Chemical Kinetics

1. Average and Instantaneous Rate of Reaction

Disappearance of Reactants :

• For a reactant undergoing change: Reactant → Product.
• The concentration decreases with time.

Where C₁, C₂ are concentrations at times t₁, t₂.

Appearance of Products :

• For a product formed: Reactant → Product.
• The concentration increases with time.

2. Rate Expression and Stoichiometry

• For a general reaction aA + bB → cC + dD:

(+) indicates products and (−) indicates reactants.

3. Factors Affecting Rate of Reaction

1. Concentration :

• Increasing reactant concentration increases the rate of reaction.
• Reason: Number of reactant molecules per unit volume increases, leading to an increase in the number of effective collisions.

2. Temperature :

• Increasing temperature increases the kinetic energy (K.E) of molecules.
• More molecules acquire threshold energy, increasing effective collisions and thus the rate.

3. Temperature & Maxwell-Boltzmann Distribution :

• For temperatures T₃ > T₂ > T₁:
• The Maxwell-Boltzmann distribution shows the fraction of molecules vs energy. The area under the curve to the right of the threshold energy (E₀) increases as temperature increases, representing more molecules with sufficient energy for effective collisions.

4. Catalyst :

• A positive catalyst provides an alternate pathway with lower activation energy (E'_a).
• E'_a < E_a.

Energy profile (PE vs progress of reaction): the uncatalysed path has the higher peak (E_a); the catalysed path has a lower peak (E′_a). Both are measured from the reactant level.

5. Surface Area :

• Increasing surface area increases the rate. This is particularly relevant for solid reactants or solid catalysts.
• Example: Fragmentation of a large solid reactant B into smaller particles increases available surface area for reaction with liquid A.

6. Presence of Light :

• Sunlight (photons) can increase the rate of photochemical reactions.
• H₂ + Cl₂ → 2HCl

4. Rate Law

• The rate law relates the reaction rate to the concentration of reactants (and sometimes products).
• For the reaction: A + 2B → Product
• rate ∝ [A]^x[B]^y ⇒ rate = k[A]^x[B]^y

5. Rate Law for Elementary Reactions

• For a single-step reaction like 1A + 2B → Product:

• Molecularity = number of reactant molecules involved in an elementary reaction.
• For elementary reactions, Molecularity = Order.
• In the example above, Molecularity = 1 + 2 = 3; Order = 1 + 2 = 3.

6. Rate Law for Net Chemical Reactions

• For a net reaction A + 2B → Product, the rate law is given by:

• For net reactions, molecularity has no significance.
• Order (x + y) must be determined experimentally.
• Reaction mechanism example:
  • Step 1: B + B → B* (Intermediate/activated complex)
  • Step 2: B* + A → Product
• Order w.r.t. A = x; Order w.r.t. B = y; Overall order = x + y.

7. Unit of Rate Constant

• For a reaction A → Product, rate = -d[A]/dt = k[A]ⁿ (where n is the order).
• Unit of k = (unit of rate) / (unit of conc)ⁿ.

• Specific cases:

8. Rate Laws for Complex Reactions (Mechanism)

• For a multi-step reaction, the slowest step is the Rate Determining Step (RDS).
• The rate law of the net reaction is determined by the RDS. If the RDS involves an intermediate, express the intermediate in terms of reactants using the equilibrium constant (K_eq) of the preceding fast steps.

9. Zero Order Reaction

• General form: A → Product
• Rate law:
  rate = k[A]⁰ = k
• Differential rate law:
  -d[A]dt = k
• Integrated rate equation:
  [A]₀ - [A]_t = kt

[A]₀ = initial concentration; [A]_t = concentration remaining after time t.

10. Time calculations

• Completion time (t_100%): At t = t_100%, [A]_t = 0.
  t_100% = [A]₀k
• Half-life time (t_1/2): At t = t_1/2, [A]_t = [A]₀2.
  t_1/2 = [A]₀2k
• Three-quarter life (t_75%): At t = t_75%, [A]_t = [A]₀4.
  t_75% = 3[A]₀4k
• Relation between t_3/4 and t_1/2:
  t_3/4 = 32 t_1/2

11. Examples of Zero Order Reactions

1. Photochemical Reactions :

• H₂(g) + Cl₂(g) → 2HCl(g)
• Br₂(g) → 2Br(g)
• Rate ∝ intensity of light (rate ∝ [intensity]¹).

2. Surface Catalysed Reactions :

• 2NH₃(g) →Fe/W/Mo/Pt N₂(g) + 3H₂(g)
• 2HI(g) →Au H₂(g) + I₂(g)
• Note: Decomposition can only take place on the catalyst surface. Once the surface is completely covered, further increase in concentration has no effect on rate.

12. Graphical Representations

• Graphs for rate = k[A]⁰:
  • Rate vs. time: Horizontal line.
  • Rate vs. [A]: Horizontal line.
  • t_1/2 vs. [A]₀: Linear line passing through origin (slope = 12k).
• Graph for [A]₀ - [A]_t = kt:
  • ([A]₀ - [A]_t) vs. time: Straight line through origin (slope = k).
• Graph for [A]_t = [A]₀ - kt:
  • [A]_t vs. time: Downward sloping straight line (intercept on y-axis = [A]₀, intercept on x-axis = [A]₀k, slope = -k).

13. Half-life of Zero Order Reaction

• For a general reaction: aA → Product with rate law rate = k[A]⁰:

where a is the stoichiometric coefficient of the reactant.

• Special Cases:

14. Half-life in Stoichiometric Reactions

• For a reaction: 2A + 3B → Product with rate = k [A]⁰ [B]⁰.
• Assuming 1 L volume with initial moles a and b:

• If the half-lives are equal ((t_1/2)_A = (t_1/2)_B), then a4k = b6k ⇒ ab = 23.
• Conclusion: Irrespective of the order, if the initial amounts of reactants are in their stoichiometric ratio, their half-lives are the same.

15. First Order Kinetics

• i) Completion time (t_100%): [A]_t = 0.

• Note: When given "after a long time", assume [A]_t ≃ 0.
• ii) Half-life time (t_1/2 or t_50%): [A]_t = [A]₀2.

• Note: Half-life for first order is independent of the initial amount of reactant.

16. Time for Completion of Reaction

• For a first-order reaction: t = 2.303k log [A]₀[A]_t

• t(99%) : [A]_t = [A]₀100 ⇒ t(99%) = 203 t(50%)
• t(90%) : [A]_t = [A]₀10 ⇒ t(90%) = 103 t(50%)

Summary of Half-life Relationships :

17. Graphical Representations

• Rate = k[A] = k[A]₀ e^−kt
• [A]_t = [A]₀ e^−kt
• kt = ln([A]₀ / [A]_t) ⇔ ln[A]_t = ln[A]₀ − kt

• Plot of Rate vs Concentration: Straight line passing through origin, slope = k.
• Plot of Rate vs Time: Exponential decay curve starting at y-intercept k[A]₀.
• Plot of [A]_t vs Time: Exponential decay curve starting at y-intercept [A]₀.
• Plot of ln[A]_t vs Time: Straight line with negative slope = −k.
• Plot of ln([A]₀ / [A]_t) vs Time: Straight line passing through origin, slope = k.
• Plot of t(1/2) vs [A]₀: Horizontal line.

18. First Order Kinetics: Gas Phase Reactions

Case 1: A(g) → Products

• For reaction A(g) → Products, where P₀ is the initial pressure of A at t=0 and P_t is the pressure of A at time t:

Case 2: A(g) → 2B(g) + C(g)

• Let P₀ be the initial pressure of the system at t=0.
• Let P_t be the total pressure of the system at time t.

• Total pressure P_t = (P₀ − x) + 2x + x = P₀ + 2x.
• Solving for x: x = P_t − P₀2.
• Pressure of A at time t: P_A = P₀ − x = P₀ − P_t − P₀2 = 3P₀ − P_t2.

19. General Property Method

• If x is a property (e.g., pressure) common to both reactants and products, and we are provided with values x₀ (t=0), x(t) (t=t), and x_∞ (t=∞), then irrespective of the specific reaction:

• For two different times t₁ and t₂:

20. Decomposition of H₂O₂

• Reaction : H₂O₂ → H₂O + 12 O₂
• Volume of KMnO₄ consumed is proportional to the concentration of H₂O₂ remaining.

where V₀ = volume of KMnO₄ at t = 0, and V_t = volume of KMnO₄ at time t.

21. Acid Catalyzed Hydrolysis of Ester

• Reaction : CH₃COOCH₂CH₃ + H₂O →^(H+) CH₃COOH + CH₃CH₂OH
• Pseudo first order rate constant (k_app) depends on [H⁺] and [H₂O].
• If concentration of H⁺ is higher, the reaction rate is faster (e.g., H₂SO₄ provides 2 H⁺ ions per molecule, while HCl provides 1).

where V₀, V_t, and V_∞ are volumes of NaOH/titrant used at t=0, t=t, and t=∞ respectively.

22. Inversion of Cane Sugar

• Reaction : C₁₂H₂₂O₁₁ + H₂O → C₆H₁₂O₆ + C₆H₁₂O₆ (Sucrose → Glucose + Fructose)
• Optical rotation (measured by PPL) changes over time as the mixture transitions from Dextro(+) to Leavo(-).
• r₀ = optical rotation at t = 0
• r_t = optical rotation at t = t
• r_∞ = optical rotation at t = ∞
• The rate follows pseudo first order kinetics : rate = k_app[sucrose].

23. Half-Life for nth Order Reaction

• For a general reaction: A → Product
• Rate law: −d[A]/dt = k[A]ⁿ
• Integrating the rate law:

• For half-life (t_1/2), the concentration of reactant [A]_t = [A]₀/2. Substituting this:

• General proportionality: t_1/2 ∝ 1/[A]₀ⁿ⁻¹
• For first-order (n=1), the half-life is constant: t_1/2 = ln 2 / k ≈ 0.693 / k.

24. First Order Kinetics: Optical Rotation

• For reactions following first order kinetics measured by optical rotation (r_t), the rate constant is given by:

where r₀, r_t, and r_∞ are the optical rotations at time 0, t, and infinity, respectively.

• This formula applies to the inversion of cane sugar and sucrose hydrolysis.

25. Reversible First Order Reaction

• Consider the reversible reaction: A ⇌ B (with forward rate constant k_f and backward rate constant k_b).

• Rate expression: d[B]dt = k_f[A] − k_b[B]
• At equilibrium (t = t_eq): d[B]dt = 0   ⇒   k_f[A] = k_b[B]
• k_fk_b = [B]_eq[A]_eq = x_eqa − x_eq    ⇒    x_eq = k_fk_f + k_b a

26. Parallel First Order Kinetics

• For the parallel decomposition of A: A → B (rate constant k₁) and A → C (rate constant k₂).
• The total rate of disappearance of A is the sum of the rates of the two parallel paths:

• Overall rate constant: k_net = k₁ + k₂
• Overall half-life:

Concentration of products :

• By integration of the rate expressions d[B]/dt = k₁[A] and d[C]/dt = k₂[A]:

27. Stoichiometric Branching (General Case)

• For reaction: A → n₁B (k₁) and A → n₂C (k₂).
• Rates: 1n₁ d[B]dt = k₁[A]  and  1n₂ d[C]dt = k₂[A].
• The ratio of product concentrations at any time t is:

• Percentage yield:

28. Parallel Reactions (Side Reactions)

Material Balance :

• For a reactant A decomposing into products via multiple pathways:
  A →^k₁ B
  A →^k₂ C
• If the reaction starts with only A, the stoichiometry dictates the distribution of products:
  [A]₀ = [A]_t + [B]_tn₁ + [C]_tn₂
• Where n₁ and n₂ are the stoichiometric coefficients of B and C respectively.

Rate of Production Ratio :

• For the parallel reactions:
  A →^k₁ n₁B
  A →^k₂ n₂C
• The ratio of the rate of production of products B and C is given by:
  d[B]/dtd[C]/dt = n₁k₁[A]n₂k₂[A] = n₁k₁n₂k₂
• This ratio is independent of time and independent of [A]_t.

Product Composition :

• For simultaneous 1st order reactions where A →^k₁ B and A →^k₂ C, the percentage of B formed is:
  % B (by mole) = 100 k₁k₁ + k₂

29. Sequential Reaction

• For the sequential reaction A →^k₁ B →^k₂ C:

30. Concentration of B and C at time t

• Concentration of B at time t:

• Concentration of C at time t:

31. Maximum Concentration of B

• Time at which [B]_t is maximum (t_max):

• Value of maximum concentration [B]_max:

32. Concentration vs Time Graph (A → B → C)

• The graph shows [A] decaying from [A]₀, [B] increasing from 0 to a peak and then decreasing, and [C] starting from 0 and plateauing.
• A vertical dashed line at t_max aligns with the peak of curve [B].

33. Steady State Approximation

• For a consecutive reaction scheme: A →k₁ B →k₂ C
• If the rate of formation of intermediate B is assumed to be zero (Steady State Approximation), the concentration of B is:

34. Consecutive Reactions: A → k₁ B → k₂ C

• Case: k₁ >> k₂
• Concentration profiles:

• [A]_t decays exponentially from [A]₀ to zero.
• [B]_t starts at zero, rises to a peak, and then decays towards zero.
• [C]_t starts at zero and rises to approach [A]₀.

35. Sequential Reaction : Case 2 (k₂ >> k₁)

• Reaction : A →^k₁ B →^k₂ C
• [A]_t = [A]₀ e^-k₁t
• [B]_t ≃ 0
• [C]_t = [A]₀ - [A]₀ e^-k₁t = [A]₀ (1 - e^-k₁t)
• Graph description: [A]_t curve decreases from [A]₀. [C]_t curve starts at 0 and increases to [A]₀ asymptotically. [B]_t curve remains near the x-axis.

36. Second & Third Order Reactions

Second Order (n=2) :

• kt = 1(2-1) [ 1[A]_t^2-1 - 1[A]₀^2-1 ] = 1[A]_t - 1[A]₀
• For t_1/2, [A]_t = [A]₀/2 ⇒ t_1/2 = 1k[A]₀

Third Order (n=3) :

• kt = 1(3-1) [ 1[A]_t^3-1 - 1[A]₀^3-1 ] = 12 [ 1[A]_t² - 1[A]₀² ]
• For t_1/2, [A]_t = [A]₀/2 ⇒ t_1/2 = 32k[A]₀²

37. Graphical Representation

• n=2: Rate = k[conc]² (Linear graph, y=rate, x=[conc]², slope=k)
• n=3: Rate = k[conc]³ (Linear graph, y=rate, x=[conc]³, slope=k)
• n=2: t_1/2 = 1k[A]₀ (Linear graph, y=t_1/2, x=1/[A]₀, slope=1/k)
• n=3: Linear graph, y=t_1/2, x=1/[A]₀², slope=3/2k
• n=2: 1[A]_t = kt + 1[A]₀ (Linear graph, y=1/[A]_t, x=time, slope=k, intercept=1/[A]₀)
• n=3: 1[A]_t² = 2kt + 1[A]₀² (Linear graph, y=1/[A]_t², x=time, slope=2k, intercept=1/[A]₀²)

38. Temperature Dependence

• Temperature coefficient (T.C) = k_T+10°k_T ≃ 2-3

39. Collision Theory

• Product is formed due to collisions among reactant molecules.
• Every collision does not lead to product formation; only effective collisions result in products.
• Effective collisions require:
  • (a) Sufficient energy
  • (b) Proper orientation
• To form products, reactant molecules must overcome two barriers:
  • Energy barrier
  • Orientation barrier

40. Arrhenius Equation

• The Arrhenius equation describes the dependence of the rate constant on temperature:

where k = rate constant, T = absolute temperature (K), A = frequency factor (unit of A = unit of k), E_a = activation energy.

• A represents the effective collisions per unit volume per unit time.
• e^−E_a/RT represents the fraction of molecules having energy more than or equal to the threshold energy.

41. Temperature Dependence of Rate Constant

• As temperature (T) changes, the rate constant (k) varies:
• If T → 0, then k → 0.
• If T → ∞, then k → A.

42. Linear Forms of Arrhenius Equation

43. Calculation of Activation Energy

• Activation energy can be calculated using the graphical method by plotting the linear forms of the Arrhenius equation:
• Plot of lnk vs 1/T: A straight line with slope = −E_a/R.
• Plot of logk vs 1/T: A straight line with slope = −E_a/(2.303R).

44. Pressure Relation for First Order Gaseous Reaction

• Reaction: A(g) → nB(g)
• At t = 0: P_A = P₀, P_B = 0, P_total = P₀
• At time t: P_A = P₀e^−kt, P_B = n(P₀ − P₀e^−kt)
• Total Pressure (P_total) = P_A + P_B

45. Catalyst Effect

• A catalyst provides an alternative path for a reaction, lowering the activation energy (E_a → E_a').
• Effect on Potential Energy : E_a decreases, rate of reaction increases.
• Enthalpy Change : ΔH = (E_a)_f − (E_a)_b = (E_a')_f − (E_a')_b (remains unchanged).
• Effect on Rate Constant (k) :
  • k = A e^−E_a/RT (uncatalyzed)
  • k_cat = A e^−E_a'/RT (catalyzed)

46. Net Activation Energy

• For a multi-step complex reaction with net rate constant k_net:

• For k_net = k₁² k₃^1/2k₂³, the net activation energy is E_net = 2E₁ + 12E₃ − 3E₂.

47. Reaction Mechanism & Catalysis

• Reaction profile: The slow step determines the magnitude of the activation energy hill.
• A → B (slow, endothermic) followed by fast steps indicates a high initial activation energy.
• Catalyst effects:
  • Lowers E_a for both forward and backward reactions.
  • Does not alter the Gibbs energy change (ΔG) or enthalpy change (ΔH) of the reaction.
  • Does not make a non-spontaneous reaction spontaneous.

48. Energy Profile and Catalysis

• In a two-step reaction mechanism, the activation energy of the forward reaction is E₁ + E₂ (measured from the reactant level to the highest peak). If the product lies above the reactant, the product is less stable (endothermic).
• Catalyst: A positive catalyst provides an alternative path for the reaction with lower activation energy, thus lowering the peak of the energy profile diagram.

49. Activation Energy Calculation

• Arrhenius equation for two temperatures: ln(k₂k₁) = E_aR (T₂ − T₁T₁T₂).

50. Nuclear Forces

• Postulated to be attractive forces between proton-neutron, neutron-neutron, and proton-proton within the nucleus.
• 10²¹ times stronger than electrostatic forces.
• Operate over a short range.
• They do not follow the inverse square law.

51. Binding Energy (B.E.)

• The energy released that stabilizes the nucleus by binding nucleons together.

If Δm is in grams, E is in MeV per g-atom.

52. Stability of Nucleus

• B.E. per nucleon = 28.144 = 7.03 MeV.
• Conclusion: Higher binding energy per nucleon indicates greater nuclear stability.
• Higher mass defect (Δm) corresponds to higher Energy (E).

53. Neutron-Proton (n/p) Ratio

• For atomic number < 20, the stable ratio is n/p = 1.
• Examples: ₁H¹ (n/p = 0); ₁H³ (n/p = 2).
• The n/p ratio increases from 1.0 to 1.5 as atomic number increases.
• Elements with n/p > 1.5 (atomic number ≥ 84) are radioactive, undergo spontaneous disintegration, and lie outside the belt of stability.

Graph: A plot of "Number of neutrons" vs "Number of protons" shows a stable region curve where the n/p ratio deviates from the n=p line as Z increases.

54. Disintegration and β-emission

• Nature of disintegration depends on the n/p ratio.
• If n/p is high, β-emission occurs to reduce the ratio (n decreases, p increases):

• Examples:
  • ₆C¹⁴ → ₇N¹⁴ + ₋₁e⁰ (n/p ratio changes from 1.3 to 1)
  • ₁₁Na²⁴ → ₁₂Mg²⁴ + ₋₁e⁰ (n/p ratio changes from 13/11 to 1)

55. Alpha Emission

• Emission of alpha particle (⁴₂He) is a form of natural radioactivity.
• Effect on nucleus: Atomic number decreases by 2; Mass number decreases by 4.
• Example: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
• Alpha particle = ⁴₂He⁺⁺. It captures electrons from the atmosphere to convert into Helium gas.
• Soddy-Fajan displacement rule: After alpha emission, the element shifts two places to the left in the periodic table (not applicable for f-block elements).

56. Positron & K-Electron Capture

• Positron emission (artificial): ⁰₊₁e. The n/p ratio increases.
• Examples:
  • ¹¹₆C → ¹¹₅B + ⁰₊₁e
  • ¹³₇N → ¹³₆C + ⁰₊₁e
• K-electron capture (artificial): An electron from the K-shell is captured by the nucleus.
• Example: ¹⁹⁴₇₉Au + ⁰_-1e → ¹⁹⁴₇₈Pt

57. Beta Emission

• Beta particle (⁰_-1e) emission involves the process: ¹₀n → ¹₁H + ⁰_-1e.
• Effect on nucleus: No change in mass number; Atomic number increases by 1.
• Example: ⁴⁰₁₉K → ⁴⁰₂₀Ca + ⁰_-1e
• Beta decay results in isobaric elements. The element shifts 1 unit to the right in the periodic table (not valid for f-block elements).

58. Calculations of Emissions

• Number of alpha-particles emitted = change in mass number4.
• Alpha-decay produces isodiapheres (nuclides with the same difference between neutron and proton numbers).
• Example: ²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He. (For both: n − p = 138 − 88 = 50 and 136 − 86 = 50).
• Successive decay: Emission of 1 alpha and 2 beta particles in succession produces an isotope of the parent element.

59. γ-Emission

• It is electromagnetic radiation; charge-less and mass-less.
• No change in atomic number, mass number, or position in the periodic table.
• Only a change in the energy state of the nucleus is observed.

60. Radioactivity

• Defined as the spontaneous disintegration of nuclei of certain elements by the ejection of α- or β-particles.
• Rate is characteristic for each particular active isotope.
• Radioactive changes are spontaneous and are not controlled by temperature, pressure, or the nature of chemical combination.
• Heavy elements from Bismuth (At. No. 83) through Uranium possess radioactive properties.

61. Radioactive Decay Kinetics

• Radioactive decay follows first-order kinetics:

• Disintegration rate (Activity) is given by:

• Integrating −dN/dt = λN gives:

where λ = decay constant, N_t = no. of active nuclei left after time t.

• Similarly for activity:

where A_t = activity after time t.

62. Comparison of α, β and γ Rays

γ-Rays are electromagnetic rays with very short wavelength (approx. 0.05 Å).

63. Properties of α, β, and γ Rays

64. Units of Radioactivity

• Curie (c): 1 c = 3.7 × 10¹⁰ dps.
• Rutherford (rd): 1 rd = 10⁶ dps.
• Becquerel (Bq): 1 Bq = 1 dps (SI unit).

Conversion Relations :

65. Radioactive Disintegration Series

66. Quick Formula Revision