Chemical Kinetics: Reaction Rates, Stoichiometry, and Experimental Measurement

Fundamentals of Chemical Kinetics

• Definition: Chemical kinetics is the study of the rates at which chemical processes occur, the pathways (mechanisms) by which they happen, and the factors that influence these speeds.

• Core Objectives of Chemical Kinetics:

  • Reaction Velocity: Determining the rate or speed at which reactant molecules are converted into products.

  • Influencing Factors: Investigating environmental and chemical variables that alter reaction rates.

  • Reaction Mechanism: Elucidating the detailed, step-by-step molecular sequence by which a reaction transforms reactants into products.

• Practical Applications:

  • Pharmacokinetics: Understanding how rapidly a pharmaceutical drug or medicine acts within the human body.

  • Atmospheric Chemistry: Evaluating whether the rate of ozone formation in the upper atmosphere is in balance with its rate of depletion.

Factors Affecting Reaction Rates

• Physical State of Reactants:

  • In order for a reaction to occur, reactant molecules must physically collide with one another.

  • Homogeneity: The more homogeneous a mixture of reactants is (e.g., liquid solution vs. solid-liquid interface), the more rapidly the molecules can collide and react.

• Concentration of Reactants:

  • As the concentration of reactant species increases, the total number of molecules per unit volume increases.

  • This higher density of particles directly increases the frequency of collisions between reactant molecules.

• Temperature:

  • Increasing the temperature provides reactant molecules with greater thermal kinetic energy, causing them to move faster.

  • Faster-moving molecules collide more frequently and with greater kinetic energy, increasing the proportion of collisions that successfully overcome the energy threshold for reaction.

• Presence of a Catalyst:

  • Catalysts increase reaction rates by altering the reaction mechanism to provide a path with a lower activation energy.

  • Catalysts participate in the reaction steps but are regenerated intact and are not consumed during the course of the overall reaction.

Mathematical Definition and Measurement of Reaction Rates

• Determination of Reaction Rates:

  • Reaction rates are determined experimentally by monitoring the change in concentration of either a reactant or a product as a function of time.

  • Basic Definition: The ratio of the observed concentration change to the time interval required for that change.

  • Mathematical Expression:         $\text{Rate} = \frac{\Delta \text{concentration}}{\Delta t} = \frac{M}{s}$

  • Units of Measurement:

    • Concentration: Expressed in molarity ($M$), defined as moles per liter ($M = \text{moles/Liter}$ or $mol\,dm^{-3}$).

    • Time: Expressed in seconds ($s$), minutes ($\text{min}$), hours ($\text{h}$), or other appropriate time units.

    • Overall Rate Units: Typically molarity per second ($M/s$ or $M\,s^{-1}$).

• Generic Reaction Analysis ($A \rightarrow B$):

  • Concentration Trends: As time progresses, the concentration of reactant $A$ decreases while the concentration of product $B$ increases.

  • Delta Notation: The Greek symbol $\Delta$ signifies "change in" or "difference in".         $\Delta [A] = [A]_{\text{final}} - [A]_{\text{initial}}$

  • Rate of Disappearance of Reactant $A$:         $\text{Rate of disappearance of } A = -\frac{\Delta [A]}{\Delta t}$

  • Sign Convention: Reaction rates are always defined as positive quantities. Because the concentration of reactant $A$ decreases over time, $\Delta [A]$ is a negative value; an explicit negative sign is added to make the rate positive.

  • Rate of Appearance of Product $B$:         $\text{Rate of appearance of } B = \frac{\Delta [B]}{\Delta t}$

• Average Rate vs. Instantaneous Rate:

  • Average Rate: The concentration change averaged over a specific, extended period of time ($\Delta t$).

  • Rate Deceleration: Average rates decrease over time because as reactants are converted to products, fewer reactant molecules remain, resulting in fewer collisions.

  • Graphical Representation: A plot of reactant concentration ($[C_4H_9Cl]$) versus time yields a downward curved line.

  • Instantaneous Rate: The reaction speed at one specific instant in time, calculated as the slope of the straight line tangent to the concentration-versus-time curve at that given point.

  • Initial Rate Benchmark: Because all reactions slow down as time passes, the instantaneous rate near the beginning of the reaction (the initial rate) serves as the standard comparative measure of reaction speed.

Reaction Rates and Stoichiometry

• 1:1 Stoichiometric Ratio:

  • For reactions such as $C_4H_9Cl(aq) + H_2O(l) \rightarrow C_4H_9OH(aq) + HCl(aq)$, the molar ratio of $C_4H_9Cl$ to $C_4H_9OH$ is $1:1$

  • The rate of disappearance of reactant equals the rate of appearance of product:         $\text{Average rate} = -\frac{\Delta [C_4H_9Cl]}{\Delta t} = \frac{\Delta [C_4H_9OH]}{\Delta t}$

• Unequal Stoichiometric Ratios:

  • For a reaction such as $2A \rightarrow B$:

    • Two moles of $A$ react for every one mole of $B$ formed.

    • The concentration of $A$ decreases twice as fast as the concentration of $B$ increases.

    • Half as much time is required to produce the same numerical concentration change in $B$ compared to $A$

    • Stoichiometric Rate Expression:             $\text{Rate} = -\frac{1}{2}\frac{\Delta [A]}{\Delta t} = \frac{\Delta [B]}{\Delta t}$

• General Stoichiometric Rate Formula:

  • For any generalized chemical reaction equation:         $aA + bB \rightarrow cC + dD$

  • The relative overall rate expressed in terms of reactants and products normalized by their balanced stoichiometric coefficients ($a$, $b$, $c$, $d$) is:         $\text{Rate} = -\frac{1}{a}\frac{\Delta [A]}{\Delta t} = -\frac{1}{b}\frac{\Delta [B]}{\Delta t} = \frac{1}{c}\frac{\Delta [C]}{\Delta t} = \frac{1}{d}\frac{\Delta [D]}{\Delta t}$

Sample Stoichiometric Calculations and Rate Expressions

• Methane Combustion Reaction:

  • Chemical equation:         $CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(g)$

  • Rate expression:         $\text{Rate} = -\frac{\Delta [CH_4]}{\Delta t} = -\frac{1}{2}\frac{\Delta [O_2]}{\Delta t} = \frac{\Delta [CO_2]}{\Delta t} = \frac{1}{2}\frac{\Delta [H_2O]}{\Delta t}$

• Dinitrogen Pentoxide Decomposition:

  • Chemical equation:         $2N_2O_5(g) \rightarrow 4NO_2(g) + O_2(g)$

  • Given rate of decomposition of $N_2O_5$:         $-\frac{\Delta [N_2O_5]}{\Delta t} = 4.2 \times 10^{-7}\,M/s$

  • Stoichiometric rate relation:         $\text{Rate} = -\frac{1}{2}\frac{\Delta [N_2O_5]}{\Delta t} = \frac{1}{4}\frac{\Delta [NO_2]}{\Delta t} = \frac{\Delta [O_2]}{\Delta t}$

  • Calculation for Rate of Appearance of $O_2$:         $\frac{\Delta [O_2]}{\Delta t} = \frac{1}{2}\left(-\frac{\Delta [N_2O_5]}{\Delta t}\right) = \frac{1}{2} \times (4.2 \times 10^{-7}\,M/s) = 2.1 \times 10^{-7}\,M/s$

  • Calculation for Rate of Appearance of $NO_2$:         $\frac{\Delta [NO_2]}{\Delta t} = 4 \times \left(\frac{1}{2}\left(-\frac{\Delta [N_2O_5]}{\Delta t}\right)\right) = 2 \times (4.2 \times 10^{-7}\,M/s) = 8.4 \times 10^{-7}\,M/s$

• Nitric Oxide Oxidation Reaction:

  • Chemical equation:         $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$

  • Stoichiometric relationship: The concentration of $NO$ decreases twice as fast as that of $O_2$

  • Rate expression options:         $\text{Rate} = -\frac{1}{2}\frac{\Delta [NO]}{\Delta t} = -\frac{\Delta [O_2]}{\Delta t} = \frac{1}{2}\frac{\Delta [NO_2]}{\Delta t}$

Detailed Case Study: Reaction of Bromine with Formic Acid

• Reaction Overview and Visual Tracking:

  • Chemical equation:         $Br_2(aq) + HCOOH(aq) \rightarrow 2Br^-(aq) + 2H^+(aq) + CO_2(g)$

  • Visual progression: Reactant $Br_2(aq)$ exhibits a prominent red color, while products ($Br^-$, $H^+$, and $CO_2$) are completely colorless.

  • Measurement methodology: The decreasing intensity of red color is quantitatively tracked over time using a spectrometer to measure $[Br_2]$.

• Average Rate Calculations:

  • Average rate equation:         $\text{Average rate} = -\frac{\Delta [Br_2]}{\Delta t}$

  • Time dependence: The calculated average rate decreases as the elapsed time interval grows longer.

  • Data evaluation between $t = 0\,s$ and $t = 350\,s$:

    • Initial concentration of $Br_2$ at $0\,s$: $0.012\,M$

    • Final concentration of $Br_2$ at $350\,s$: $0.00400\,M$

    • Calculated average rate:             $\text{Average rate} = -\frac{0.00400\,M - 0.012\,M}{350\,s - 0\,s} = -\frac{-0.008\,M}{350\,s} = 2.3 \times 10^{-5}\,M/s$

• Instantaneous Rate Determinations:

  • At time $t = 100\,s$:         $\text{Instantaneous rate} = 2.96 \times 10^{-5}\,M/s$

  • At time $t = 200\,s$:         $\text{Instantaneous rate} = 2.09 \times 10^{-5}\,M/s$

• Rate Law and Rate Constant ($k$):

  • Proportionality: The reaction rate is directly proportional to the molar concentration of bromine:         $\text{Rate} \propto [Br_2]$         $\text{Rate} = k[Br_2]$

  • Rate Constant ($k$): The proportionality constant between the reaction rate and the reactant concentration.

  • Properties of $k$:

    • The value of $k$ is independent of the reactant concentration.

    • The physical units of $k$ depend directly on the overall order of the chemical reaction.

  • Unit Derivation for First-Order Reaction:         $k = \frac{\text{Rate}}{[Br_2]} = \frac{M/s}{M} = \frac{1}{s} = s^{-1}$