the rate constant at the other temperature, so our estimation will be limited in accuracy.

3. Estimate the rate at 308 K using the activation energy

Assuming the activation energy doesn't change significantly in the given temperature range and the pre-exponential factor is the same, we can approximate the new rate at 308 K by using a ratio of initial rates at both temperatures: \(\frac{Rate\: at\: 308\: K}{Rate\: at\: 298\: K} = \frac{k_{308}}{k_{298}}\) However, without the activation energy or the rate constants at both temperatures, we cannot directly estimate the new rate at 308 K. In conclusion, without additional information about the activation energy, or the rate constants at both temperatures, it is not possible to estimate the amount of oxygen gas produced by an identical solution in 100 s at 308 K.

Key concepts

Rate of Reaction

Understanding the rate of a chemical reaction is crucial in predicting how quickly reactants will convert into products. This rate can depend on various factors such as temperature, concentration of the reactants, surface area, and the presence of a catalyst. Typically, the rate is expressed as the change in concentration of a reactant or product over a specific time interval. In the exercise, the initial rate is calculated using the volume of oxygen gas produced over time (100 seconds).

For a reaction converting hydrogen peroxide to oxygen, knowing that 12 mL of oxygen gas is produced in 100 seconds at a certain temperature allows us to calculate the rate as 0.12 mL/s. This number represents how fast the reaction is occurring, which is a pivotal piece of data when experimenting or scaling up chemical processes for industrial applications.

Arrhenius Equation

The Arrhenius equation is a formula that relates the rate of a chemical reaction to temperature, providing significant insight into the temperature dependence of reaction rates. It is written as k = Ae^{-(Ea/RT)}, where k represents the rate constant of the reaction, A is the pre-exponential factor reflecting the number of correct orientations, Ea is the activation energy of the reaction, R is the gas constant, and T is the absolute temperature in Kelvin.

Applied to our exercise, the Arrhenius equation could help estimate the rate constant at a new temperature if the activation energy were known. Even without exact values for Ea and A, we could apply the concept to understand that as temperature increases, the rate constant k typically increases, explaining why reactions generally occur faster at higher temperatures.

Activation Energy

Activation energy, denoted as Ea, is the minimum energy needed for reactants to transform into products during a chemical reaction. It serves as a barrier that reactants must overcome to result in a successful reaction. A higher activation energy means that the reactants need more energy to start the reaction, which usually results in a slower reaction rate.

In the provided exercise, the activation energy is a key value that we need for predicting the new rate at a higher temperature. It also signifies the sensitivity of the reaction rate to temperature changes. Unfortunately, without the actual value of Ea, we can't accurately calculate the reaction's behavior at 308 K using the Arrhenius equation. Nevertheless, comprehending the role of activation energy is essential in chemical kinetics.

Gas Laws

Gas laws describe the behavior of gases in relation to pressure, volume, temperature, and quantity. These principles are critical when studying reactions that produce or consume gas, as in the decomposition of hydrogen peroxide in the exercise. The most commonly referenced gas laws are Boyle's Law (pressure and volume), Charles's Law (volume and temperature), and the Ideal Gas Law, which combines these relationships into one equation: PV = nRT.

Although gas laws weren't directly applied in the exercise solution, they could provide additional insight or even a different approach to predicting the volume change of gas at different temperatures. Remember that in chemical kinetics, gas laws augment our understanding when dealing with reactions involving gaseous substances, offering another angle from which to analyze and predict reaction behaviors.