The first-order rate constant for the decomposition of N2O5, 2 N2O5(g) → 4 NO2(g) + O2(g), at 70 °C is 6.82 * 10-3 s-1. Suppose we start with 0.0250 mol of N2O5(g) in a volume of 2.0 L. (b) How many minutes will it take for the quantity of N2O5 to drop to 0.010 mol?

Verified step by step guidance

First, identify the type of reaction and the order. This is a first-order reaction, which means the rate of reaction is directly proportional to the concentration of the reactant.

Use the first-order rate equation: \( [A] = [A]_0 e^{-kt} \), where \([A]\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time.

Calculate the initial concentration \([A]_0\) of \(N_2O_5\) using the formula \([A]_0 = \frac{n}{V}\), where \(n\) is the initial moles (0.0250 mol) and \(V\) is the volume (2.0 L).

Substitute the known values into the first-order rate equation: \(0.010 = 0.0125 e^{-6.82 \times 10^{-3} \times t}\). Solve for \(t\) by isolating \(e^{-kt}\) and taking the natural logarithm of both sides.

Convert the time \(t\) from seconds to minutes by dividing by 60, since the rate constant \(k\) is given in \(s^{-1}\).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 integrated rate laws to calculate the concentration of reactants 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 how long it takes for the concentration of a reactant to decrease to a specific value, which is essential for solving the given problem.

Concentration and Molarity

Concentration, often expressed in molarity (M), is defined as the number of moles of solute per liter of solution. In this problem, we need to calculate the initial concentration of N2O5 using the formula M = moles/volume. Understanding how to convert between moles and molarity is crucial for applying the integrated rate law and determining the time required for the concentration to drop.

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Textbook Question

As described in Exercise 14.41, the decomposition of sulfuryl chloride (SO₂Cl₂) is a first-order process. The rate constant for the decomposition at 660 K is 4.5 × 10^-2 s^-1. (a) If we begin with an initial SO₂Cl₂ pressure of 450 torr, what is the partial pressure of this substance after 60 s?

Textbook Question

As described in Exercise 14.41, the decomposition of sulfuryl chloride (SO₂Cl₂) is a first-order process. The rate constant for the decomposition at 660 K is 4.5 × 10^-2 s^-1. (b) At what time will the partial pressure of SO₂Cl₂ decline to one-tenth its initial value?

Textbook Question

The first-order rate constant for the decomposition of N₂O₅, 2 N₂O₅(g) → 4 NO₂(g) + O₂(g), at 70°C is 6.82×10^-3 s^-1. Suppose we start with 0.0250 mol of N₂O₅(g) in a volume of 2.0 L. (a) How many moles of N₂O₅ will remain after 5.0 min?

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Textbook Question