Kinetics

CHEM 205 - Physical Chemistry Study Notes

Course Content

• Part I: Thermodynamics

• Part II: Spectroscopy

• Part III: Kinetics

Thermodynamics & Chemical Kinetics

• Thermodynamics indicates the direction and extent of chemical changes but does not inform how or how fast a reaction occurs.   - Example: Diamond to Graphite; $  ext{ΔG} < 0$ indicates a spontaneous process.

1. Chemical Kinetics

• Topics Covered:   - Reaction rates and implications.   - How to speed up or slow down a reaction process.   - Reaction mechanisms.

Key Definitions:

• A. Reaction Rate: The change in concentration of a reactant/product over time.

• B. Rate Law: An equation that relates the reaction rate with the concentration of reactants.

• C. Rate Constant (k): The proportionality constant in the rate law that is specific to a reaction at a given temperature.

• D. Reaction Order: The exponent to which the concentration of a reactant is raised in the rate law.

1.1 Reaction Rate

• The reaction rate (v) is defined mathematically:   $ext{Rate} = v = - rac{d[A]}{dt}$ for reaction $A + 2B ightarrow C$.

• The stoichiometry of the reaction is essential:   $- rac{1}{2} rac{d[B]}{dt} = rac{d[C]}{dt}$.

Measuring Rate

• Measure concentration of [A] as a function of time or other quantities such as pressure (for gases) or absorbance (for solutions).   - Example of pressure: $P = rac{nRT}{V}$.   - Example of absorbance: $ext{Absorbance} = ext{εlc}$ where ε is the molar absorptivity, l is path length, and c is concentration.

Characteristics of Rate

• Rate is positive by definition:   - Average reaction rate for example:
      - Reaction: $2 ext{HI} ightarrow ext{H}_2 + ext{I}_2$     - Concentration change: From 4.0 mM to 3.5 mM in 100 seconds gives:       $ext{Average Rate} = - rac{ ext{Δ[HI]}}{ ext{Δt}} = - rac{3.5 - 4.0}{100} = 5.0 imes 10^{-3} ext{mMs}^{-1}$

Factors Affecting Rate

• Nature of reactants/products.

• Concentrations of reactants/products.

• Temperature: a rise of 10°C often results in a doubling of the reaction rate.

• Catalysts: Compounds that increase the reaction rate without being consumed in the reaction.

• Inhibitors: Compounds that decrease the reaction rate without being consumed in the reaction.

Importance of Factors to Mechanism

• The ultimate aim in kinetics is to determine the reaction mechanism.

1.2 Rate Law

• The rate of a reaction can often be expressed as a power law:   $rac{d[A]}{dt} = k[A]^a[B]^b[C]^c$ where:   - k = rate constant, always positive,   - a, b, c = orders of the reaction for constituents A, B, C.

Example Rate Law

• For the reaction: $ext{S}_2 ext{O}_8^{2-} + 3 ext{I}^{-} ightarrow 2 ext{SO}_4^{2-} + ext{I}_3^{-}$
    - Rate Law:     $rac{d[ ext{SO}_4^{2-}]}{dt} = k[ ext{S}_2 ext{O}_8^{2-}][ ext{I}^{-}]^3$.
     - The stoichiometric coefficients do not equal the reaction order.

Rate Constant Units

• When concentrations are in $ ext{mol L}^{-1}$ and time in seconds, rates can be denoted as:   - First order: $k$ in $ ext{L mol}^{-1} ext{s}^{-1}$,   - Second order: $k$ in $ ext{mol}^{-1} ext{L s}^{-1}$,   - Zeroth order: $k$ in $ ext{mol L}^{-1} ext{s}^{-1}$.

1.2.1 Reaction Order

• The reaction order in a component is the exponent of its concentration in the rate law.

• The overall reaction order is the sum of all individual orders (e.g., $n = a + b + c$).

• Reaction orders can be negative or fractional.

1.3 Integrated Rate Laws

Definition

• Integrated rate laws express the concentration of reactants or products at any time after the start of a reaction.

• They usually arise by integrating a fundamental rate law.

Zero-th Order Reaction

• For a zero-th order reaction:
    $[A] = [A]<em>0 - kt$;   - Half-life: $t</em>{1/2} = rac{[A]_0}{2k}$.

First Order Reaction

• For a first-order reaction:
    $ext{ln}[ ext{A}] = ext{ln}[ ext{A}]<em>0 - kt$;   - Half-life: $t</em>{1/2} = rac{0.693}{k}$, independent of initial concentration.

Second Order Reaction

• For a second-order reaction:
    $rac{1}{[ ext{A}]} = rac{1}{[ ext{A}]<em>0} + kt$;   - Half-life:   $t</em>{1/2} = rac{1}{k[ ext{A}]_0}$, dependent on initial concentration.

Example of Half-life Calculation:
In a first-order reaction with $k = 1.0 ext{s}^{-1}$, $t_{1/2} = rac{0.693}{1.0} = 0.693 ext{s}.$

2. Temperature Dependence of Reaction Rates

Arrhenius Equation

• Proposed by Svante Arrhenius, it states that the rate constant (k) is a function of temperature (T) given by:   $k = Ae^{- rac{E_a}{RT}}$;   - Where:     - A: pre-exponential factor, characteristic of the reaction.     - $E_a$: activation energy.     - R: universal gas constant ($8.314 J ext{mol}^{-1} ext{K}^{-1}$).

Effect of Activation Energy

• A higher $E_a$ leads to a reaction rate highly sensitive to temperature changes.

• Lower $E_a$ indicates a reaction rate less sensitive to temperature.

4. Catalysis

Catalyst Overview

• Definition: A substance that changes the rate of a reaction without undergoing a permanent change itself.

• Catalysts provide a new pathway with lower activation energy.

Types of Catalysts

1. Heterogeneous Catalysts:    - Present in a different phase from the reactants (e.g., solid catalyst with gas reactants).

2. Homogeneous Catalysts:    - Present in the same phase as the reactants (e.g., reactions in solutions).

Enzyme Catalysis

• Enzymes are biological catalysts:   $E + S ⇌ ES → E + P$ where:   - S: substrate,   - P: product,   - ES: enzyme-substrate complex.

6. Conclusion - Practical Applications

Radiocarbon Dating

• Based on first-order kinetics, with a half-life of around 5750 years.

• Measures remaining 14C in ancient samples to determine age.

Example Calculation: If a bone has a measured CPM of 71, and a standard of 100 CPM, calculate its age: $ext{Age} = rac{- ext{ln}( rac{71}{100})}{k}$; Where k is derived from the half-life.