Kinetics: Chemical Reaction Rates and Rate Laws

Chemical Kinetics Overview

• The study of chemical kinetics addresses the fundamental question: "How fast does a reaction happen?"

• Kinetics is the fourth pillar in understanding chemical processes, building upon previous concepts:

  • Chapter 7: What happens? (Reaction types and products).

  • Chapter 12: Will it happen? (Thermodynamics and spontaneity).

  • Chapter 13: How much will it happen? (Equilibrium).

  • Chapter 17: How fast does it happen? (Kinetics).

• Everyday Life Relevance:

  • Determining the rate at which a car rusts.

  • Measuring ozone depletion and repair rates in the atmosphere.

  • Predicting the time required for food to cook (e.g., how long until dinner is ready).

  • The economic principle of "Time is Money," emphasizing the importance of reaction speed in industrial processes.

• Broad Range of Rates:

  • Vision: $10^{-12}\,\text{s}$.

  • Nuclear reactions: $10^{-6}\,\text{s}$.

  • Cement drying: $10^{1}\,\text{s}$.

  • Transition of Carbon to Diamonds: $10^{9}\,\text{s}$.

17.1 Chemical Reaction Rates

• Definitions:

  • Chemical Reaction Rate: The change in the amount of a reactant or product per unit time.

  • Chemical Kinetics: The specific branch of chemistry that studies the rates of chemical processes.

  • This field is critical for revealing the molecular-level mechanism of reactions.

• The Collision Model:

  • Molecules must collide (run into each other) to react.

  • The frequency of collisions determines the reaction rate: greater frequency results in a faster rate.

  • Collisions are only effective if reactants possess:

  • Sufficient energy.

  • Correct molecular orientation.

• Mathematical Representation (Rate Expressions):

  • Rates are always expressed as positive quantities.

  • Concentration is most often measured in Molarity ($M$ or $\text{mol/L}$).

  • Rate Formula:

  • $\text{Rate} = \frac{\text{Change in Concentration}}{\text{Change in Time}} = \frac{\Delta[\,]}{\Delta(t)} = \frac{[\,]_{t2} - [\,]_{t1}}{t_2 - t_1}$

  • Units: $\text{M/s}$ or $\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$.

  • Sign Conventions:

  • For Reactants (R): $\text{rate} = -\frac{\Delta[R]}{\Delta(t)}$ (negative sign indicates concentration is decreasing).

  • For Products (P): $\text{rate} = +\frac{\Delta[P]}{\Delta(t)}$ (positive sign indicates concentration is increasing).

• Average vs. Instantaneous vs. Initial Rates:

  • Average Reaction Rate: The rate at which a reaction proceeds over a specified time interval, calculated using concentrations at the start and end of that period.

  • Instantaneous Reaction Rate: The rate at which a reaction proceeds at a specific moment in time or specific concentration.

  • Calculated via a Concentration vs. Time graph by finding the slope of a straight line tangent to the curve at the specific time point.

  • Can also be estimated by calculating the average rate over an extremely short time interval.

  • Initial Reaction Rate: The instantaneous reaction rate measured at "time zero" ($t = 0$).

• Case Study: Decomposition of Hydrogen Peroxide ($2H_2O_2(aq) \rightarrow 2H_2O(l) + O_2(g)$):

  • Observations indicate that the rate of decomposition of $H_2O_2$ decreases as the concentration of $H_2O_2$ decreases over time.

  • Sample Data at $40^{\circ}\text{C}$:

  • $t = 0.00\,\text{h}$, $[H_2O_2] = 1.000\,M$

  • $t = 6.00\text{h}$, $[H_2O_2] = 0.500\,M$

  • $t = 12.00\text{h}$, $[H_2O_2] = 0.250\,M$

  • $t = 18.00\text{h}$, $[H_2O_2] = 0.125\,M$

  • $t = 24.00\text{h}$, $[H_2O_2] = 0.0625\,M$

• Stoichiometry and Relative Rates:

  • Rates can be expressed relative to any reactant or product based on stoichiometric factors derived from the balanced equation.

  • For a general reaction $aA + bB \rightarrow cC + dD$:

  • $\text{Rate} = -\frac{1}{a} \frac{\Delta[A]}{\Delta(t)} = -\frac{1}{b} \frac{\Delta[B]}{\Delta(t)} = \frac{1}{c} \frac{\Delta[C]}{\Delta(t)} = \frac{1}{d} \frac{\Delta[D]}{\Delta(t)}$

  • Example: Monitoring $2NH_3 \rightarrow N_2 + 3H_2$ shows the slope for $H_2$ formation is three times the slope of $N_2$ formation due to stoichiometry.

  • Problem Example: $2N_2O_5(g) \rightarrow 4NO_2(g) + O_2(g)$

  • Rate of $N_2O_5$ decomposition: $-\frac{1}{2} \frac{\Delta[N_2O_5]}{\Delta(t)}$

  • Rate of $NO_2$ formation: $"+"\frac{1}{4} \frac{\Delta[NO_2]}{\Delta(t)}$

  • Rate of $O_2$ formation: $"+"\frac{\Delta[O_2]}{\Delta(t)}$

17.2 Factors Affecting Reaction Rates

• Chemical Nature of Reactants:

  • The identity of the substances determines the inherent speed. For example, alkali metals react faster with water as you move down the group in the periodic table.

• Physical State / State of Subdivision:

  • Reaction rates generally increase with increased surface area contact between reactants.

  • Example: Iron and Hydrochloric acid ($Fe(s) + 2HCl(aq) \rightarrow FeCl_2(aq) + H_2(g)$).

  • Iron powder reacts rapidly (creates bubbles of hydrogen gas quickly).

  • An iron nail reacts much more slowly because its surface area is significantly smaller.

• Temperature:

  • Reaction rates ordinarily increase as the temperature increases.

• Concentration:

  • Reaction rates ordinarily increase as the concentration of reactants increases.

  • Higher concentration leads to a higher likelihood of collisions, thus increasing the rate.

  • As reactants are consumed, collision frequency drops, causing the rate to decrease.

• Catalysts:

  • The presence of a catalyst increases the rate of reaction.

17.3 Rate Laws

• Definition: Mathematical expressions representing the relationship between the rate of a reaction and the concentration of its reactants.

• Synonyms: Differential Rate Laws, Rate Equations.

• The Rate Law Expression:

  • For a reaction $A + B \rightarrow \text{products}$:

  • $\text{rate} = k[A]^m[B]^n$

  • Rate Constant ($k$):

  • Unique for every specific reaction.

  • Independent of reactant concentration.

  • Constant at a fixed temperature.

  • Reaction Order:

  • $m$: The order with respect to reactant $A$.

  • $n$: The order with respect to reactant $B$.

  • Overall Order: The sum of all individual orders ($m + n$).

  • Note: $m$ and $n$ have no relationship to the stoichiometric coefficients in the balanced equation and must be determined experimentally.

• General Rules for Exponents/Orders:

  • Order 0: $\text{rate} \propto [A]^0$. The rate is independent of the concentration of $A$.

  • Order 1: $\text{rate} \propto [A]^1$. The rate is directly proportional to $[A]$ (doubling $[A]$ doubles the rate).

  • Order 2: $\text{rate} \propto [A]^2$. Doubling $[A]$ quadruples the rate ($2^2 = 4$).

• Methodologies for Determining Rate Laws:

  • Method of Inspection: Useful for simple or "made-up" data sets where concentration changes and rate changes are easily compared.

  • Method of Initial Rates: Uses experimental data at $t = 0$ to calculate orders ($m, n$) and the rate constant ($k$).

  • Calculation Example:

  • Assuming $\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{0.008}{0.002} = 4$

  • $\frac{[\,]_2}{[\,]_1} = \frac{1.0}{0.5} = 2$

  • If $4 = 2^x$, then $x = 2$ (second order).

  • Calculation of $k$: $k = \frac{\text{rate}}{[A]^2} = \frac{0.018\,M/s}{(1.5\,M)^2}$.

• Real-World Application: The reaction $NO(g) + O_3(g) \rightarrow NO_2(g) + O_2(g)$ is one of the processes responsible for the depletion of the stratospheric ozone layer over Antarctica during spring months.