Definition and Overview
Concept
Rate law: mathematical expression relating reaction rate to reactant concentrations. Indicates how rate varies with concentration changes.
Purpose
Describes kinetics without mechanistic details. Enables prediction of reaction speed and influence of conditions.
General form
Rate = k [A]^m [B]^n ..., where k = rate constant, m and n = orders with respect to reactants A and B.
Rate Constant (k)
Definition
k: proportionality factor linking rate and concentrations. Specific for each reaction and condition.
Units
Depends on overall order (sum of exponents). Units adjust to maintain rate units (mol L⁻¹ s⁻¹).
Temperature dependence
Arrhenius equation: k = A exp(-Ea/RT). A = frequency factor, Ea = activation energy.
Table of units by order
| Overall Order | Units of k |
|---|---|
| 0 (Zero order) | mol L⁻¹ s⁻¹ |
| 1 (First order) | s⁻¹ |
| 2 (Second order) | L mol⁻¹ s⁻¹ |
Order of Reaction
Definition
Order: exponent of concentration term in rate law. Indicates reaction sensitivity to concentration changes.
Types
Zero order: rate independent of concentration. First order: rate proportional to concentration. Second order: rate proportional to square or product of concentrations.
Overall order
Sum of individual orders. May be integer or fractional.
Significance
Determines reaction kinetics, influences integrated rate equations and half-life formulas.
Differential Rate Law
Definition
Expresses rate as function of instantaneous concentrations. Derived from experimental data.
General form
Rate = k [A]^m [B]^n ... differential form: d[A]/dt = -k [A]^m [B]^n
Use
Determines reaction order and k by measuring initial rates at varying concentrations.
Example
Rate = k [NO]^2 [Cl2]^1Differential form: -d[NO]/dt = k [NO]^2 [Cl2]Integrated Rate Law
Definition
Relates reactant concentration to time. Derived by integrating differential rate law.
Zero order
[A] = [A]₀ - k t
First order
ln[A] = ln[A]₀ - k t
Second order
1/[A] = 1/[A]₀ + k t
Table summary
| Order | Integrated Law | Half-life (t½) |
|---|---|---|
| 0 | [A] = [A]₀ - k t | [A]₀ / 2k |
| 1 | ln[A] = ln[A]₀ - k t | 0.693 / k |
| 2 | 1/[A] = 1/[A]₀ + k t | 1 / (k [A]₀) |
Application
Determines concentration at any time, calculates half-life, validates reaction order.
Experimental Determination
Initial rates method
Measure initial rate at varying reactant concentrations. Analyze rate change to deduce order and rate law.
Isolation method
Keep all but one reactant concentration constant to isolate order for single reactant.
Graphical methods
Plot concentration vs time data per integrated rate laws. Linear plots indicate reaction order.
Example
Plot ln[A] vs time linear → first order. Plot 1/[A] vs time linear → second order.
Reaction Mechanism and Rate Law
Relation
Rate law reflects rate-determining step (RDS) in mechanism. Overall order often differs from stoichiometric equation.
Elementary steps
For elementary reactions, rate law can be written from molecularity (e.g., unimolecular first order, bimolecular second order).
Complex reactions
Require mechanism elucidation. Rate law derived from slowest step or steady-state approximation.
Example
NO2 + CO → NO + CO2 rate law: Rate = k [NO2]^2, mechanism involves dimer formation.
Rate-Determining Step
Definition
Slowest step in reaction sequence. Controls overall rate.
Significance
Rate law corresponds to RDS molecularity and reactants involved.
Identification
Experimental kinetics, intermediate detection, isotope labeling.
Effect on kinetics
Alters apparent reaction order, can cause non-integer orders if equilibrium precedes RDS.
Effect of Temperature
Arrhenius equation
k = A e^(-Ea/RT). Temperature increases k exponentially.
Activation energy (Ea)
Energy barrier to reaction. Higher Ea → greater temperature sensitivity.
Graphical analysis
Plot ln k vs 1/T yields straight line with slope -Ea/R.
Practical implication
Reaction speed predictable with temperature changes. Crucial in process design.
Impact of Catalysts
Definition
Substance increasing rate without consumption. Lowers activation energy.
Effect on rate law
Can introduce new pathways, alter rate law form and rate constant.
Homogeneous vs heterogeneous
Homogeneous: catalyst in same phase as reactants. Heterogeneous: different phase, surface reactions.
Enzymatic catalysis
Highly selective, complex rate laws involving Michaelis-Menten kinetics.
Common Misconceptions
Rate law from stoichiometry
Incorrect to assume rate law exponents equal stoichiometric coefficients unless elementary step.
Order determination
Must be experimentally derived, not assumed.
Constant rate constant
k varies with temperature and catalyst presence, not universal constant.
Half-life dependence
Varies by order: constant for first order, concentration-dependent for others.
References
- Atkins, P., de Paula, J. Physical Chemistry. 10th ed., Oxford University Press, 2014, pp. 814-850.
- Laidler, K. J. Chemical Kinetics. 3rd ed., Harper & Row, 1987, pp. 120-168.
- Espenson, J. H. Chemical Kinetics and Reaction Mechanisms. McGraw-Hill, 1995, pp. 45-89.
- Steinfeld, J. I., Francisco, J. S., Hase, W. L. Chemical Kinetics and Dynamics. 2nd ed., Prentice Hall, 1999, pp. 210-265.
- Fersht, A. Structure and Mechanism in Protein Science. W. H. Freeman, 1999, pp. 33-74.