← Chemistry β€” Std 12
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Chemical Kinetics

Ch. 6Std 12

Easy Overview

Some reactions happen in a flash β€” like an explosion. Others take years β€” like rusting or the formation of geological deposits. Chemical kinetics is the study of reaction speeds. How fast does a reaction go? What factors affect that speed? And most importantly, how do we figure out the mechanism β€” the step-by-step molecular choreography β€” of a reaction? The rate of a reaction is how fast reactants disappear or products appear. For A β†’ B: rate = -d[A]/dt = d[B]/dt. The rate is not constant β€” as reactants are used up, the reaction slows down. The instantaneous rate is the slope of the concentration vs time curve at a particular moment. The initial rate (at t = 0) is often the fastest. The rate law is an experimentally determined equation: rate = k[A]^m[B]^n. You CANNOT deduce the rate law from the balanced equation β€” it must be measured. The exponents m and n define the order of the reaction. Zero order: rate is constant, independent of concentration. First order: rate ∝ [A]. Second order: rate ∝ [A]Β² or [A][B]. The rate constant k is specific for each reaction at a given temperature. Integrated rate laws give concentration as a function of time. Half-life (t₁/β‚‚) is the time for half the reactant to be consumed. For first order: t₁/β‚‚ = 0.693/k β€” constant, independent of initial concentration. For zero order: t₁/β‚‚ = [A]β‚€/2k. For second order: t₁/β‚‚ = 1/(k[A]β‚€). The Arrhenius equation describes temperature dependence: k = A e^(-Ea/RT). Activation energy (Ea) is the energy barrier molecules must overcome to react. Catalysts lower this barrier. This chapter gives you the tools to answer: how fast, how long, and at what temperature?

Rate of reaction β€” average and instantaneous

Rate of reaction = change in concentration per unit time. For A β†’ B: rate = -Ξ”[A]/Ξ”t = +Ξ”[B]/Ξ”t. The negative sign for reactants is because their concentration decreases β€” the rate itself is always positive. Average rate = (final - initial concentration) / total time. Instantaneous rate = slope of the tangent to the [A] vs t curve at a specific time. The rate generally decreases as reaction progresses because reactant concentrations drop. The initial rate (at t = 0) is often used for comparing rates under different conditions because at t = 0, concentrations are exactly known and no products have formed yet. Units of rate: mol L⁻¹ s⁻¹ (or M s⁻¹). For gas-phase reactions, pressure units may be used.

Rate law and order of reaction

Rate law = k[A]^m[B]^n. The exponents m and n (orders) are determined EXPERIMENTALLY β€” NOT from stoichiometric coefficients. Methods: (1) Initial rate method β€” vary one reactant concentration at a time while keeping others constant, measure initial rates. If doubling [A] doubles rate β†’ first order in A. If doubling [A] quadruples rate β†’ second order. If no change β†’ zero order. (2) Integrated rate method β€” plot concentration data to see which order fits (linear plot = correct order). (3) Half-life method β€” if t₁/β‚‚ is constant β†’ first order. Overall order = m + n. Example: rate = k[A]Β²[B]ΒΉ β€” overall order = 3 (third order). Units of k vary with overall order: for zero order: mol L⁻¹ s⁻¹; first order: s⁻¹; second order: L mol⁻¹ s⁻¹; third order: LΒ² mol⁻² s⁻¹.

Zero order reactions

Rate = k (constant, independent of concentration). Units of k: mol L⁻¹ s⁻¹. Integrated rate law: [A] = [A]β‚€ - kt. A plot of [A] vs t is linear with slope = -k. Half-life: t₁/β‚‚ = [A]β‚€/2k (depends on initial concentration). Zero-order reactions occur when the rate is limited by a factor other than concentration β€” like surface area in heterogeneous catalysis (decomposition of NH₃ on a hot Pt surface), or light intensity in photochemical reactions. Also occurs in enzyme-catalyzed reactions at high substrate concentration (enzyme is saturated). Example: decomposition of HI on gold surface is zero order because only HI molecules adsorbed on the surface react, and the surface is saturated.

First order reactions

Rate = k[A]. Units of k: s⁻¹ (or time⁻¹). Integrated rate law: ln[A] = ln[A]β‚€ - kt, or in log form: log[A] = log[A]β‚€ - kt/2.303. A plot of ln[A] vs t is linear with slope = -k. Half-life: t₁/β‚‚ = 0.693/k β€” CONSTANT, independent of initial concentration. This is the defining feature of first-order reactions. Examples: radioactive decay (carbon dating), decomposition of Hβ‚‚Oβ‚‚, hydrolysis of sucrose (inversion of cane sugar), decomposition of Nβ‚‚Oβ‚…. The constancy of t₁/β‚‚ makes it easy to identify first-order kinetics. After n half-lives, fraction remaining = (1/2)ⁿ. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. After 10: about 0.1%. Worked example: half-life of a reaction is 100 s. k = 0.693/100 = 0.00693 s⁻¹. Fraction remaining after 300 s (= 3 half-lives) = (1/2)Β³ = 1/8 = 12.5%.

Second order reactions

Rate = k[A]Β² or k[A][B]. Units of k: L mol⁻¹ s⁻¹. For A β†’ products (type 1): integrated rate law: 1/[A] = 1/[A]β‚€ + kt. A plot of 1/[A] vs t is linear with slope = k. Half-life: t₁/β‚‚ = 1/(k[A]β‚€) β€” depends on initial concentration (inversely proportional). For A + B β†’ products (type 2) with different initial concentrations: kt = 1/([B]β‚€ - [A]β‚€) ln([A][B]β‚€/([B][A]β‚€)). Examples: decomposition of HI, dimerization of butadiene, saponification of ethyl acetate. The dependence of t₁/β‚‚ on initial concentration is a key distinguishing feature: zero order (t₁/β‚‚ ∝ [A]β‚€), first order (t₁/β‚‚ constant), second order (t₁/β‚‚ ∝ 1/[A]β‚€).

Pseudo-first-order reactions

When one reactant is in large excess (typically 100Γ— or more), its concentration remains essentially constant during the reaction. The rate then depends only on the limiting reactant. The constant concentration of the excess reactant gets absorbed into the rate constant, making the reaction appear first order. Example: hydrolysis of ethyl acetate in excess water β€” water concentration stays ~55.5 M while ester is ~0.01 M. Rate = k[Hβ‚‚O][ester] β‰ˆ k'[ester] where k' = k[Hβ‚‚O]. Inversion of cane sugar (sucrose β†’ glucose + fructose) in excess water is another example. The rate constant determined experimentally (k') is the pseudo-first-order rate constant. To get the true second-order rate constant: k = k'/[Hβ‚‚O]. This technique simplifies kinetics experiments.

Half-life method for determining order

The relationship between half-life and initial concentration reveals the reaction order. Zero order: t₁/β‚‚ = [A]β‚€/2k β€” t₁/β‚‚ increases with [A]β‚€. First order: t₁/β‚‚ = 0.693/k β€” constant, independent of [A]β‚€. Second order: t₁/β‚‚ = 1/(k[A]β‚€) β€” t₁/β‚‚ decreases as [A]β‚€ increases. In general: t₁/β‚‚ ∝ 1/[A]β‚€^(n-1) for an nth-order reaction (n β‰  1). To determine order experimentally: measure t₁/β‚‚ at different initial concentrations. If t₁/β‚‚ increases with [A]β‚€ β†’ zero order. If t₁/β‚‚ is constant β†’ first order. If t₁/β‚‚ decreases β†’ second order (or higher). This is a powerful experimental method because half-life is easy to measure β€” just track the time when concentration drops to half.

Activation energy β€” the energy barrier

For molecules to react, they must overcome an energy barrier called activation energy (Ea). Even exothermic reactions have an Ea β€” this is why gasoline does not spontaneously ignite at room temperature despite Ξ”G being very negative. Ea is the difference between the energy of the reactants and the transition state (the highest-energy, most unstable configuration along the reaction path). The transition state has partial bonds and cannot be isolated. A potential energy diagram shows the reaction coordinate (x-axis) vs potential energy (y-axis): reactants β†’ transition state (peak) β†’ products. For exothermic reactions, products are at lower energy than reactants. The activation energy for the forward reaction (Ea_forward) is less than Ea_reverse. A high Ea means a slow reaction at room temperature. Only molecules with kinetic energy β‰₯ Ea can react when they collide.

Arrhenius equation β€” temperature dependence

k = A e^(-Ea/RT). A = frequency factor (pre-exponential factor) β€” related to collision frequency and proper orientation. e^(-Ea/RT) = fraction of molecules with energy β‰₯ Ea (the Boltzmann factor). In log form: ln k = ln A - Ea/(RT). An Arrhenius plot (ln k vs 1/T) gives a straight line: slope = -Ea/R, intercept = ln A. Two-point form: ln(kβ‚‚/k₁) = (Ea/R)(1/T₁ - 1/Tβ‚‚). This allows calculation of Ea from rate constants at two temperatures. Worked example: a reaction's rate doubles when T increases from 300 K to 310 K. ln(2) = (Ea/8.314)(1/300 - 1/310) = (Ea/8.314)(0.003333 - 0.003226) = (Ea/8.314)(0.000107). Ea = 0.693 Γ— 8.314 / 0.000107 = 53,900 J/mol = 53.9 kJ/mol. Rule of thumb: a 10Β°C increase roughly doubles the reaction rate near room temperature (for reactions with Ea β‰ˆ 50 kJ/mol). Higher Ea means stronger temperature sensitivity.

Collision theory and transition state theory

Collision theory: for a reaction to occur, molecules must collide with (1) sufficient kinetic energy (β‰₯ Ea), (2) proper orientation at the moment of collision. Rate = p Γ— Z Γ— e^(-Ea/RT), where Z = collision frequency (depends on concentration and temperature), p = steric factor (probability of correct orientation, typically 0.1 to 1). The Arrhenius frequency factor A = p Γ— Z. Not all collisions lead to reaction β€” only those meeting both energy and orientation requirements. Transition state theory: as reactants approach, bonds start to break and form, and the system passes through a high-energy unstable configuration called the activated complex or transition state. It is at the peak of the potential energy diagram. It is not a stable molecule β€” it has partial bonds and cannot be isolated. The activation energy Ea is the energy difference between reactants and the transition state. A catalyst lowers Ea by providing a different reaction pathway with a lower-energy transition state.

Catalysis β€” speeding up reactions

A catalyst increases reaction rate by providing an alternative pathway with lower activation energy. It is not consumed in the reaction and does NOT affect the equilibrium constant or Ξ”G β€” it only helps reach equilibrium faster. Homogeneous catalysts: in the same phase as reactants (acid-catalyzed hydrolysis of esters, Hβ‚‚SOβ‚„ in esterification). Heterogeneous catalysts: in a different phase (solid catalyst with gaseous reactants β€” Fe in Haber process, Vβ‚‚Oβ‚… in Contact process, Pt in catalytic converters). The catalyst provides a surface where reactant molecules adsorb, react, and desorb. Enzymes are biological catalysts with remarkable specificity β€” each enzyme catalyzes only one specific reaction. The lock and key model and induced fit model describe substrate binding to the enzyme's active site. Factors: optimum temperature (~37Β°C for human enzymes), optimum pH (pepsin ~2, trypsin ~8), substrate concentration (Michaelis-Menten kinetics: rate increases then plateaus as enzyme becomes saturated).

Key Points

  • β€’Rate = -d[R]/dt = +d[P]/dt; units: mol L⁻¹ s⁻¹
  • β€’Rate law determined experimentally, NOT from stoichiometry
  • β€’Zero order: rate = k, [A] = [A]β‚€ - kt, t₁/β‚‚ = [A]β‚€/2k
  • β€’First order: rate = k[A], ln[A] = ln[A]β‚€ - kt, t₁/β‚‚ = 0.693/k (constant!)
  • β€’Second order: rate = k[A]Β², 1/[A] = 1/[A]β‚€ + kt, t₁/β‚‚ = 1/k[A]β‚€
  • β€’Pseudo-first-order: one reactant in large excess (rate β‰ˆ k'[limiting])
  • β€’Units of k: zero order β€” mol L⁻¹ s⁻¹; first β€” s⁻¹; second β€” L mol⁻¹ s⁻¹
  • β€’t₁/β‚‚ constant only for first order β€” key to identifying reaction order
  • β€’Linear plots: [A] vs t (zero), ln[A] vs t (first), 1/[A] vs t (second)
  • β€’Activation energy Ea = minimum energy needed for reaction
  • β€’Arrhenius: k = A e^(-Ea/RT); ln k = ln A - Ea/RT
  • β€’Arrhenius plot: ln k vs 1/T β†’ slope = -Ea/R
  • β€’10Β°C rise β‰ˆ doubles rate near room temp for typical Ea (~50 kJ/mol)
  • β€’Collision theory: need collision + energy β‰₯ Ea + proper orientation
  • β€’Transition state: unstable high-energy activated complex
  • β€’Catalyst lowers Ea, does NOT affect Ξ”G or equilibrium constant
  • β€’Enzymes: biological catalysts with high specificity; Michaelis-Menten kinetics

Practice Questions

  • The half-life of a first-order reaction is 100 s. Calculate k. How much remains after 300 s?
  • What is activation energy? Draw a labeled potential energy diagram for an exothermic reaction showing reactants, products, transition state, and Ea.
  • Derive the integrated rate equation for a first-order reaction. Show that t₁/β‚‚ = 0.693/k.
  • How does temperature affect reaction rate? Using the Arrhenius equation, calculate Ea if k doubles when T increases from 300 K to 310 K.
  • Distinguish between zero, first, and second order reactions with one example each.
  • Explain collision theory. What factors determine whether a collision leads to a reaction?
  • A first-order reaction has k = 2.0 Γ— 10⁻³ s⁻¹. What is t₁/β‚‚? What percentage remains after 5 half-lives?
  • Define catalyst. Explain homogeneous and heterogeneous catalysis with examples. How does a catalyst affect Ea?