8.1 Reaction stoichiometry and kinetics

Chemical reactions are the heart of engineering processes. Understanding how they work, from the basic math to the nitty-gritty details, is crucial. This topic breaks down the numbers behind reactions and explores what makes them tick.

We'll dive into stoichiometry, reaction rates, and kinetics. These concepts help engineers figure out how much stuff reacts, how fast it happens, and what factors can speed things up or slow them down. It's all about making reactions work better for us.

Chemical Reaction Stoichiometry

Quantitative Relationships in Chemical Reactions

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• Chemical reaction stoichiometry quantifies the relationship between the amounts of reactants and products in a balanced chemical equation
  • Stoichiometric coefficients in a balanced equation represent the relative molar amounts of reactants and products
  • The mole ratio of reactants and products can be derived from the stoichiometric coefficients (e.g., 2 H2 + O2 -> 2 H2O, mole ratio of H2 to O2 is 2:1)
• The limiting reactant completely consumed in a chemical reaction determines the maximum amount of product formed
  • Excess reactants remain after the limiting reactant has been completely consumed
  • Identifying the limiting reactant is crucial for calculating the theoretical yield of a product (e.g., in the reaction 2 Al + 3 CuO -> Al2O3 + 3 Cu, if Al is limiting, it determines the amount of Cu produced)

Conversion and Yield in Chemical Reactions

• The conversion of a reactant represents the fraction or percentage of the reactant converted into products
  • Calculated by dividing the amount of reactant consumed by the initial amount of reactant
  • High conversion is desirable for maximizing product formation and minimizing unreacted reactants
• The yield of a product is the amount of product formed relative to the theoretical maximum based on stoichiometry
  • Actual yield is the experimentally obtained amount of product
  • Theoretical yield is the maximum amount of product calculated from the limiting reactant and stoichiometry
  • Percent yield = (Actual yield / Theoretical yield) × 100% (e.g., if the actual yield of a product is 80 g and the theoretical yield is 100 g, the percent yield is 80%)

Reactor Design Considerations

• Stoichiometric ratio of reactants affects reactor design and operation
  • Reactants must be supplied in the correct proportions to ensure complete conversion and avoid excess reactants
  • Off-stoichiometric ratios can lead to unconverted reactants or side reactions
• Desired conversion and yield influence reactor size, residence time, and operating conditions
  • Higher conversion and yield require larger reactors, longer residence times, or more favorable conditions (e.g., higher temperature or pressure)
  • Optimizing conversion and yield is essential for process efficiency and economics
• Side reactions and byproducts impact reactor design and downstream separation processes
  • Undesired side reactions consume reactants and reduce selectivity to the target product
  • Byproducts may require additional separation steps or disposal considerations (e.g., in the production of ethylene oxide, CO2 is a byproduct that must be removed)

Fundamentals of Chemical Kinetics

Reaction Rates and Rate Laws

• Chemical reaction kinetics studies the rates at which chemical reactions occur and the factors affecting these rates
  • Reaction rate is the change in concentration of a reactant or product per unit time (e.g., mol/L·s)
  • Factors influencing reaction rates include temperature, catalyst presence, and reactant concentrations
• The rate law relates the reaction rate to the concentrations of the reactants raised to a power (the order of the reaction)
  • General form of the rate law: Rate = k[A]^m[B]^n, where k is the rate constant, [A] and [B] are reactant concentrations, and m and n are the orders with respect to each reactant
  • The overall order of a reaction is the sum of the powers to which the reactant concentrations are raised in the rate law (e.g., if m = 1 and n = 2, the overall order is 3)

Rate Constants and Reaction Orders

• The rate constant (k) is a proportionality constant that relates the reaction rate to the reactant concentrations in the rate law
  • The units of the rate constant depend on the overall order of the reaction (e.g., for a first-order reaction, k has units of 1/s; for a second-order reaction, k has units of L/mol·s)
  • The rate constant is temperature-dependent and can be determined experimentally
• The order of a reaction with respect to each reactant is determined by the exponent in the rate law
  • Zero-order: Rate = k, the reaction rate is independent of reactant concentration
  • First-order: Rate = k[A], the reaction rate is directly proportional to the concentration of one reactant
  • Second-order: Rate = k[A]^2 or Rate = k[A][B], the reaction rate depends on the concentrations of one or two reactants squared or multiplied

Factors Affecting Reaction Rates

• Temperature strongly influences reaction rates through its effect on the rate constant
  • Higher temperatures generally increase reaction rates by providing more energy for reactant molecules to overcome the activation energy barrier
  • The Arrhenius equation relates the rate constant to temperature: k = Ae^(-Ea/RT), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the absolute temperature
• Catalysts increase reaction rates by providing an alternative reaction pathway with a lower activation energy
  • Catalysts participate in the reaction but are not consumed, allowing them to be regenerated and reused
  • Examples of catalysts include enzymes in biological systems and metal surfaces in heterogeneous catalysis (e.g., Pt in catalytic converters)
• Reactant concentrations affect reaction rates as described by the rate law
  • Increasing the concentration of reactants generally increases the reaction rate by providing more molecules for collisions and interactions
  • The effect of concentration on reaction rate depends on the order of the reaction with respect to each reactant (e.g., doubling the concentration of a reactant in a first-order reaction doubles the rate)

Rate-Determining Steps in Reactions

Reaction Mechanisms and Elementary Steps

• A reaction mechanism is a stepwise sequence of elementary reactions describing the molecular-level details of a chemical reaction
  • Elementary reactions are the individual steps in a reaction mechanism, each involving a single molecular event (e.g., collision, rearrangement, or dissociation)
  • The overall balanced equation for a reaction can be obtained by summing the elementary steps in the mechanism
• Elementary steps have simple rate laws based on their molecularity (the number of molecules involved in the step)
  • Unimolecular steps have first-order rate laws (e.g., A -> products, Rate = k[A])
  • Bimolecular steps have second-order rate laws (e.g., A + B -> products, Rate = k[A][B])
  • Termolecular steps are rare due to the low probability of three molecules colliding simultaneously

Rate-Determining Steps and Overall Reaction Rates

• The rate-determining step (RDS) is the slowest step in a reaction mechanism and controls the overall rate of the reaction
  • The RDS is often the step with the highest activation energy or the lowest rate constant
  • The overall rate of the reaction cannot exceed the rate of the RDS
• The RDS can be identified by comparing the rates of the individual elementary steps in the mechanism
  • The step with the slowest rate (lowest rate constant or highest activation energy) is the RDS
  • The RDS can be determined experimentally by measuring the effect of reactant concentrations on the overall reaction rate
• The rate law for the overall reaction is determined by the rate-determining step and its molecularity
  • If the RDS is unimolecular, the overall rate law will be first-order
  • If the RDS is bimolecular, the overall rate law will be second-order (e.g., if the RDS is A + B -> products, the overall rate law is Rate = k[A][B])

Factors Influencing the Rate-Determining Step

• In a multi-step reaction mechanism, the RDS can change depending on the reaction conditions
  • Temperature changes can alter the relative rates of elementary steps, potentially causing a different step to become rate-determining
  • Pressure changes can affect the rates of elementary steps involving gaseous species, leading to a shift in the RDS
• Concentration changes can also influence the RDS by altering the relative rates of elementary steps
  • Increasing the concentration of a reactant involved in the RDS can increase the overall reaction rate
  • Changing the concentration of a reactant not involved in the RDS may have little effect on the overall rate
• Identifying the RDS and understanding the factors that influence it are crucial for optimizing reaction conditions and improving overall reaction rates

Reaction Order and Rate Constants

Integrated Rate Laws

• Integrated rate laws relate the concentration of a reactant or product to time for reactions of different orders
  • Zero-order: [A] = [A]0 - kt, where [A]0 is the initial concentration and t is time
  • First-order: ln([A]/[A]0) = -kt or [A] = [A]0e^(-kt)
  • Second-order: 1/[A] = 1/[A]0 + kt
• Integrated rate laws can be used to determine the order of a reaction by plotting concentration data versus time and analyzing the resulting graph
  • Zero-order reactions yield a linear plot of [A] vs. t
  • First-order reactions yield a linear plot of ln[A] vs. t
  • Second-order reactions yield a linear plot of 1/[A] vs. t

Reaction Half-Lives

• The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value
  • For a first-order reaction, the half-life is independent of the initial concentration and can be calculated using the rate constant: t1/2 = ln(2)/k
  • For a second-order reaction, the half-life depends on the initial concentration and can be calculated using the rate constant and initial concentration: t1/2 = 1/(k[A]0)
• The concept of half-life is useful for predicting the time required for a reaction to reach a certain extent of completion
  • After one half-life, 50% of the reactant has been consumed; after two half-lives, 75% has been consumed; after three half-lives, 87.5% has been consumed, and so on
  • The number of half-lives (n) required to reach a specific concentration can be calculated using the equation: [A] = [A]0(1/2)^n

Experimental Determination of Rate Laws

• The method of initial rates can be used to determine the order of a reaction with respect to each reactant
  • The initial rates of the reaction are measured at different initial concentrations of one reactant while keeping the concentrations of other reactants constant
  • The order with respect to the varied reactant can be determined by comparing the ratios of the initial rates and concentrations (e.g., doubling the concentration and observing the effect on the initial rate)
• The Arrhenius equation can be used to determine the activation energy and pre-exponential factor from experimental data
  • The rate constant is measured at different temperatures, and ln(k) is plotted versus 1/T
  • The slope of the resulting line is -Ea/R, and the y-intercept is ln(A)
  • Knowing the activation energy and pre-exponential factor allows for the prediction of rate constants at different temperatures