The decomposition of NO2(g) occurs by the following...

The decomposition of $\mathrm{NO}_{2}(g)$ occurs by the following bimolecular elementary reaction: $$ 2 \mathrm{NO}_{2}(g) \longrightarrow 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) $$ The rate constant at $273 \mathrm{K}$ is $2.3 \times 10^{-12} \mathrm{L} \mathrm{mol}^{-1} \mathrm{s}^{-1}$ and the activation energy is $111 \mathrm{kJ} / \mathrm{mol} .$ How long will it take for the concentration of $\mathrm{NO}_{2}(g)$ to decrease from an initial partial pressure of 2.5 atm to 1.5 atm at $500 . \mathrm{K} ?$ Assume ideal gas behavior.

The decomposition of $\mathrm{NO}_{2}(g)$ occurs by the following bimolecular elementary reaction:
$$
2 \mathrm{NO}_{2}(g) \longrightarrow 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g)
$$
The rate constant at $273 \mathrm{K}$ is $2.3 \times 10^{-12} \mathrm{L} \mathrm{mol}^{-1} \mathrm{s}^{-1}$
and the activation energy is $111 \mathrm{kJ} / \mathrm{mol} .$ How long will it take for the concentration of $\mathrm{NO}_{2}(g)$ to decrease from an initial partial pressure of 2.5 atm to 1.5 atm at $500 . \mathrm{K} ?$ Assume ideal gas behavior.

Key Concepts

Bimolecular Elementary Reaction

An elementary reaction is one in which the rate law can be directly deduced from the stoichiometry of the reaction. In the case of a bimolecular elementary reaction, two reactant molecules collide to yield products, which means that the reaction rate is proportional to the product of the concentrations (or pressures, for gases) of the two reactants. This direct relation simplifies the kinetic analysis since the molecularity matches the reaction order.

Second Order Kinetics

When a reaction is bimolecular, it follows second order kinetics. The rate law for a second order reaction shows that the reaction rate is proportional to the square of the concentration of a single reactant (or the product of the concentrations of two different reactants). The integrated rate law for second order kinetics provides a relationship between concentration and time, allowing one to determine the time required for a concentration change under given conditions.

Arrhenius Equation

The Arrhenius equation describes how the rate constant of a chemical reaction depends on temperature and activation energy. It allows one to calculate the rate constant at a new temperature provided the activation energy is known, by accounting for the exponential increase in the rate constant with temperature. This concept is crucial in kinetic analyses that involve reactions performed under different thermal conditions.

Ideal Gas Behavior

Assuming ideal gas behavior implies that the partial pressures of gases are directly proportional to their molar concentrations. This relationship allows one to interchange pressures and concentrations in kinetic analyses for gaseous reactions, thereby simplifying calculations involving rate laws and reaction rates when pressure data is provided.

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