Reaction Rate and Rate Expressions

Reaction Rate

• Reaction rate measures how quickly reactants are used up or products are formed.
• Reactions typically slow down over time.

Factors that Influence Reaction Rate

• Reactant concentration affects reaction rate.
• Temperature impacts reaction rate.
• Catalysts influence reaction rate.
• Surface area is a factor in reaction rate.

Rate Expression Explanation

• For a reaction: $aA + bB \rightarrow cC + dD$
• The rate expression is: rate $= -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$
• Reactant rates are expressed as negative values.
• Product rates are expressed as positive values.

Example Rate Expression

• For the reaction $2HI(g) \rightarrow H_2(g) + I_2(g)$
• The rate expression is: rate $= -\frac{1}{2}\frac{\Delta[HI]}{\Delta t} = \frac{\Delta[H_2]}{\Delta t} = \frac{\Delta[I_2]}{\Delta t}$

Instantaneous Reaction Rate

• Instantaneous rate is the rate at a specific point in time during the reaction.
• It is equivalent to the slope of the curve at that particular point.

Rate Law Basics

• A rate law shows the relationship between reaction rate and reactant concentrations.
• For a reaction $aA + bB \rightarrow cC + dD$
• The rate law is expressed as: rate $= k[A]^m[B]^n$
• 'k' represents the rate constant.
• 'm' is the order of reaction with respect to reactant A.
• 'n' is the order of reaction with respect to reactant B.
• 'm + n' gives the overall reaction order.
• The values of 'm' and 'n' must be found through experiments.
• The rate constant 'k' changes with temperature.

Understanding Reaction Order

• Reaction order is the exponent that indicates how reactant concentration affects rate.
• The overall reaction order is the sum of all the exponents.
• Rate = k represents a zero-order reaction.
• Rate = k[A] represents a first-order reaction.
• Rate = k[A]^2 represents a second-order reaction.

Methods for Determining Rate Laws

• Initial rates are used to determine rate laws.

Example Problem

• Reaction: $NH_4^+(aq) + NO_2^-(aq) \rightarrow N_2(g) + 2H_2O(l)$
• Experimental data is used to find the rate law.
• Comparing experiments allows determination of reaction orders (m and n).
• The rate law is found to be rate $ = k[NH_4^+][NO_2^-]$.
• Substituting values into the rate law, k is: $k = 2.7 \times 10^{-4} M^{-1}s^{-1}$.

Integrated Rate Laws Overview

• Integrated rate laws illustrate the relationship between reactant concentrations and time.

Zero-Order Reactions Explained

• Rate equation: rate $= -\frac{\Delta[A]}{\Delta t} = k$
• Integrated rate law: $[A]_t = -kt + [A]_0$
• $[A]_t$ is the concentration of A at time t.
• $[A]_0$ is the initial concentration of A.
• Plotting [A] against t yields a straight line with a slope of -k.

First-Order Reactions Explained

• Rate equation: rate $= -\frac{\Delta[A]}{\Delta t} = k[A]$
• Integrated rate law: $ln[A]_t = -kt + ln[A]_0$
• Can be written as: $ln[A]_t - ln[A]_0 = -kt$
• Or: $ln\frac{[A]_t}{[A]_0} = -kt$
• A graph of ln[A] vs. t produces a straight line with a slope of -k.

Second-Order Reactions Explained

• Rate equation: rate $= -\frac{\Delta[A]}{\Delta t} = k[A]^2$
• Integrated rate law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
• Graphing $\frac{1}{[A]}$ against t yields a linear plot with a slope of k.

Understanding Half-Life

• Half-life ($t_{1/2}$) is the time for a reactant's concentration to decrease to half its initial value.

Equations for Half-Life

• Zero order: $t_{1/2} = \frac{[A]_0}{2k}$
• First order: $t_{1/2} = \frac{0.693}{k}$
• Second order: $t_{1/2} = \frac{1}{k[A]_0}$

Basic Collision Model

• Reaction depends on molecular collisions.
• Collisions must occur with enough energy.
• Molecules must collide with the proper orientation.

Understanding Activation Energy

• Activation Energy ($E_a$) is the minimum energy needed for a reaction to occur.

Arrhenius Equation Defined

• The Arrhenius equation mathematically relates rate constant to temperature and activation energy.
• The equation is: $k = Ae^{-E_a/RT}$
• 'k' is the rate constant.
• 'A' is the frequency factor.
• '$E_a$' is the activation energy.
• 'R' is the gas constant (8.314 J/mol⋅K).
• 'T' is the temperature in Kelvin.
• The equation can also be expressed as: $ln(k) = ln(A) - \frac{E_a}{RT}$
• Or: $ln(\frac{k_1}{k_2}) = \frac{E_a}{R}(\frac{1}{T_2} - \frac{1}{T_1})$

Reaction Mechanism Overview

• A reaction mechanism outlines the series of steps in a reaction.

Elementary Step Clarification

• An elementary step is a single step within a reaction mechanism.

Molecularity Explained

• Molecularity is the number of molecules participating in an elementary step.
  • Unimolecular: one molecule.
  • Bimolecular: two molecules.
  • Termolecular: three molecules.

Rate-Determining Step Defined

• The rate-determining step is the slowest step in the mechanism.
• This step dictates the rate law for the overall reaction.

About Intermediates

• Intermediates are species formed and consumed within the reaction mechanism.
• They do not appear in the overall balanced equation.

Catalyst Role

• A catalyst increases reaction rate without being consumed.
• It provides an alternate mechanism with lower activation energy.
  • Homogeneous catalysts are in the same phase as reactants.
  • Heterogeneous catalysts are in a different phase from reactants.