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Define the rate of reaction in terms of changing concentrations for $$ a \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D} $$

Short Answer

Expert verified
The rate is equal to \(-\frac{1}{a} \frac{d[\mathrm{A}]}{dt} = -\frac{1}{b} \frac{d[\mathrm{B}]}{dt} = \frac{1}{c} \frac{d[\mathrm{C}]}{dt} = \frac{1}{d} \frac{d[\mathrm{D}]}{dt}\).

Step by step solution

01

Understand the Reaction Equation

The given chemical reaction is \( a \mathrm{~A} + b \mathrm{~B} \rightarrow c \mathrm{C} + d \mathrm{D} \). This reaction involves reactants \( \mathrm{A} \) and \( \mathrm{B} \), and products \( \mathrm{C} \) and \( \mathrm{D} \), with corresponding stoichiometric coefficients \( a, b, c, \) and \( d \) respectively.
02

Define Rate of Reaction

The rate of reaction refers to the speed at which the concentration of a reactant or product changes over time.
03

Rate in Terms of Reactants

For the reactants \( \mathrm{A} \) and \( \mathrm{B} \), the rate of reaction can be expressed as the negative change in concentration of the reactants per unit time. This is written as:\[ \text{Rate} = -\frac{1}{a} \frac{d[\mathrm{A}]}{dt} = -\frac{1}{b} \frac{d[\mathrm{B}]}{dt} \]
04

Rate in Terms of Products

For the products \( \mathrm{C} \) and \( \mathrm{D} \), the rate of reaction is the positive change in concentration of the products per unit time, which is given by:\[ \text{Rate} = \frac{1}{c} \frac{d[\mathrm{C}]}{dt} = \frac{1}{d} \frac{d[\mathrm{D}]}{dt} \]
05

Combine Definitions

Since the rate of reaction is the same regardless of whether we measure it in terms of reactants or products, we can write:\[ \text{Rate} = -\frac{1}{a} \frac{d[\mathrm{A}]}{dt} = -\frac{1}{b} \frac{d[\mathrm{B}]}{dt} = \frac{1}{c} \frac{d[\mathrm{C}]}{dt} = \frac{1}{d} \frac{d[\mathrm{D}]}{dt} \]

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Key Concepts

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

Concentration Change
In any chemical reaction, changes in concentrations of reactants and products are pivotal. These concentration changes help us understand how fast a reaction proceeds. Imagine it as watching ingredients disappear and new ones form in your cooking. In chemistry, this is quantified by observing how the quantity of a substance in moles per liter, decreases or increases over time.
For example, in the reaction \( a \mathrm{~A} + b \mathrm{~B} \rightarrow c \mathrm{C} + d \mathrm{D} \), concentration changes for reactants \( \mathrm{A} \) and \( \mathrm{B} \) mean they get used up, while products \( \mathrm{C} \) and \( \mathrm{D} \) are formed. By measuring these changes, we define the rate of reaction.
Think of the rate as a speedometer for chemical reactions. It's calculated as the rate at which the concentration of a reactant or product changes over time.
Stoichiometry
Stoichiometry is like the recipe of a chemical reaction. It tells us the proportions of reactants and products involved. When you look at \( a \mathrm{~A} + b \mathrm{~B} \rightarrow c \mathrm{C} + d \mathrm{D} \), stoichiometric coefficients \( a, b, c, \) and \( d \) directly impact how we calculate reaction rates.
Each coefficient indicates the amount required or produced in the reaction. For instance, if \( a = 2 \), you need 2 moles of \( \mathrm{A} \) to fully react as per the chemical equation. Similarly, if \( d = 3 \), production of 3 moles of \( \mathrm{D} \) is indicated.
  • Reactant coefficients show how much is consumed.
  • Product coefficients illustrate how much is produced.
Thus, stoichiometry helps us understand and predict how much of each substance is involved in or produced from a reaction.
Reactants and Products
Reactants and products are core ingredients in any chemical equation. Reactants, like our \( \mathrm{A} \) and \( \mathrm{B} \), are substances consumed during the reaction. Products, such as \( \mathrm{C} \) and \( \mathrm{D} \), are the new substances formed.
Understanding these roles is crucial. In the context of the earlier reaction, when you start with reactants, they undergo transformation to yield products. Essentially, reactants are the starting line and products are the finish line.
  • The reactants decrease in concentration as they convert to products.
  • The products increase in concentration as they are formed from reactants.
Monitoring how reactants turn into products gives us insight into the reaction’s dynamics.
Chemical Kinetics
Chemical kinetics is the study of the rates at which these chemical processes occur. It's like a window into how molecules interact and change over time to transform from reactants to products. In kinetics, we focus on understanding speeds and mechanisms involved.
For the reaction \( a \mathrm{~A} + b \mathrm{~B} \rightarrow c \mathrm{C} + d \mathrm{D} \), chemical kinetics addresses questions like how fast does \( \mathrm{A} \) and \( \mathrm{B} \) turn into \( \mathrm{C} \) and \( \mathrm{D} \), and under what conditions this process is optimal or slowed down.
Key aspects of kinetics include:
  • Calculating rates of reactions through concentration changes over time.
  • Understanding the role of temperature, catalysts, and other conditions affecting rates.
This branch of chemistry helps us not only predict but also control the outcomes of chemical reactions effectively.

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Most popular questions from this chapter

OBJECTIVE. Draw energy-level diagrams for catalyzed and uncatalyzed reactions. A catalyst decreases the activation energy of a particular exothermic reaction by \(15 \mathrm{~kJ} / \mathrm{mol}\), from 40 to \(25 \mathrm{~kJ} / \mathrm{mol}\). Assuming that the reaction is exothermic, that the mechanism has only one step, and that the products differ from the reactants by \(40 \mathrm{~kJ}\), sketch approximate energy-level diagrams for the catalyzed and uncatalyzed reactions.

Explain how temperature influences the rate of an uncatalyzed exothermic reaction and a catalyzed (lower- \(E_{a}\) ) exothermic reaction.

OBJECTIVE. Calculate the concentration-time behavior for a first-order reaction from the rate law and the rate constant. When formic acid is heated, it decomposes to hydrogen and carbon dioxide in a first-order decay. $$ \mathrm{HCOOH}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) $$ At \(550^{\circ} \mathrm{C}\), the half-life of formic acid is 24.5 minutes. (a) What is the rate constant, and what are its units? (b) How many seconds are needed for formic acid, initially \(0.15 M\), to decrease to 0.015 M?

Two reactions have activation energies of 45 and \(40 \mathrm{~kJ} / \mathrm{mol}\), respectively. Which reaction shows the greater increase in rate with an increase in temperature?

When methyl bromide reacts with hydroxide ion, methyl alcohol and bromide ion form. $$ \mathrm{CH}_{3} \mathrm{Br}+\mathrm{OH}^{-} \rightarrow \mathrm{CH}_{3} \mathrm{OH}+\mathrm{Br}^{-} $$ Consider the two mechanisms that follow, and write the expected rate law for each. (a) A two-step mechanism with the rate limited by the dissociation of methyl bromide: $$ \begin{array}{l} \mathrm{CH}_{3} \mathrm{Br} \stackrel{\text { Slow }}{\longrightarrow} \mathrm{H}_{3} \mathrm{C}^{+}+\mathrm{Br}^{-} \\ \mathrm{H}_{3} \mathrm{C}^{+}+\mathrm{OH}^{-} \stackrel{\text { Fast }}{\longrightarrow} \mathrm{CH}_{3} \mathrm{OH} \end{array} $$ (b) Formation of a transition state followed by fast rearrangement: $$ \mathrm{OH}^{-}+\mathrm{CH}_{3} \mathrm{Br} \rightarrow \mathrm{CH}_{3} \mathrm{OH}+\mathrm{Br}^{-} $$

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