Given that the first-order rate constant for the overall...

Given that the first-order rate constant for the overall decomposition of $\mathrm{N}_{2} \mathrm{O}_{5}$ is $k=\left(4.3 \times 10^{13} \mathrm{s}^{-1}\right) \mathrm{e}^{-103000 / R T},$ calculate $(a)$ the half-life at $-10^{\circ} \mathrm{C}$ and $(b)$ the time required for $90 \%$ reaction at $50^{\circ} \mathrm{C}$. The activation energy is in $\mathrm{J} \mathrm{mol}^{-1}$.

Key Concepts

Reaction Completion Time Calculation

For first-order reactions, the time required for a specified fraction of a reactant to react can be calculated using the integrated rate law, resulting in an expression like t = -ln(remaining fraction) / k. For example, determining the time for 90% completion involves solving for t when the remaining fraction is 0.10. This approach is critical for predicting how long a reaction must proceed to reach a desired level of conversion.

Half-Life in First-Order Reactions

The half-life of a first-order reaction, defined as the time required for half of the reactant to be consumed, is given by t1/2 = ln 2 / k. This expression is independent of the initial concentration of the reactant, making the half-life a convenient parameter for characterizing the kinetics of first-order processes.

First-Order Reaction Kinetics

In first-order reactions, the rate of reaction is directly proportional to the concentration of a single reactant. This linear dependence simplifies the mathematics of chemical kinetics, leading to an exponential decay in reactant concentration over time. First-order kinetics is described by the differential equation d[A]/dt = -k[A], where k is the first-order rate constant.

Temperature Conversion

When applying the Arrhenius equation and other kinetic formulas, temperature must be expressed in Kelvin. Converting temperatures from Celsius to Kelvin is achieved by adding 273.15 to the Celsius value. This conversion is essential because the kinetic energy of molecules, which drives chemical reactions, is directly related to temperature measured on the Kelvin scale.

Arrhenius Equation

The Arrhenius equation describes how the rate constant (k) of a chemical reaction depends exponentially on the inverse of the temperature (in Kelvin) and the activation energy of the reaction. It is expressed as k = A exp(-Ea/RT), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. This relationship is fundamental in understanding temperature effects on reaction rates.

Activation Energy

Activation energy is the minimum energy barrier that must be overcome for reactants to convert into products during a chemical reaction. In the context of the Arrhenius equation, a higher activation energy implies a lower rate constant at a given temperature, as fewer molecules have sufficient energy to react. This concept is crucial for predicting how changes in temperature influence the reaction rate.

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