Question

For the first order reaction $$ 2 \mathrm{~N}_{2} \mathrm{O}_{5}(\mathrm{~g}) \rightarrow 4 \mathrm{NO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) $$ (A) the concentration of the reactant decreases exponentially with time. (B) the half-life of the reaction decreases with increasing temperature. (C) the half-life of the reaction depends on the initial concentration of the reactant. (D) the reaction proceeds to \(99.6 \%\) completion in eight half-life duration.

Short answer

A) True, B) False, C) False, D) True. The concentration decreases exponentially (A), but the half-life does not depend on initial concentration (C), nor does it decrease with increasing temperature (B). The reaction nearly completes after eight half-lives (D).

Step-by-step solution

- Analyzing Given Information

Understand that the reaction is a first-order reaction, which implies that the rate of the reaction is directly proportional to the concentration of the reactant. The general form of a first-order reaction can be given by the rate law: rate = -d[Reactant]/dt = k[Reactant], where k is the rate constant.

- Determine the Nature of Reactant Concentration Change

Since the rate of reaction is proportional to the concentration of the reactant, the concentration of the reactant changes exponentially with time. This is characterized by the equation [Reactant] = [Reactant]_0 * e^{-kt}, where [Reactant]_0 is the initial concentration.

- Evaluate the Half-Life Dependency on Temperature

According to the Arrhenius equation, the rate constant (k) increases with increasing temperature. Since the half-life of a first-order reaction is given by t_1/2 = ln(2)/k, an increase in k due to an increase in temperature will decrease the half-life of the reaction.

- Investigate the Half-Life Concentration Dependency

For a first-order reaction, the half-life is defined as t_1/2 = ln(2)/k and is constant. It does not depend on the initial concentration of the reactant.

- Assess the Completion Level After Eight Half-Lives

After one half-life, 50% of the reactant remains. After each successive half-life, half of the remaining reactant is consumed. After eight half-lives, the fraction of reactant remaining is (1/2)^8 = 1/256. Therefore, the reaction proceeds to completion by 1 - (1/256) = 255/256 or approximately 99.6%.

Key concepts

Rate Law

When studying chemical reactions, scientists use the rate law to state the relationship between the rate of a reaction and the concentration of the reactants. In a first-order reaction, like the decomposition of dinitrogen pentoxide (\r(2 N_2O_5 \rightarrow 4 NO_2 + O_2\r)), the rate at which the reaction progresses is directly proportional to the single reactant's concentration present.\r
\rThe rate law for a first-order reaction is expressed mathematically as \r\(\text{rate} = -\frac{d[\text{Reactant}]}{dt} = k[\text{Reactant}]\).\r Here, \r\(k\)\r is the rate constant, a signature value for each reaction at a given temperature, and \r\([\text{Reactant}]\)\r stands for the concentration of the reactant. The negative sign indicates that the concentration of the reactant decreases over time. Understanding the rate law is crucial for predicting how changes in conditions affect how fast a reaction will occur.

Half-Life of a Reaction

Half-life, in the context of a chemical reaction, is the time it takes for half of the reactant to be used up or converted into products. For a first-order reaction, the half-life is uniquely constant and is not dependent on the initial concentration of the reactant. This means that no matter how much reactant you start with, it will always take the same amount of time for half of it to react.\r
\rThe half-life for a first-order reaction can be calculated using the formula \r\(t_{1/2} = \frac{\text{ln}(2)}{k}\).\r Since \r\(k\)\r is in the denominator, if the rate constant increases, the half-life decreases. What's more, the half-life remains the same throughout the reaction, signifying that it's a powerful tool to understand the reaction dynamics without the need to track the entire reaction process.

Arrhenius Equation

The rate of a chemical reaction is temperature-dependent, and this relationship is described by the Arrhenius equation, a pivotal concept in chemical kinetics. The Arrhenius equation is \r\(k = A e^{-\frac{E_a}{RT}}\)\r, where \r\(k\)\r is the rate constant, \r\(A\)\r is the frequency factor (how often molecules collide), \r\(E_a\)\r is the activation energy (minimum energy required for a reaction to occur), \r\(R\)\r is the gas constant, and \r\(T\)\r is the temperature in kelvins.\r
\rA key takeaway is that as the temperature increases, the rate constant \r\(k\)\r also increases, implying reactions go faster at higher temperatures. And since the half-life of a first-order reaction is inversely related to the rate constant (\r\(t_{1/2} \backsim \frac{1}{k}\)\r), a rise in temperature will reduce the half-life, corroborating option (B) in the exercise which states the half-life decreases with increasing temperature.

Exponential Decay

Exponential decay is a pattern of decrease over time that is often observed in the context of first-order reactions. This type of decay indicates that the rate at which a substance decreases is proportional to its current amount. For first-order reactions, this is exemplified through the equation \r\([\text{Reactant}] = [\text{Reactant}]_0 e^{-kt}\)\r where \r\([\text{Reactant}]_0\)\r is the initial concentration and \r\(e^{-kt}\)\r represents the exponential decay over time.\r
\rIn practice, this means that no matter how close the reaction gets to completion, there will always be a small fraction of the original reactant left, approaching zero but never quite reaching it completely. Understanding exponential decay is crucial for making predictions about how long a reaction will take to reach a certain point of completion—something very valuable in industries that rely on precise chemical processes.