Chemical Kinetics and Chemical Equilibrium: Study Notes

Chemical Kinetics

Rate Laws and Rate Constants

Chemical kinetics studies the speed of chemical reactions and the factors that affect this speed. The rate law expresses the relationship between the rate of a reaction and the concentration of its reactants.

• General Rate Law: For a reaction $aA + bB \rightarrow cC$, the rate law is $\text{Rate} = k[A]^m[B]^n$ where $k$ is the rate constant, and $m$, $n$ are the reaction orders with respect to A and B.

• Determining Rate Law: Use experimental data to compare how changes in concentration affect the rate.

• Units of Rate Constant: Depend on the overall order of the reaction. For example, for a second-order reaction, $k$ has units of $\text{M}^{-1}\text{s}^{-1}$.

• Example: If doubling [A] doubles the rate, the reaction is first order in A.

Integrated Rate Laws

Integrated rate laws relate reactant concentration to time. The form depends on the reaction order.

• First Order: $[A]_t = [A]_0 e^{-kt}$ or $\ln[A]_t = \ln[A]_0 - kt$

• Second Order: $\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$

• Zero Order: $[A]_t = [A]_0 - kt$

• Half-life ($t_{1/2}$): For first order, $t_{1/2} = \frac{0.693}{k}$; for second order, $t_{1/2} = \frac{1}{k[A]_0}$

• Example: For a first-order reaction with $k = 5.36 \times 10^{-5}\ \text{s}^{-1}$, $t_{1/2} = \frac{0.693}{5.36 \times 10^{-5}}$

Activation Energy and the Arrhenius Equation

The activation energy ($E_a$) is the minimum energy required for a reaction to occur. The Arrhenius equation relates the rate constant to temperature:

• $k = A e^{-E_a/(RT)}$

• Taking logarithms: $\ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T}$

• Plotting $\ln k$ vs. $1/T$ gives a straight line with slope $-E_a/R$

• Example: If $k$ increases with temperature, $E_a$ can be calculated using two $k$ values at different $T$.

Reaction Mechanisms and Molecularity

A reaction mechanism is a sequence of elementary steps that make up the overall reaction. The molecularity of an elementary step is the number of reactant particles involved.

• Unimolecular: Involves one molecule

• Bimolecular: Involves two molecules

• Termolecular: Involves three molecules (rare)

• The rate-determining step is the slowest step in the mechanism.

Chemical Equilibrium

Equilibrium Constant Expressions

At equilibrium, the rates of the forward and reverse reactions are equal. The equilibrium constant ($K$) quantifies the ratio of product to reactant concentrations at equilibrium.

• For a reaction $aA + bB \rightleftharpoons cC + dD$, $K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

• Only gases and aqueous species are included; solids and pure liquids are omitted.

• Example: For $\text{FeO(s)} + \text{H}_2(g) \rightleftharpoons \text{Fe(s)} + \text{H}_2\text{O}(g)$, $K = \frac{[\text{H}_2\text{O}]}{[\text{H}_2]}$

Le Châtelier’s Principle

Le Châtelier’s Principle predicts how a system at equilibrium responds to disturbances.

• Concentration: Adding reactants/products shifts equilibrium to consume the added species.

• Pressure/Volume: Increasing pressure (by decreasing volume) shifts equilibrium toward the side with fewer moles of gas.

• Temperature: For endothermic reactions, increasing temperature favors products; for exothermic, it favors reactants.

• Example: For $2\text{NOBr}(g) \rightleftharpoons 2\text{NO}(g) + \text{Br}_2(g)$, increasing pressure shifts equilibrium toward fewer moles of gas.

Relationship Between $K_c$ and $K_p$

For gaseous reactions, $K_p$ (in terms of partial pressures) and $K_c$ (in terms of concentrations) are related:

• $K_p = K_c(RT)^{\Delta n}$, where $\Delta n$ is the change in moles of gas ($\text{moles of products} - \text{moles of reactants}$)

Manipulating Equilibrium Constants

When reactions are reversed or multiplied, the equilibrium constant changes:

• Reversing a reaction: $K_{\text{new}} = 1/K_{\text{original}}$

• Multiplying coefficients by $n$: $K_{\text{new}} = (K_{\text{original}})^n$

• Adding reactions: Multiply their $K$ values.

Sample Table: Types of Rate Laws and Integrated Forms

Order	Rate Law	Integrated Rate Law	Half-life Expression
Zero	Rate = k	$[A]_t = [A]_0 - kt$	$t_{1/2} = \frac{[A]_0}{2k}$
First	Rate = k[A]	$\ln[A]_t = \ln[A]_0 - kt$	$t_{1/2} = \frac{0.693}{k}$
Second	Rate = k[A]^2	$\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$	$t_{1/2} = \frac{1}{k[A]_0}$

Sample Table: Factors Affecting Equilibrium

Change	Effect on Equilibrium
Increase [Reactant]	Shifts right (toward products)
Increase [Product]	Shifts left (toward reactants)
Increase Pressure (decrease volume)	Shifts toward fewer moles of gas
Increase Temperature (endothermic)	Shifts right (toward products)
Increase Temperature (exothermic)	Shifts left (toward reactants)

Graphical Methods for Determining Reaction Order

• Plot $[A]$ vs. time: Linear for zero order

• Plot $\ln[A]$ vs. time: Linear for first order

• Plot $1/[A]$ vs. time: Linear for second order

Summary of Key Terms

• Rate Law: Mathematical expression relating rate to concentrations

• Rate Constant (k): Proportionality constant in the rate law

• Activation Energy ($E_a$): Minimum energy required for reaction

• Equilibrium Constant (K): Ratio of product to reactant concentrations at equilibrium

• Le Châtelier’s Principle: Predicts response to disturbances in equilibrium

Additional info: These notes expand on the concepts tested in the provided questions, including definitions, formulas, and examples relevant to General Chemistry topics of kinetics and equilibrium.