Chapter 14: Chemical Kinetics III – Rate Laws, Mechanisms, and Catalysis

Chapter 14: Chemical Kinetics III

Rate Laws and Rate Constants: Initial Rates

Rate laws are mathematical expressions that relate the rate of a chemical reaction to the concentration of its reactants. They must be determined experimentally for each reaction.

• General Rate Law: For a reaction A → B, the rate law is given by: $\text{Rate} = k[\text{A}]^x[\text{B}]^y$ where k is the specific rate constant, and x, y are the orders with respect to each reactant.

• Reaction Order: The sum of the exponents in the rate law gives the overall reaction order.

• Example: If $\text{Rate} = k[\text{NH}_4^+][\text{NO}_2^-]$, the reaction is first order in each reactant and second order overall.

Method of Initial Rates

This experimental method determines the rate law by comparing the initial rates of reaction for different starting concentrations of reactants.

• Choose two experiments where one reactant's concentration changes while the other remains constant.

• Set up the ratio: $\frac{\text{conc ratio}^{\text{exponent}}}{\text{rate ratio}}$

• Example: For $\text{Rate} = k[\text{NO}_2^-]^x[\text{NH}_4^+]^y$, use experimental data to solve for x and y.

An Example of How Concentration Affects Rate

The rate law shows the relationship between rate and concentration for all reactants.

• Given: $\text{Rate} = k[\text{NH}_4^+][\text{NO}_2^-]$

• To find k, substitute experimental values: $k = \frac{\text{Rate}}{[\text{NH}_4^+][\text{NO}_2^-]}$

• Example calculation: $k = \frac{5.4 \times 10^{-4} \text{ M/s}}{(0.0100 \text{ M})(0.200 \text{ M})} = 2.7 \times 10^{-1} \text{ M}^{-1}\text{s}^{-1}$

Integrated Rate Laws

Integrated rate laws relate the concentration of reactants to time, allowing for the determination of reaction order and rate constants from concentration vs. time data.

• Rate Law: Relates rate, rate constant (k), and concentration.

• Integrated Rate Law: Relates time and concentration.

Using the Full Data Set

Graphical analysis of concentration vs. time data provides a more accurate determination of rate laws and constants.

• Instantaneous rate at a given time is the slope of the tangent to the concentration vs. time curve.

First-Order Reactions

First-order reactions depend linearly on the concentration of one reactant.

• General form: $\text{Rate} = k[\text{A}]$

• Integrated form: $\ln[\text{A}] = -kt + \ln[\text{A}]_0$

• Plotting $\ln[\text{A}]$ vs. time yields a straight line with slope $-k$.

• Example: Conversion of methyl isonitrile to acetonitrile.

Second-Order Reactions

Second-order reactions depend on the square of one reactant or the product of two reactant concentrations.

• General form: $\text{Rate} = k[\text{A}]^2$ or $\text{Rate} = k[\text{A}][\text{B}]$

• Integrated form: $\frac{1}{[\text{A}]} = \frac{1}{[\text{A}]_0} + kt$

• Plotting $1/[\text{A}]$ vs. time yields a straight line with slope $k$.

• Example: Decomposition of $\text{NO}_2$.

Zero Order Reactions

Zero-order reactions have rates independent of reactant concentration.

• General form: $\text{Rate} = k$

• Integrated form: $[\text{A}] = -kt + [\text{A}]_0$

• Plotting $[\text{A}]$ vs. time yields a straight line with slope $-k$.

Half-Life of Reactions

The half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease by half.

• First-order: $t_{1/2} = \frac{0.693}{k}$

• Second-order: $t_{1/2} = \frac{1}{k[\text{A}]_0}$ (depends on initial concentration)

Temperature and Rate: Activation Energy and the Arrhenius Equation

Reaction rates generally increase with temperature due to increased molecular energy and collision frequency.

• Rate constant $k$ is temperature dependent.

• Arrhenius Equation: $k = Ae^{-E_a/RT}$

• Linearized form: $\ln k = -\frac{E_a}{R}\frac{1}{T} + \ln A$

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

Collision Model

Based on kinetic molecular theory, molecules must collide to react. The rate depends on collision frequency and proper orientation.

• More collisions → higher reaction rate.

• Bonds are broken and formed during collisions.

• Proper alignment is necessary for effective collisions.

Orientation Factor

Not all collisions are effective; molecules must have the correct orientation to react. This factor is not directly predictable except in simple systems.

Activation Energy Model

The minimum energy required for a reaction to occur is the activation energy ($E_a$). Reactants must overcome this barrier to form products.

Transition State (Activated Complex)

The transition state is the highest energy arrangement of atoms during a reaction, corresponding to the activation energy. It is not a stable intermediate.

Adding $E_a$ to Reaction Coordinate Diagrams

Energy diagrams plot the energy of the system as the reaction progresses, showing reactants, products, and the transition state.

Reassessing Effect of Temperature

At higher temperatures, a greater fraction of molecules have enough energy to overcome $E_a$, increasing the reaction rate.

Determining Activation Energy

• Arrhenius equation relates $k$, $E_a$, and $T$.

• Plotting $\ln k$ vs. $1/T$ (Arrhenius plot) gives $E_a$ as the slope ($-E_a/R$).

• Equation for comparing two temperatures: $\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

Reaction Mechanisms

Mechanisms describe the sequence of steps by which a reaction occurs. Each step is called an elementary reaction.

• Molecularity: Number of molecules involved in an elementary step.

• Unimolecular: one molecule; Bimolecular: two molecules; Termolecular: three molecules.

Molecularity	Elementary Reaction	Rate Law
Unimolecular	A → products	Rate = k[A]
Bimolecular	A + B → products	Rate = k[A][B]
Termolecular	A + B + C → products	Rate = k[A][B][C]

Termolecular Possibilities

Termolecular steps are rare and slower than unimolecular or bimolecular steps. Most mechanisms involve only unimolecular or bimolecular steps.

What Limits the Rate?

The overall reaction rate is determined by the slowest step, called the rate-determining step (RDS).

Requirements of a Plausible Mechanism

• Rate law must be derived from the RDS.

• Each step must balance and intermediates must be used up.

• Stoichiometry must match the overall reaction.

• Catalysts are used and regenerated.

Mechanisms with Slow or Fast Initial Steps

• Slow initial step: The rate law is determined by the first (slowest) step.

• Fast initial step: The equilibrium method is used to derive the rate law.

• Intermediates are produced in one step and consumed in another; they are not reactants or products.

Catalysis

A catalyst increases the rate of a reaction by providing an alternative pathway with lower activation energy. It is not consumed in the reaction.

• Homogeneous Catalysts: Catalyst and reactants are in the same phase (often dissolved in the same solvent).

• Heterogeneous Catalysts: Catalyst is in a different phase (e.g., solid catalyst with gaseous reactants).

• Enzymes: Biological catalysts with specific active sites for substrates. The lock-and-key model explains enzyme specificity, though active sites may be flexible.