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Chapter 13: Problem 26

Can the average rate and instantaneous rate of a chemical reaction ever be the same?

Short Answer

Expert verified
Answer: Yes, the average rate and instantaneous rate of a chemical reaction can be the same if the reaction rate remains constant over the given time interval. In most chemical reactions, the rate changes over time, so the average and instantaneous rates will generally differ.

Step by step solution

01

Definition of Average Rate

The average rate of a chemical reaction measures the change in concentration of a reactant or product over a specific time interval. It's calculated as the change in concentration divided by the change in time.
02

Definition of Instantaneous Rate

The instantaneous rate of a chemical reaction represents the rate at which a reactant or product concentration changes at a specific moment in time. It can be thought of as the limit of the average rate as the time interval approaches zero. In mathematical terms, it's the derivative of the concentration with respect to time.
03

Comparing Average and Instantaneous Rates

Since the average rate measures the change in concentration over a specific time interval and the instantaneous rate measures the change in concentration at an exact moment in time, they are generally different. However, under certain conditions, they can be equal.
04

Conditions for Equal Average and Instantaneous Rates

If the reaction rate remains constant over the given time interval, the average rate and instantaneous rate will be the same. This is true because if the rate does not change, the concentration will change linearly with time, and the derivative of a linear function is constant. To sum up, the average rate and instantaneous rate of a chemical reaction can be the same only if the reaction rate remains constant over the given time interval. In most chemical reactions, the rate changes over time, so the average and instantaneous rates will generally differ.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Rate
The average rate of a chemical reaction is a way to express how fast reactants turn into products over a certain period of time. Imagine you're watching a race. Instead of noting a sprinter's speed at any given second, you calculate their average speed over the whole race. For chemical reactions, this is done by looking at concentration changes over a specific time frame.
  • Formula: \( \text{Average Rate} = \frac{\text{Change in Concentration}}{\text{Change in Time}} \)
  • Useful for predicting how much reactant is used or product is made over minutes or hours.

Think of it as a broad overview that tells us about the progression of a reaction and helps in planning further experiments.
Instantaneous Rate
The instantaneous rate zooms in on a chemical reaction at a specific point in time. It's like using a speedometer in a car that tells you the exact speed at that moment, rather than the average speed for the entire trip.
  • Mathematically, it鈥檚 the slope of the tangent to the concentration-time curve at a particular time.
  • Given by the derivative \( \frac{d[A]}{dt} \), where \([A]\) is the concentration of a reactant or product.

This helps in understanding the dynamics of the reaction precisely at that moment, and it's essential for kinetic studies.
Reaction Rate Constant
The reaction rate constant, often symbolized as \(k\), is a key parameter in the rate equation of a reaction. It's unique to each reaction and depends on factors like temperature and the presence of a catalyst.
  • Appears in the rate law: \( \text{Rate} = k[A]^m[B]^n \)
  • Determines how quickly a reaction reaches completion.

Having a large \(k\) indicates a fast reaction, while a small value signifies a slow one. It's crucial for calculating both average and instantaneous rates.
Concentration Changes
Concentration changes are central to understanding chemical reaction rates. When concentrations of reactants or products shift, they directly impact how fast the reaction is progressing.
  • A decrease in reactant concentration can slow down the reaction over time.
  • An increase in product concentration generally indicates the reaction is proceeding.

By observing these changes, chemists can infer reaction progress and predict future rates. This information helps optimize conditions for desired outcomes in both industrial and laboratory settings.

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Most popular questions from this chapter

Under what circumstances is the activation energy of a reaction proceeding in the forward direction greater than the activation energy of it happening in reverse?

The rate constant for the reaction of ozone with oxygen atoms was determined at four temperatures. Calculate the activation energy and frequency factor \(A\) for the reaction $$\mathrm{O}(g)+\mathrm{O}_{3}(g) \rightarrow 2 \mathrm{O}_{2}(g)$$ given the following data: $$\begin{array}{cc}T(\mathrm{K}) & k\left[\mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\right] \\\250 & 2.64 \times 10^{-4} \\\\\hline 275 & 5.58 \times 10^{-4} \\\\\hline 300 & 1.04 \times 10^{-3} \\\\\hline 325 & 1.77 \times 10^{-3} \\\\\hline\end{array}$$

The rate laws for the thermal and photochemical decomposition of \(\mathrm{NO}_{2}\) are different. Which of the following mechanisms are possible for the thermal decomposition of \(\mathrm{NO}_{2},\) and which are possible for the photochemical decomposition of \(\mathrm{NO}_{2}\) ? For the thermal decomposition, Rate \(=k\left[\mathrm{NO}_{2}\right]^{2},\) and for the photochemical decomposition, Rate \(=k\left[\mathrm{NO}_{2}\right]\). a. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{4}(g)\) \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \stackrel{\text { fast }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g)\) \(\mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g) \stackrel{\text { fast }}{\mathrm{N}_{2} \mathrm{O}_{2}(g)} \stackrel{\mathrm{fast}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{2}(g)+\mathrm{O}_{2}(g)\) \(\quad \quad \mathrm{NO}(g)\) b. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{NO}_{3}(g)\) \(\mathrm{NO}_{3}(g) \stackrel{\mathrm{fast}}{\longrightarrow} \mathrm{NO}(g)+\mathrm{O}_{2}(g)\) c. \(\quad \mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{N}(g)+\mathrm{O}_{2}(g)\) \(\begin{aligned} \mathrm{N}(g)+& \mathrm{NO}_{2}(g) \frac{\mathrm{fast}}{\mathrm{N}_{2} \mathrm{O}_{2}(g)} \mathrm{N}_{2} \mathrm{O}_{2}(g) \\ & \stackrel{\text { fast }}{\longrightarrow} \mathrm{NO}(g) \end{aligned}\)

The rate law for the reaction of NO with \(\mathrm{Cl}_{2}\) (Rate \(\left.=k[\mathrm{NO}]\left[\mathrm{Cl}_{2}\right]\right)\) is the same as that for the reaction of \(\left.\mathrm{NO}_{2} \text { with } \mathrm{F}_{2} \text { (Rate }=k\left[\mathrm{NO}_{2}\right]\left[\mathrm{F}_{2}\right]\right) .\) Is it possible that these reactions have similar mechanisms?

Does a catalyst affect both the rate and the rate constant of a reaction?
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