Chemical Kinetics Notes

Chemical Kinetics

Reaction Rate

The reaction rate is defined as the change in the concentration of a reactant or product per unit time.
It's measured by monitoring the disappearance of a reactant or the appearance of a product over time.

Defining Reaction Rate

For reactants, a negative sign is placed in front of the definition.
The rate of a generic reaction $A \rightarrow B$ can be expressed as:
- The rate at which $[A]$ decreases.
- The rate at which $[B]$ increases.
Rate units are typically molar per second, $M/s$ or $M \cdot s^{-1}$ .
Slower reactions may be expressed as $M/min$ or $M/hr$ .

Factors Affecting Reaction Rates

Particle Size of Solid Reactants (Surface Area):
- Influences how well the reactants come into contact with each other.
Concentration of Reactants:
- Influences how often reactant particles collide.
Temperature:
- Influences the kinetic energy with which reactant particles collide.
Nature (Potential Energy) of the Reactants
Presence of a Catalyst:
- Influences the energy needed to initiate the reaction.

Measuring Reaction Rates

Reaction rates can be measured as:
- Average rates
- Instantaneous rates
Average Rate: The concentration is measured at two times, and the change in concentration is divided by the change in time.
Instantaneous Rate: Measured by plotting concentration versus time and determining the slope of a tangent to that plot.
Instantaneous rates differ greatly as the overall rate of the reaction slows.

Instantaneous Rate

The rate is the slope of the tangent:
$Rate = -\frac{\Delta[A]}{\Delta t}$
$Rate = \frac{\Delta[B]}{\Delta t}$

Initial Rates

Chemists often determine instantaneous rates as close as possible to the start of the reaction (time zero); these are initial rates.
This is important because the rate of reaction slows over time as the reactant concentration decreases.

Example 15.1: Expressing Reaction Rates

Consider the balanced chemical equation:
In the first 10.0 seconds of the reaction, the concentration of $I^-$ drops from 1.000 M to 0.868 M
- Calculate the average rate of this reaction in this time interval.
- Determine the rate of change in the concentration of $H^+$ during this time interval.

Stoichiometry and Reaction Rate

For a reaction such as $A \rightarrow B$ , there is a 1:1 relationship between the disappearance of A and the appearance of B.
However, if the reaction is $2A \rightarrow B$ , then A disappears at twice the rate that B appears.
Many reactions involve more than one reactant or product, such as the reaction, $A \rightarrow B + 2C$ .
For the reaction, $aA \rightarrow bB$ , where a and b are stoichiometric coefficients:

The Rate Law

The rate of a reaction is directly proportional to the concentration of each reactant raised to a power.
For the reaction the rate law would have the form given below.
- $n$ is called the order; usually, it is an integer that determines rate dependence on reactant concentration.
- $k$ is called the rate constant.

Reaction Rates and Rate Laws

Rate law is the mathematical description for the dependence of a reaction rate upon the concentration of its reactants
- The general rate law for a reaction $aA \rightarrow products$ is
$n$ is called the order; usually, it is an integer that determines rate dependence on reactant concentration.
- $k$ is called the rate constant.

Reaction Order, n

Common values of n are 0, 1, and 2, although negative and fractional values are possible.
- When $n = 0$ , the reaction is zero order with respect to A.
- When $n = 1$ , the reaction is first order with respect to A.
- When $n = 2$ , the reaction is second order with respect to A.
When a reaction has multiple reactants, each reactant has its own order, and the overall reaction order is the sum of the orders with respect to each reactant.

Rate Law: aA + bB → Products

General rate law is
Determine the values of k, n, and m experimentally.
If n and m are both equal to 1, then the reaction is first order with respect to each reactant and second order overall.

Rate and Order of Reactions

In zero-order reactions, $n = 0$ , changing the concentration of the reactant has no effect on the rate of the reaction.
In first-order reactions, $n = 1$ , the rate law is simply, $rate = k[A]$ .
- If $[A]$ is doubled, the reaction rate also doubles.
- When $[A]$ drops to one-half of the initial concentration, the reaction rate is one-half the initial rate.
In a second-order reaction with a single reactant,
- $n = 2$
- $rate = k[A]^2$
- When the concentration of A is doubled, the rate of the reaction increases by a factor of $2^2 = 4$ . Thus, the rate is quadrupled.
- When the concentration of A drops to one-half of its initial concentration, the reaction rate is one-fourth of its initial rate.

Integrated Rate Laws

The rate law shows the relationship between rate and concentration.
It is useful to have an equation relating concentration with time.
Using calculus we can obtain the integrated rate law that shows the relationship between the concentration of A and the time of the reaction.

Half-Life

The half-life ( $t_{1/2}$ ) of a reaction is the time required for the concentration of the reactant to fall to half its initial value.
The half-life of the reaction depends on the order of the reaction.
For a first-order reaction, the half-life is constant and independent of concentration.

Rate Laws Summarized

Order	Rate Law	Units of k	Integrated Rate Law	Straight-Line Plot	Half-Life Expression
0	Rate = k[A]0 = k	M/s	$[A]<em>t = -kt + [A]</em>0$	linear is $[A]$ versus time,	$t<em>{1/2} = \frac{[A]</em>0}{2k}$
1	Rate = k[A]	s-1	$ln[A]<em>t = -kt + ln[A]</em>0$ $ln \frac{[A]<em>t}{[A]</em>0} = -kt$	linear is $ln[A]$ versus time	$t_{1/2} = \frac{0.693}{k}$
2	Rate = k[A]2	M-1s-1	$\frac{1}{[A]<em>t} = kt + \frac{1}{[A]</em>0}$	linear is $\frac{1}{[A]}$ versus time	$t<em>{1/2} = \frac{1}{k[A]</em>0}$
2	Rate = k[A][B]	M-1s-1
3	Rate = k[A]2[B]	M-2s-1

Graphical Determination of the Rate Law for A Gives Product

Plots of $[A]$ versus time, versus time allow determination of whether a reaction is zero, first, or second order.
Whichever plot gives a straight line determines the order with respect to $[A]$ .

The Effect of Temperature on Rate

The Arrhenius equation shows the relationship: $k = A e^{-\frac{E_a}{RT}}$
- T is the temperature in Kelvin
- R is the gas constant in energy units, $8.3124 J/(mol \cdot K)$
- A is called the frequency factor, the number of times the reactant energy approaches the activation energy per time
- Ea is the activation energy, the minimum energy needed to start the reaction

Reaction Energy Profile

Illustrates the activation energy required for a reaction.

Activation Energy and the Activated Complex

There is an energy barrier to almost all reactions.
The activation energy is the amount of energy needed to convert reactants into the activated complex.
- Also known as transition state
The activated complex is a chemical species with partially broken and partially formed bonds.
- Always very high in energy because of its partial bonds

Arrhenius Plots

The Arrhenius equation can be algebraically solved to give the following form:

Collision Theory

In collision theory, for a reaction to take place, the reacting molecules must collide with one another.
Once molecules collide, they may react together or they may not, depending on two factors:
- Whether the collision has enough energy to “break the bonds holding reactant molecules together”
- Whether the reacting molecules collide in the proper orientation for new bonds to form

The Frequency Factor of the Arrhenius Equation

The Arrhenius equation includes a term, A, called the frequency factor.
The frequency factor can be broken into two terms that relate to the two factors that determine whether a collision will be effective.

Effective Collisions

Collisions that lead to reaction are called effective collisions.
The higher the frequency of effective collisions, the faster the reaction rate.
The molecules must be aligned in a very specific way for the reaction to occur.

Reaction Mechanisms

The reaction mechanism is the series of individual chemical steps by which an overall chemical reaction occurs.

Rate Laws of Elementary Steps

Elementary Step	Molecularity	Rate Law
A gives products	1	Rate = k [A]
A + A gives products	2	Rate = k [A]^2
A + B gives products	2	Rate = k [A] [B]
A + A + A gives products	3 (rare)	Rate = k [A]^3
A + A + B gives products	3 (rare)	Rate = k [A]^2 [B]
A + B + C gives products	3 (rare)	Rate = k [A] [B] [C]

Rate-Determining Step

In most mechanisms, one step occurs much slower than the other steps.
We call the slowest step in the mechanism the rate-determining step (RDS).
- The slowest step has the largest activation energy.
The rate law of the rate-determining step determines the rate law for the overall reaction.

Catalysts

Catalysts are substances that affect the rate of a reaction without being consumed.
Catalysts work by providing an alternative mechanism for the reaction with a lower activation energy.

Homogeneous and Heterogeneous Catalysis

Homogeneous catalysts are in the same phase as the reactant particles.
- They hold one reactant molecule in proper orientation for reaction to occur when the collision takes place, sometimes helping to start breaking bonds.
Heterogeneous catalysts are in a different phase than the reactant particles.
- They react with one of the reactant molecules to form a more stable activated complex with a lower activation energy.