Calculating Reaction Rates, Collision Theory, and Reaction Mechanisms

Calculating Reaction Rates Practice

• Reaction components and chemical equation: Reaction of solid zinc with aqueous hydrochloric acid to produce hydrogen gas and aqueous zinc chloride.

  • Chemical Equation: $Zn_{(s)} + 2HCl_{(aq)} \rightarrow H_{2(g)} + ZnCl_{2(aq)}$

• Experimental Setup and Data:

  • Initial conditions: A piece of zinc is dropped into $1.00\,dm^3$ ($1.00\text{ L}$) of $0.100\text{ M } HCl$.

  • Time vs. Mass Data:

  • $0\text{ s}$: $0.016\text{ g}$

  • $4\text{ s}$: $0.014\text{ g}$

  • $8\text{ s}$: $0.012\text{ g}$

  • $12\text{ s}$: $0.010\text{ g}$

  • $16\text{ s}$: $0.008\text{ g}$

  • $20\text{ s}$: $0.006\text{ g}$

• Rate Calculations and Conceptual Questions:

  • Average rate of reaction in grams of $Zn$ consumed per second:

  • $\text{Rate of Reaction} = -\frac{\Delta \text{mass}}{\Delta t} = -\left(\frac{0.006\text{ g} - 0.016\text{ g}}{20\text{ s}}\right) = 5 \times 10^{-4}\text{ g\,s}^{-1}$

  • Average rate of reaction in moles of $Zn$ consumed per second:

  • Molar mass of $Zn$ ($M$) = $65.38\text{ g\,mol}^{-1}$

  • Formula: $n = \frac{m}{M}$

  • Rate: $\frac{5 \times 10^{-4}\text{ g\,s}^{-1}}{65.38\text{ g\,mol}^{-1}} = 7.65 \times 10^{-6}\text{ mol\,s}^{-1} \approx 8 \times 10^{-6}\text{ mol\,s}^{-1}$

  • Effect on $[HCl]$ as reaction proceeds: Concentration decreases as $HCl$ is consumed by reacting with zinc.

  • Effect on $[ZnCl_2]$ as reaction proceeds: Concentration increases because $ZnCl_2$ is a product formed in the reaction.

• Reaction of Magnesium with Hydrochloric Acid:

  • Chemical Description: Reaction of magnesium metal with dilute hydrochloric acid to form hydrogen gas and aqueous magnesium chloride.

  • Balanced Chemical Equation: $Mg_{(s)} + 2HCl_{(aq)} \rightarrow H_{2(g)} + MgCl_{2(aq)}$

  • Rate Stoichiometry Problems:

  • Given: Rate of consumption of magnesium = $5.0 \times 10^{-9}\text{ mol\,s}^{-1}$

  • Rate of consumption of $HCl$: $5.0 \times 10^{-9}\text{ mol\,s}^{-1} \times \left(\frac{2\text{ mol } HCl}{1\text{ mol } Mg}\right) = 1.0 \times 10^{-8}\text{ mol\,s}^{-1}\text{ } HCl$

  • Rate of production of $H_2$: $5.0 \times 10^{-9}\text{ mol\,s}^{-1} \times \left(\frac{1\text{ mol } H_2}{1\text{ mol } Mg}\right) = 5.0 \times 10^{-9}\text{ mol\,s}^{-1}\text{ } H_2$

  • Mass of Magnesium consumed in $5.0\text{ minutes}$:

  • Time conversion: $5.0\text{ min} = 300\text{ s}$

  • Step 1 (Moles consumed): $n_{Mg} = 5.00 \times 10^{-9}\text{ mol\,s}^{-1} \times 300\text{ s} = 1.5 \times 10^{-6}\text{ mol}$

  • Step 2 (Mass calculation): $m = n \times M = 1.5 \times 10^{-6}\text{ mol} \times 24.305\text{ g\,mol}^{-1} = 3.645 \times 10^{-5}\text{ g} \approx 3.65 \times 10^{-5}\text{ g}$

Collision Theory & Rate Factors

• Requirements for Chemical Reaction (Collision Theory):

  • For two substances to react, chemical bonds must break and form. Reacting species must:

  • Collide with one another.

  • Have correct orientation to overcome activation energy barrier.

  • Possess enough energy (kinetic energy equal to or greater than activation energy).

  • It is very rare for all required criteria to happen simultaneously during a collision.

• Key Energy Definitions:

  • Kinetic Energy: The energy of movement; temperature is a measurement of average kinetic energy.

  • Activation Energy ($E_a$): The minimum amount of energy required for a chemical reaction to occur.

• Potential Energy (PE) Diagrams:

  • Exothermic Reaction:

  • Potential energy of reactants is higher than potential energy of products.

  • Peak of graph represents the transition state.

  • Activation energy ($E_a$) is the energy difference between reactants and the transition state.

  • Enthalpy change ($\Delta H$) is negative (releases heat to surroundings).

  • Endothermic Reaction:

  • Potential energy of reactants is lower than potential energy of products.

  • Peak of graph represents the transition state.

  • Activation energy ($E_a$) is the large energy difference between reactants and the transition state.

  • Enthalpy change ($\Delta H$) is positive (absorbs heat from surroundings).

• Activation Energy for Reversible Reactions:

  • Reversible System Example: $CO_{(g)} + NO_{2(g)} \rightleftharpoons CO_{2(g)} + NO_{(g)}$

  • Reaction profile elements:

  • Reactants: $CO_{(g)} + NO_{2(g)}$

  • Peak: Activated complex

  • Forward Activation Energy ($E_{a(fwd)}$): Energy barrier from reactants to activated complex.

  • Reverse Activation Energy ($E_{a(rev)}$): Energy barrier from products to activated complex.

  • Products: $CO_{2(g)} + NO_{(g)}$

  • Enthalpy change ($\Delta H$): Difference in potential energy between reactants and products.

Summary of Factors Affecting Rate

• Factors Affecting Reaction Rates Table:

  • Concentration:

  • Effect on Rate: Rate increases as concentration ($[\quad]$) increases ($\text{Rate} \uparrow\text{ as }[\quad] \uparrow$).

  • Reason: More reactant molecules present per unit volume leading to a higher frequency/chance of collisions.

  • Pressure (Gases Only):

  • Effect on Rate: Rate increases as pressure increases ($P \uparrow$).

  • Reason: Decreased volume (Boyle's Law) forces particles closer together, creating a higher chance of collision.

  • Surface Area:

  • Effect on Rate: Rate increases as surface area increases (Rate $\uparrow$ as Surface Area $\uparrow$).

  • Reason: Exposes more solid/liquid particles to the outer boundary, increasing collision opportunities (e.g., sugar dissolving faster when finely ground).

  • Temperature:

  • Effect on Rate: Rate increases as temperature increases ($T \uparrow$).

  • Reason: Increasing $T$ adds kinetic energy, making particles move faster with more energy to overcome activation energy ($E_a$).

  • Catalyst:

  • Effect on Rate: Rate increases with addition of a catalyst.

  • Reason: Provides an alternate reaction pathway with a lower activation energy ($E_a$), enabling a greater proportion of molecules to possess sufficient energy to react.

Mechanistic Details of Rate Factors

• Fundamental Determinants of Reaction Rate:

  • Determined by frequency of particle collisions.

  • Determined by proportion of collisions that result in broken bonds, which depends on collision energy and collision orientation.

  • Any factor influencing collision frequency or collision energy/orientation affects reaction rate.

• Detailed Mechanism of Factors:

  • Temperature:

  • Higher temperature causes reactant particles to move faster, giving them more kinetic energy.

  • Faster movement increases collision frequency.

  • Higher kinetic energy means a larger proportion of collisions meet or exceed activation energy ($E_a$) to break bonds, further increasing rate.

  • Concentration and Pressure:

  • An increase in concentration of solutions or pressure of gases increases number of particles in a given volume.

  • More particles in a given space yields more total collisions, increasing rate.

  • Surface Area:

  • Increasing surface area of solids or liquids exposes more surface particles to collisions.

  • Greater number of exposed particles leads to more collisions, increasing rate.

  • Stirring:

  • Stirring a heterogeneous reaction mixture (mixture with two or more phases) increases collision rate between reactant particles, increasing overall reaction rate.

  • Catalysts:

  • Catalysts lower activation energy ($E_a$) required to break bonds by orienting molecules in specific alignments that weaken existing bonds.

  • Lower activation energy means a larger proportion of collisions have sufficient energy to react, increasing rate.

  • Catalysts are not consumed or permanently changed in chemical reactions.

  • Biological catalysts are called enzymes.

• Comprehensive Impact Matrix:

  • Increasing temperature: Increases collision number; does not change $E_a$ but increases kinetic energy so more particles overcome $E_a$ ; increases reaction rate.

  • Increasing pressure of gases: Increases collision number; no effect on collision energy; increases reaction rate.

  • Increasing concentration of solutions: Increases collision number; no effect on collision energy; increases reaction rate.

  • Increasing surface area of solid/liquid reactants: Increases collision number; no effect on collision energy; increases reaction rate.

  • Stirring heterogeneous reaction mixtures: Increases collision number; no effect on collision energy; increases reaction rate.

  • Using a catalyst: No effect on collision number; lowers $E_a$ so more particles can overcome it; increases reaction rate.

• Summary Matching of Factors to Primary Effects:

  • Temperature: Affects the speed of particles, thereby influencing both frequency and kinetic energy of collisions.

  • Pressure and concentration: Affects the number of particles in a given volume, thereby influencing collision frequency.

  • Surface area and stirring: Affects the quantity of reactant particles exposed to collisions, thereby influencing collision frequency.

  • Catalyst: Lowers energy barrier required for collisions to break existing bonds.

Rate Laws

• General Rate Law Expression:

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

  • Rate Law formula: $r = k [A]^x [B]^y$

• The Rate Constant ($k$):

  • Value varies with temperature only.

  • If $k$ is very large, reaction proceeds very rapidly.

  • If $k$ is very small, reaction proceeds very slowly.

• Reaction Orders:

  • Overall Order: Sum of individual reactant exponents in the rate law ($x + y$).

  • Order with Respect to Specific Reactants: Expressed as "$x$th order for reactant $A$" and "$y$th order for reactant $B$".

• Order Characteristics Table:

  • 0th Order (Zero Order):

  • Effect on Rate: Rate is completely unaffected by concentration of any reactant ($A + B \rightarrow AB$).

  • Rate Law Equation: $r = k [A]^0 [B]^0 \Rightarrow r = k$

  • Concentration vs. Time Graph: Linear downward slope with constant negative gradient.

  • 1st Order (First Order):

  • Effect on Rate: Rate is directly proportional to concentration of a single reactant.

  • Rate Law Equation: $r = k [A]^1$ or $r = k [B]^1$

  • Concentration vs. Time Graph: Exponential decay curve.

  • 2nd Order (Second Order):

  • Effect on Rate: Rate is proportional to concentration of both reactants combined or proportional to the square of concentration of one reactant ($[\quad]^2$).

  • Rate Law Equation: $r = k [A]^1 [B]^1$, $r = k [A]^2$, or $r = k [B]^2$

  • Concentration vs. Time Graph: Steep exponential decay curve.

Reaction Mechanisms

• Important Mechanism Terms:

  • Reaction Mechanism: A series of elementary steps that make up an overall chemical reaction (analogous to Hess' Law steps).

  • Elementary Step: A single simple step/reaction in a multi-step reaction mechanism that adds up to the overall reaction.

  • Reaction Intermediate: A chemical species produced in one elementary step and subsequently consumed in a later elementary step.

  • Catalyst: A substance that is consumed in an early elementary step and subsequently regenerated as a product in a later step (remains unconsumed overall).

• Example Reaction Mechanism:

  • Overall Reaction: $2N_2O_{(g)} \rightarrow 2N_{2(g)} + O_{2(g)}$

  • Elementary Step 1: $N_2O \rightarrow N_2 + O$

  • Elementary Step 2: $N_2O + O \rightarrow N_2 + O_2$

• Molecularity of Elementary Steps:

  • Elementary steps typically involve 1 or 2 colliding molecules.

  • Unimolecular: Involves 1 reactant molecule in the elementary step.

  • Equation: $A \rightarrow \text{products}$

  • Elementary Rate Law: $r = k [A]$

  • Bimolecular: Involves 2 reactant molecules colliding in the elementary step.

  • Equation: $A + B \rightarrow \text{products} \implies r = k [A][B]$

  • Equation: $2A \rightarrow \text{products} \implies r = k [A]^2$

• Rate-Determining Step (RDS):

  • Reaction rates are determined experimentally.

  • For a reaction mechanism, the slowest elementary step dictates the overall rate of reaction.

  • Slowest step is termed the Rate-Determining Step (RDS).

  • Stoichiometric coefficients of reactants in the RDS equation determine the rate law for the overall reaction.

Oxidation of HBr Mechanism Problem

• Overall Reaction System:

  • Balanced overall equation: $4HBr_{(g)} + O_{2(g)} \rightarrow 2H_2O_{(g)} + 2Br_{2(g)}$

  • Proposed multi-step mechanism:

  • Step 1: $HBr_{(g)} + O_{2(g)} \rightarrow HOOBr_{(g)}$

  • Step 2: $HOOBr_{(g)} + HBr_{(g)} \rightarrow 2HOBr_{(g)}$

  • Step 3: $2\left(HOBr_{(g)} + HBr_{(g)} \rightarrow H_2O_{(g)} + Br_{2(g)}\right) \Rightarrow 2HOBr_{(g)} + 2HBr_{(g)} \rightarrow 2H_2O_{(g)} + 2Br_{2(g)}$

• Analysis Questions and Solutions:

  • Elementary Steps Addition Verification:

  • Adding all three steps together:       $\begin{aligned} HBr + O_2 &\rightarrow HOOBr \\ HOOBr + HBr &\rightarrow 2HOBr \\ 2HOBr + 2HBr &\rightarrow 2H_2O + 2Br_2 \end{aligned}$

  • Canceling intermediate species ($HOOBr$ and $2HOBr$) on both sides yields: $4HBr + O_2 \rightarrow 2H_2O + 2Br_2$

  • Result: Yes, elementary steps add up to the overall balanced equation.

  • Elementary Step Rate Laws:

  • Step 1 Rate Law: $r_1 = k_1 [HBr][O_2]$

  • Step 2 Rate Law: $r_2 = k_2 [HOOBr][HBr]$

  • Step 3 Rate Law: $r_3 = k_3 [HOBr]^2 [HBr]^2$

  • Identification of Intermediates:

  • Chemical species $HOOBr$ and $HOBr$ are intermediates because both are produced in one step and consumed in a subsequent step.

  • Identification of Catalysts:

  • There are no catalysts in this reaction, as no chemical species is introduced as an initial reactant and subsequently regenerated as a final product.

  • Rate-Determining Step Deduction:

  • Experimental finding: Overall reaction is first order with respect to $HBr$ and first order with respect to $O_2$ ($r_{exp} = k [HBr]^1 [O_2]^1$).

  • Comparison: Rate law of Step 1 ($r_1 = k_1 [HBr][O_2]$) matches experimental rate law.

  • Conclusion: Step 1 must be the slow rate-determining step (RDS) that dictates the overall reaction rate.