The first-order rate constant for the decomposition of N2O5, 2 N2O51g2¡4 NO21g2 + O21g2, a t 70 C i s 6.82 * 10-3 s-1. Suppose we start with 0.0250 mol of N2O51g2 in a volume of 2.0 L. (a) How many moles of N2O5 will remain after 5.0 min?

Verified step by step guidance

1. Identify the type of reaction and the order: The problem states that the decomposition of \( \text{N}_2\text{O}_5 \) is a first-order reaction.
2. Use the first-order rate equation: \( [A]_t = [A]_0 e^{-kt} \), where \([A]_t\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time.
3. Calculate the initial concentration \([A]_0\): \([A]_0 = \frac{\text{moles of } \text{N}_2\text{O}_5}{\text{volume}} = \frac{0.0250 \text{ mol}}{2.0 \text{ L}}\).
4. Convert the time from minutes to seconds: \(5.0 \text{ min} = 5.0 \times 60 \text{ s/min} = 300 \text{ s}\).
5. Substitute the known values into the first-order rate equation to find \([A]_t\): \([A]_t = [A]_0 e^{-6.82 \times 10^{-3} \text{ s}^{-1} \times 300 \text{ s}}\).

Key Concepts

First-Order Kinetics

First-order kinetics refers to a reaction rate that is directly proportional to the concentration of one reactant. In this case, the decomposition of N2O5 follows first-order kinetics, meaning the rate of reaction can be expressed as rate = k[N2O5], where k is the rate constant. This relationship allows us to use the integrated rate law to calculate the concentration of N2O5 over time.

Integrated Rate Law

The integrated rate law for a first-order reaction is given by the equation ln([A]0/[A]) = kt, where [A]0 is the initial concentration, [A] is the concentration at time t, k is the rate constant, and t is time. This equation allows us to determine the concentration of a reactant at any given time, which is essential for solving the problem of how many moles of N2O5 remain after a specified duration.

Molarity and Moles

Molarity (M) is defined as the number of moles of solute per liter of solution. To find the remaining moles of N2O5 after a certain time, we first need to calculate the initial molarity using the initial moles and volume. Understanding the relationship between moles, volume, and molarity is crucial for converting between these units and applying the integrated rate law effectively.