Question

The gas-phase decomposition of peroxyacetyl nitrate (PAN) obeys first-order kinetics: $$ \mathrm{CH}_{3} \mathrm{COOONO}_{2} \longrightarrow \mathrm{CH}_{3} \mathrm{COOO}+\mathrm{NO}_{2} $$ with a rate constant of \(4.9 \times 10^{-4} \mathrm{~s}^{-1} .\) Calculate the rate of decomposition (in \(M / \mathrm{s}\) ) if the concentration of PAN is 0.55 ppm by volume. Assume STP conditions.

Short answer

The rate of decomposition is \(1.20 \times 10^{-11} \text{ M/s}\).

Step-by-step solution

Convert ppm to Molarity

Under Standard Temperature and Pressure (STP), 1 mole of gas occupies 22.4 L. Thus, to convert parts per million (ppm) to molarity (M), we use the formula:\[\text{Molarity (M)} = \frac{\text{ppm} \times 1 \text{ atm}}{22.4 \times 10^6} \]Plugging in the given values:\[\text{Molarity} = \frac{0.55 \times 1}{22.4 \times 10^6} = 2.455 \times 10^{-8} \text{ M}.\]

Apply First-Order Rate Law

The rate of decomposition for a first-order reaction is given by the equation:\[\text{Rate} = k [A]\]where \(k\) is the rate constant and \([A]\) is the concentration in molarity. Here, \(k = 4.9 \times 10^{-4} \text{ s}^{-1}\) and \([A] = 2.455 \times 10^{-8} \text{ M}\).

Calculate the Rate of Decomposition

Using the formula and values from the previous steps, we substitute to find the rate:\[\text{Rate} = 4.9 \times 10^{-4} \times 2.455 \times 10^{-8} = 1.20295 \times 10^{-11} \text{ M/s}\]Therefore, the rate of decomposition is \(1.20 \times 10^{-11} \text{ M/s}\).

Key concepts

Rate of Decomposition

The rate of decomposition tells us how fast a chemical reaction occurs, specifically how quickly a reactant is transformed into products. For first-order reactions, this rate depends directly on the concentration of the reactant. In our case, the decomposition of peroxyacetyl nitrate (PAN) is a first-order reaction. Therefore, the rate of this reaction can be calculated using the equation: \[ \text{Rate} = k [A] \] where:

• \( k \) is the rate constant, which has been given as \( 4.9 \times 10^{-4} \text{ s}^{-1} \).
• \([A]\) is the concentration of PAN in molarity, which we previously converted from ppm.

Using this relationship, we find that the rate of decomposition is \( 1.20 \times 10^{-11} \text{ M/s} \).
This rate tells us the number of moles of PAN that decompose per second in each liter.

Peroxyacetyl Nitrate

Peroxyacetyl nitrate, often abbreviated as PAN, is a significant pollutant often found in photochemical smog. It is a reactive compound that can have various harmful effects on both the environment and human health, as it is a strong eye irritant and can damage vegetation. PAN is formed from the reaction of volatile organic compounds and nitrogen oxides in the presence of sunlight, making it a component of urban air pollution.

Understanding its rate of decomposition helps predict its behavior in the atmosphere. As it breaks down into acetic acid and nitrogen dioxide, knowing its reaction kinetics allows us to understand how long it will persist in the air and its potential environmental impact.

Standard Temperature and Pressure

Standard Temperature and Pressure (STP) is a reference point used in chemistry to provide a common temperature and pressure conditions for experiments and reactions. STP is defined as a temperature of 0 degrees Celsius (273.15 K) and a pressure of 1 atmosphere (atm). These conditions are essential for consistency when comparing chemical data across different experiments.

At STP, 1 mole of an ideal gas occupies 22.4 liters. This relationship is crucial when converting between volumes of gases, like understanding how the ppm (parts per million) concentration of a gas can be translated into molarity, as done in our example with peroxyacetyl nitrate.

PPM to Molarity Conversion

Understanding how to convert ppm (parts per million) to molarity is essential in chemistry, as it allows us to work with concentrations that are more relevant to reaction equations. PPM expresses the number of parts of a substance in a million parts of the whole. For gases, under STP conditions, this is related to the volume they occupy.

To convert ppm to molarity, use the following formula:

• \[ \text{Molarity (M)} = \frac{\text{ppm} \times 1 \text{ atm}}{22.4 \times 10^6} \]

This equation accounts for the volume that a mole of gas would occupy at STP.
In our exercise, a concentration of 0.55 ppm was converted to a molarity of \(2.455 \times 10^{-8} \text{ M}\), making it possible to use this value to calculate the rate of decomposition as per the first-order kinetics formula.