Unit 2: Comprehensive Study Guide to Chemical Kinetics

The Concept of Reaction Rates

Chemical kinetics is the branch of physical chemistry that studies the rates of chemical reactions, the factors affecting these rates, and the mechanism by which reactions proceed.

Reaction Rate: The rate of a chemical reaction is defined as the change in concentration of a reactant or product per unit time. It indicates how fast a reaction is proceeding.

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

The rate of reaction can be expressed as:

Rate = - (1/a) * (Δ[A]/Δt) = - (1/b) * (Δ[B]/Δt) = + (1/c) * (Δ[C]/Δt) = + (1/d) * (Δ[D]/Δt)

The negative sign for reactants indicates their concentration decreases over time, while the positive sign for products indicates their concentration increases.

Factors Affecting Reaction Rates

• Temperature: Increasing the temperature generally increases the reaction rate. This is because higher temperatures lead to more frequent and energetic collisions between reactant molecules, increasing the fraction of molecules that possess energy equal to or greater than the activation energy.
• Pressure: For reactions involving gaseous reactants, increasing the pressure generally increases the reaction rate. Higher pressure means a higher concentration of gas molecules, leading to more frequent collisions. Pressure has a negligible effect on reactions in liquid or solid phases.
• Catalyst: A catalyst is a substance that increases the rate of a chemical reaction without being consumed in the reaction itself. Catalysts provide an alternative reaction pathway with a lower activation energy.
• Concentration of Reactants: Generally, increasing the concentration of reactants increases the reaction rate. A higher concentration means more reactant molecules per unit volume, leading to more frequent collisions.
• Surface Area of Reactants: For reactions involving solid reactants, increasing the surface area increases the reaction rate. More surface area allows for more contact points for reactions to occur.
• Nature of Reactants: The chemical nature of the reactants significantly influences the reaction rate. For example, ionic reactions are typically very fast, while reactions involving covalent bond breaking and formation are slower.

Order and Molecularity of a Reaction

Order of a Reaction

Order of a Reaction: The order of a reaction with respect to a particular reactant is the power to which its concentration term is raised in the experimentally determined rate law. The overall order of a reaction is the sum of the orders with respect to all the reactants. It is an experimental quantity and can be an integer, a fraction, or even zero.

For a reaction aA + bB → products, the rate law might be Rate = k[A]^x[B]^y, where:

• x is the order with respect to A
• y is the order with respect to B
• (x + y) is the overall order of the reaction
• k is the rate constant.

Molecularity of a Reaction

Molecularity of a Reaction: The molecularity of an elementary reaction is the number of reactant species (atoms, ions, or molecules) that collide simultaneously in the rate-determining step to bring about the chemical reaction. It is always a whole number and cannot be zero or fractional.

• Unimolecular reactions: Molecularity = 1 (e.g., A → products)
• Bimolecular reactions: Molecularity = 2 (e.g., A + B → products or 2A → products)
• Termolecular reactions: Molecularity = 3 (e.g., A + B + C → products or 2A + B → products). These are rare due to the low probability of three species colliding simultaneously.

Comparison: Order vs. Molecularity

Integrated Rate Equations

Integrated rate equations relate the concentration of reactants (or products) to time. They are useful for determining the order of a reaction and for calculating concentrations at any given time.

Zero-Order Reactions

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

Differential Rate Law: Rate = -d[A]/dt = k[A]⁰ = k

Integration of this gives the integrated rate equation:

[A]_t = [A]₀ - kt

Where:

• [A]_t = concentration of reactant A at time t
• [A]₀ = initial concentration of reactant A
• k = rate constant (units: concentration/time, e.g., mol L^-1 s^-1)

A plot of [A]_t versus time (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 the rate of reaction depends on the first power of the concentration of one reactant.

Differential Rate Law: Rate = -d[A]/dt = k[A]¹

Integration of this gives the integrated rate equation:

ln[A]_t = ln[A]₀ - kt
or
[A]_t = [A]₀e^-kt
or
k = (1/t) ln([A]₀/[A]_t)

Where:

• [A]_t = concentration of reactant A at time t
• [A]₀ = initial concentration of reactant A
• k = rate constant (units: 1/time, e.g., s^-1)

A plot of ln[A]_t versus time (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 the rate of reaction depends on the second power of the concentration of one reactant, or on the first power of the concentrations of two different reactants.

Differential Rate Law (for Rate = k[A]²): Rate = -d[A]/dt = k[A]²

Integration of this gives the integrated rate equation:

1/[A]_t = 1/[A]₀ + kt

Where:

• [A]_t = concentration of reactant A at time t
• [A]₀ = initial concentration of reactant A
• k = rate constant (units: L mol^-1 s^-1)

A plot of 1/[A]_t versus time (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)

Half-life (t_1/2): The half-life 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

When [A]_t = [A]₀/2, t = t_1/2

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

When [A]_t = [A]₀/2, t = t_1/2

ln([A]₀ / ([A]₀/2)) = kt_1/2
ln(2) = kt_1/2
t_1/2 = ln(2) / k = 0.693 / k

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

Second-Order Half-Life (for Rate = k[A]²)

When [A]_t = [A]₀/2, t = t_1/2

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

General Methods for Determination of Order of a Reaction

Initial Rate Method

This method involves measuring the initial rate of reaction at different initial concentrations of the reactants, while keeping the concentrations of other reactants constant. By observing how the initial rate changes with varying concentrations, the order with respect to each reactant can be determined.

1. Perform a series of experiments where the initial concentration of one reactant is varied, while others are kept constant.
2. Measure the initial rate for each experiment.
3. Compare the rates and concentrations to determine the order. For example, if doubling [A] doubles the rate (with [B] constant), the reaction is first order with respect to A. If doubling [A] quadruples the rate, it's second order.

Integrated Rate Law Method (Graphical Method)

This method involves plotting experimental concentration-time data according to the integrated rate equations for different orders (zero, first, and second). The plot that yields a straight line indicates the correct order of the reaction.

• Plot [A] vs. time: If linear, it's zero order.
• Plot ln[A] vs. time: If linear, it's first order.
• Plot 1/[A] vs. time: If linear, it's second order.

Half-Life Method

This method involves determining the half-life of a reaction at different initial concentrations. The dependence of t_1/2 on [A]₀ reveals the order.

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

Concept of Activation Energy and its Calculation from Arrhenius Equation

Concept of Activation Energy (E_a)

Activation Energy (E_a): The minimum amount of energy that reactant molecules must possess to undergo an effective collision and transform into products. It represents an energy barrier that must be overcome for a reaction to occur.

Reactant molecules must collide with sufficient kinetic energy to break existing bonds and form new ones. This minimum energy threshold is the activation energy. The state of molecules at the peak of the energy barrier is called the transition state or activated complex.

Arrhenius Equation

The Arrhenius equation quantitatively describes the effect of temperature on the rate constant (k) of a reaction.

k = Ae^-E_a/RT

Where:

• k = rate constant
• A = Arrhenius pre-exponential factor or frequency factor (related to the frequency of collisions and the probability of favorable orientation)
• E_a = activation energy (in J mol^-1 or kJ mol^-1)
• R = universal gas constant (8.314 J mol^-1 K^-1)
• T = absolute temperature (in Kelvin)

Calculation of Activation Energy from Arrhenius Equation

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

ln k = ln (Ae^-E_a/RT)
ln k = ln A - E_a/RT

This equation is in the form of a straight line (y = mx + c), where y = ln k, x = 1/T, m = -E_a/R, and c = ln A.

Therefore, a plot of ln k versus 1/T yields a straight line with a slope of -E_a/R. From the slope, E_a can be calculated:

E_a = -Slope × R

If rate constants (k₁ and k₂) are known at two different temperatures (T₁ and T₂), the activation energy can be calculated using the two-point form of the Arrhenius equation:

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

Theories of Reaction Rates

Collision Theory of Bimolecular Reactions

Collision theory explains reaction rates based on the collisions between reactant molecules. Its main postulates for a bimolecular reaction are:

1. Reactant molecules must collide with each other for a reaction to occur.
2. The colliding molecules must possess a certain minimum amount of kinetic energy, called the activation energy (E_a). Collisions with energy less than E_a are ineffective.
3. The colliding molecules must have the proper orientation relative to each other at the moment of collision. Only a small fraction of collisions occur with the correct orientation.

According to collision theory, the rate of reaction is proportional to the number of effective collisions per unit time.

Rate = Z_AB * f * P

• Z_AB = collision frequency (total number of collisions between A and B per unit volume per second)
• f = fraction of molecules with energy ≥ E_a (given by e^-E_a/RT)
• P = steric factor or probability factor (fraction of collisions with correct orientation)

Limitations: It assumes molecules are hard spheres and does not account for the exact nature of the bonds breaking and forming, and sometimes predicts rate constants that are too high because it simplifies the steric factor (P).

Activated Complex Theory (Transition State Theory) of Bimolecular Reactions

Activated Complex Theory (ACT), also known as Transition State Theory (TST), focuses on the formation of an unstable intermediate species called the