Unit 2: Chemical Kinetics: An Exam-Ready Study Resource

Chemical kinetics is the branch of physical chemistry that deals with the study of the rates of chemical reactions, the factors affecting the rates, and the mechanism by which reactions proceed. Understanding chemical kinetics is crucial for predicting how fast a reaction will occur and optimizing reaction conditions in various applications.

The Concept of Reaction Rates

The rate of a reaction refers to the change in concentration of a reactant or product per unit time. It quantifies how quickly reactants are consumed and products are formed.

Rate = Change in concentration / Change in time = Δ[Concentration] / Δt

For a general reaction: aA + bB → cC + dD

Rate = -1/a * d[A]/dt = -1/b * d[B]/dt = +1/c * d[C]/dt = +1/d * d[D]/dt

• The negative sign for reactants indicates their concentration decreases over time.
• The positive sign for products indicates their concentration increases over time.
• The stoichiometric coefficients (a, b, c, d) are used to normalize the rates, ensuring the reaction rate is independent of the species chosen.

Units of reaction rate are typically moles per liter per second (mol L^-1 s^-1) or molar per second (M s^-1).

Factors Affecting Reaction Rates

Several factors can influence how fast a chemical reaction proceeds. Understanding these factors allows for the control and optimization of reaction processes.

Effect of Temperature

Increasing the temperature generally increases the rate of a chemical reaction.

• Increased Kinetic Energy: Higher temperatures mean reactant molecules possess greater average kinetic energy.
• Increased Collision Frequency: Molecules move faster, leading to more frequent collisions.
• Increased Fraction of Effective Collisions: More significantly, a larger fraction of molecules will have energy equal to or greater than the activation energy (E_a), leading to more effective collisions that result in product formation.

Effect of Pressure

The effect of pressure is most significant for reactions involving gaseous reactants.

• Increased Concentration: For gaseous reactions, increasing the pressure means that the gas molecules are forced into a smaller volume, effectively increasing their concentration.
• Increased Collision Frequency: Higher concentration leads to more frequent collisions between reactant molecules, thus increasing the reaction rate.
• Pressure typically has a negligible effect on reactions involving only liquids or solids.

Effect of a Catalyst

A catalyst is a substance that increases the rate of a chemical reaction without being consumed in the reaction itself.

• Lowering Activation Energy: Catalysts provide an alternative reaction pathway with a lower activation energy.
• No Change in Equilibrium: A catalyst speeds up both the forward and reverse reactions equally, thus it does not change the position of equilibrium, only the rate at which equilibrium is attained.

Other Factors

• Concentration of Reactants: Generally, increasing the concentration of reactants leads to a higher reaction rate because it increases the frequency of collisions between reactant molecules.
• Surface Area of Reactants: For reactions involving solids, increasing the surface area (e.g., by crushing a solid into a powder) increases the contact points between reactants, thereby increasing the reaction rate.
• Nature of Reactants: The chemical nature of the reactants (e.g., bond strengths, molecular complexity) inherently influences how fast they react. For instance, ionic reactions are often very fast, while covalent reactions can be slower.

Order and Molecularity of a Reaction

Order and molecularity are two important concepts used to describe the kinetics of chemical reactions, though they refer to different aspects.

Order of a Reaction

The order of a reaction with respect to a reactant is the exponent to which its concentration term is raised in the experimentally determined rate law. The overall order of a reaction is the sum of the exponents of the concentration terms in the rate law.

• Experimental Value: Reaction order is always determined experimentally; it cannot be predicted from the stoichiometry of the balanced chemical equation.
• Can Be Zero, Integer, or Fractional: The order can be 0, 1, 2, 3, or even a fraction (e.g., 0.5, 1.5).
• Rate Law: For a reaction A + B → Products, if the rate law is Rate = k[A]x[B]y, then 'x' is the order with respect to A, 'y' is the order with respect to B, and (x+y) is the overall order.

Molecularity of a Reaction

The molecularity of a reaction is defined as the number of reacting species (atoms, ions, or molecules) that collide simultaneously in an elementary step of a chemical reaction.

• Theoretical Value: Molecularity is a theoretical concept determined from the mechanism of an elementary reaction, not an overall reaction.
• Always an Integer: Molecularity must be a whole number (1, 2, or 3). It cannot be zero, fractional, or greater than three (due to the very low probability of more than three molecules colliding simultaneously in an effective manner).
• Elementary Steps Only: Molecularity applies only to elementary reactions, not to complex reactions that occur in multiple steps.

Comparison of Order and Molecularity

Integrated Rate Equations

Integrated rate equations relate the concentration of reactants to time. They are useful for predicting concentrations at specific times or determining the time required for a reaction to reach a certain extent.

Zero Order Reactions

A reaction is zero order if its rate is independent of the concentration of the reactant.

Rate = -d[A]/dt = k[A]⁰ = k

Integrated Rate Equation:

[A]_t = [A]₀ - kt

Where:

• [A]t is the concentration of reactant A at time t.
• [A]0 is the initial concentration of reactant A.
• k is the rate constant for a zero order reaction (units: mol L^-1 s^-1).

Graphical Representation: A plot of [A]t versus t yields a straight line with a slope of -k and a y-intercept of [A]0.

First Order Reactions

A reaction is first order if its rate is directly proportional to the concentration of one reactant.

Rate = -d[A]/dt = k[A]¹

Integrated Rate Equation:

ln [A]_t = ln [A]₀ - kt or ln([A]₀/[A]_t) = kt

Where:

• [A]t is the concentration of reactant A at time t.
• [A]0 is the initial concentration of reactant A.
• k is the rate constant for a first order reaction (units: s^-1).

Graphical Representation: A plot of ln [A]t versus t yields a straight line with a slope of -k and a y-intercept of ln [A]0.

Second Order Reactions

A reaction is second order if its rate is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants.

Rate = -d[A]/dt = k[A]² or Rate = k[A][B]

For the case where Rate = k[A]2:

Integrated Rate Equation:

1/[A]_t = 1/[A]₀ + kt

Where:

• [A]t is the concentration of reactant A at time t.
• [A]0 is the initial concentration of reactant A.
• k is the rate constant for a second order reaction (units: L mol^-1 s^-1).

Graphical Representation: A plot of 1/[A]t versus t yields a straight line with a slope of k and a y-intercept of 1/[A]0.

Half-life of a Reaction (t_1/2)

The half-life (t_1/2) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value.

Zero Order Half-life

Using [A]t = [A]0 - kt, at t = t_1/2, [A]t = [A]0/2.

[A]₀/2 = [A]₀ - kt_1/2 => kt_1/2 = [A]₀ - [A]₀/2 => kt_1/2 = [A]₀/2

t_1/2 = [A]₀ / 2k

For a zero-order reaction, the half-life is directly proportional to the initial concentration of the reactant.

First Order Half-life

Using ln [A]t = ln [A]0 - kt, at t = t_1/2, [A]t = [A]0/2.

ln ([A]₀/2) = ln [A]₀ - kt_1/2 => kt_1/2 = ln [A]₀ - ln ([A]₀/2) => kt_1/2 = ln ([A]₀ / ([A]₀/2)) => kt_1/2 = ln 2

t_1/2 = ln 2 / k = 0.693 / k

For a first-order reaction, the half-life is constant and independent of the initial concentration of the reactant.

Second Order Half-life

Using 1/[A]t = 1/[A]0 + kt, at t = t_1/2, [A]t = [A]0/2.

1/([A]₀/2) = 1/[A]₀ + kt_1/2 => 2/[A]₀ = 1/[A]₀ + kt_1/2 => kt_1/2 = 2/[A]₀ - 1/[A]₀ => kt_1/2 = 1/[A]₀

t_1/2 = 1 / (k[A]₀)

For a second-order reaction, the half-life is inversely proportional to the initial concentration of the reactant.

Determination of Order of a Reaction

The order of a reaction cannot be determined from stoichiometry but must be found experimentally using various methods.

Initial Rate Method

This method involves measuring the initial rate of a reaction at different initial concentrations of reactants while keeping others constant. By comparing how the initial rate changes with changes in reactant concentration, the order with respect to each reactant can be determined.

• Set up experiments where the concentration of one reactant is varied, and all others are kept constant.
• Measure the initial rate for each experiment.
• Determine the exponent 'x' for Rate = k[A]x by observing how the rate changes as [A] changes.

Integrated Rate Law Method (Graphical Method)

This method involves plotting experimental concentration-time data according to the integrated rate laws for different orders and seeing which plot yields a straight line.

• Plot [A] vs t. If linear, it's zero order.
• Plot ln [A] vs t. If linear, it's first order.
• Plot 1/[A] vs t. If linear, it's second order.

The rate constant k can be determined from the slope of the linear plot.

Half-life Method

The half-life of a reaction depends on the order of the reaction and sometimes on the initial concentration. By observing the half-life, the order can be deduced.

• If t_1/2 is constant and independent of initial concentration, the reaction is first order.
• If t_1/2 is inversely proportional to [A]0, the reaction is second order.
• If t_1/2 is directly proportional to [A]0, the reaction is zero order.

Ostwald's Isolation Method

In this method, the concentration of all reactants except one is taken in large excess. This makes the reaction rate effectively independent of the reactants in excess, allowing the determination of the order with respect to the isolated reactant. This process is repeated for each reactant to find the overall order.

Activation Energy and Arrhenius Equation

Activation Energy (E_a)

Activation energy (E_a) is the minimum amount of energy that reacting species must possess to undergo a chemical reaction. It represents an energy barrier that must be overcome for reactants to transform into products.

• Molecules must collide with energy greater than or equal to E_a to form an activated complex (transition state) and subsequently products.
• A higher E_a means a slower reaction rate, as fewer molecules possess the required energy.

Arrhenius Equation

The Arrhenius equation mathematically describes the temperature dependence of the rate constant k.

k = A e^{-Ea / RT}

Where:

• k is the rate constant.
• A is the Arrhenius factor or pre-exponential factor, which represents the frequency of collisions with proper orientation.
• Ea is the activation energy (in J mol^-1 or kJ mol^-1).
• R is the ideal gas constant (8.314 J mol^-1 K^-1).
• T is the absolute temperature (in Kelvin).

Calculation from Arrhenius Equation:

Taking the natural logarithm of both sides of the Arrhenius equation:

ln k = ln (A e^{-Ea / RT}) => ln k = ln A + ln (e^{-Ea / RT}) => ln k = ln A - E_a / RT

This equation is in the form of a straight line y = mx + c if ln k is plotted against 1/T. The slope of this line will be -Ea/R, and the y-intercept will be ln A. Thus, E_a can be calculated from the slope.

For two different temperatures (T₁ and T₂) and their corresponding rate constants (k₁ and k₂):

ln k₁ = ln A - E_a / RT₁
ln k₂ = ln A - E_a / RT₂

Subtracting the first equation from the second:

ln k₂ - ln k₁ = (ln A - E_a / RT₂) - (ln A - E_a / RT₁)
ln (k₂/k₁) = E_a / RT₁ - E_a / RT₂
ln (k₂/k₁) = (E_a/R) * (1/T₁ - 1/T₂)

This equation allows for the calculation of activation energy if rate constants at two different temperatures are known.

Theories of Reaction Rates

Two main theories explain how chemical reactions occur and why they have certain rates.

Collision Theory

Collision theory, primarily developed for gaseous reactions, explains reaction rates based on molecular collisions.

• Postulates:
  1. Reactant molecules must collide with each other for a reaction to occur.
  2. The colliding molecules must possess a minimum amount of energy, known as the activation energy (E_a), to break existing bonds and form new ones.
  3. The colliding molecules must have the proper orientation relative to each other at the moment of impact for an effective collision to take place.
• Rate Expression: The rate of a bimolecular reaction according to collision theory is proportional to the collision frequency (Z), the fraction of collisions with sufficient energy (e^-Ea/RT), and a steric factor (P).

Rate = P Z_AB e^-Ea/RT

• Steric Factor (P): Represents the probability that collisions have the correct orientation. For simple reactions, P is close to 1; for complex molecules, it can be much less than 1.
• Limitations: Collision theory is oversimplified. It assumes molecules are hard spheres and doesn't fully account for the complexity of molecular structures and how they interact during a collision. It struggles to accurately predict rates for complex molecules where the steric factor is hard to estimate.

Activated Complex Theory (Transition State Theory)

Activated Complex Theory (ACT), also known as Transition State Theory (TST), provides a more sophisticated approach to understanding reaction rates.

• Concept of Activated Complex/Transition State: According to ACT, during a reaction, reactants pass through a high-energy, unstable intermediate structure called the activated complex or transition state. This complex exists at the peak of the potential energy diagram.
• Reaction Pathway: Reactants approach each other, their bonds begin to stretch and break, and new bonds start to form, leading to the formation of the activated complex. This complex then rapidly decomposes to form products.
• Activation Energy: The activation energy (E_a) is the difference in energy between the reactants and the activated complex. Overcoming this energy barrier is necessary for the reaction to proceed.
• Key Idea: The theory assumes that there is an equilibrium between the reactants and the activated complex. The rate of reaction is then determined by the rate at which the activated complex transforms into products.

Comparison of Collision Theory and Activated Complex Theory (Qualitative Treatment)