The reaction AB(aq) → A(g) + B(g) is second order in AB and has a rate constant of 0.0118 M^-1 s^-1 at 25.0 °C. A reaction vessel initially contains 250.0 mL of 0.100 M AB that is allowed to react to form the gaseous product. The product is collected over water at 25.0 °C. How much time is required to produce 200.0 mL of the products at a barometric pressure of 755.1 mmHg? (The vapor pressure of water at this temperature is 23.8 mmHg.)

Verified step by step guidance

Identify the initial concentration of AB, which is 0.100 M, and the initial volume of the solution, which is 250.0 mL.

Use the ideal gas law to determine the moles of gas produced. First, calculate the pressure of the gas by subtracting the vapor pressure of water from the barometric pressure: \( P_{gas} = 755.1 \text{ mmHg} - 23.8 \text{ mmHg} \).

Convert the volume of the gas produced (200.0 mL) to liters and use the ideal gas law \( PV = nRT \) to find the moles of gas, where \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \) and \( T = 298 \text{ K} \).

Determine the change in concentration of AB using the stoichiometry of the reaction and the moles of gas produced. Since the reaction is second order, use the integrated rate law for a second-order reaction: \( \frac{1}{[AB]_t} = \frac{1}{[AB]_0} + kt \).

Solve for the time \( t \) using the integrated rate law, where \( k = 0.0118 \text{ M}^{-1} \text{ s}^{-1} \), \([AB]_0 = 0.100 \text{ M}\), and \([AB]_t\) is the concentration of AB after the reaction.

Key Concepts

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

Reaction Order

The order of a reaction indicates how the rate of the reaction depends on the concentration of the reactants. In this case, the reaction is second order in AB, meaning that the rate is proportional to the square of the concentration of AB. Understanding reaction order is crucial for determining how changes in concentration affect the rate and for applying the appropriate rate laws to calculate time or concentration changes.

Rate Constant

The rate constant (k) is a proportionality factor in the rate law that relates the rate of a reaction to the concentrations of the reactants. For a second-order reaction, the units of k are M^-1 s^-1. The value of the rate constant is temperature-dependent, and knowing it allows for the calculation of the reaction rate and the time required to reach a certain concentration of products.

Gas Laws and Vapor Pressure

Gas laws describe the behavior of gases under various conditions of temperature and pressure. In this scenario, the vapor pressure of water at 25.0 °C must be considered when calculating the volume of gaseous products collected over water. The total pressure of the gas collected is the sum of the partial pressures of the gas and the water vapor, which is essential for determining the actual volume of the gaseous products produced.

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Open Question

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Open Question

The reaction 2 H2O2(aq) → 2 H2O(l) + O2(g) is first order in H2O2 and under certain conditions has a rate constant of 0.00752 s⁻¹ at 20.0 °C. A reaction vessel initially contains 150.0 mL of 30.0% H2O2 by mass solution (the density of the solution is 1.11 g/mL). The gaseous oxygen is collected over water at 20.0 °C as it forms. What volume of O2 forms in 85.0 seconds at a barometric pressure of 742.5 mmHg? (The vapor pressure of water at this temperature is 17.5 mmHg.)

Textbook Question

Consider this energy diagram:

a. How many elementary steps are involved in this reaction?

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Textbook Question

Consider this energy diagram:

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b. Label the reactants, products, and intermediates.