Chemical Kinetics

Rate of a Chemical Reaction

**Definition:** The rate of a chemical reaction is the change in concentration of a reactant or product per unit time, expressed as:

**Rate of disappearance of reactant R:**

$$r_{av} = -\frac{\Delta[R]}{\Delta t}$$

**Rate of appearance of product P:**

$$r_{av} = +\frac{\Delta[P]}{\Delta t}$$

The negative sign for reactants is used because their concentration decreases, and we want rate to be positive. For products, concentration increases, so no negative sign is needed.

**Units of Rate:** mol L⁻¹ s⁻¹ (or mol L⁻¹ min⁻¹, mol L⁻¹ h⁻¹, atm s⁻¹ for gaseous reactions)

**Average Rate vs Instantaneous Rate:**

**Average Rate (r_av):** Calculated over a finite time interval (Δt is measurable). Useful for getting overall picture of reaction progress. Cannot predict rate at a specific instant.

**Instantaneous Rate (r_inst):** Rate at a specific moment in time. Calculated when Δt approaches zero:

$$r_{inst} = -\frac{d[R]}{dt} = \frac{d[P]}{dt}$$

Mathematically obtained using calculus (derivative). Graphically found by drawing tangent to concentration vs time curve at a specific time point and finding slope of tangent.

**Important Point:** Average rate always decreases with time as reactant concentration decreases, but instantaneous rate at any moment gives accurate picture of how fast reaction proceeds at that moment.

**Expression for Reactions with Unequal Stoichiometric Coefficients:**

For reaction: **aA + bB → cC + dD**

$$\text{Rate} = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = +\frac{1}{c}\frac{\Delta[C]}{\Delta t} = +\frac{1}{d}\frac{\Delta[D]}{\Delta t}$$

Example: For **2HI(g) → H₂(g) + I₂(g)**

$$\text{Rate} = -\frac{1}{2}\frac{\Delta[HI]}{\Delta t} = \frac{\Delta[H_2]}{\Delta t} = \frac{\Delta[I_2]}{\Delta t}$$

This ensures rate of reaction is same regardless of which species we follow (elimination of stoichiometric ambiguity).

Rate Law and Rate Expression

**Definition:** The **rate law** (or rate expression) is a mathematical expression showing how reaction rate depends on concentration of reactants at constant temperature.

**General Form:**

$$\text{Rate} = k[A]^x[B]^y$$

where:

**k** = rate constant (independent of concentration but depends on temperature and catalyst)

**[A], [B]** = molar concentrations of reactants

**x, y** = exponents determined experimentally (NOT from stoichiometric coefficients)

**Differential Form:**

$$-\frac{d[R]}{dt} = k[A]^x[B]^y$$

**Critical Point:** Rate law CANNOT be predicted from balanced equation alone; it must be determined experimentally by studying how initial rate changes with initial concentrations.

**Example:** For reaction **2NO(g) + O₂(g) → 2NO₂(g)**

Initial rate data shows:

When [NO] doubles (keeping [O₂] constant), rate increases 4-fold → order with respect to NO = 2

When [O₂] doubles (keeping [NO] constant), rate doubles → order with respect to O₂ = 1

Therefore: **Rate = k[NO]²[O₂]**

Note: Exponents (2 and 1) differ from stoichiometric coefficients (2 and 1 in this case match, but not always). For **CHCl₃ + Cl₂ → CCl₄ + HCl**, experimentally determined rate = k[CHCl₃][Cl₂]^(1/2), where exponent ½ ≠ stoichiometric coefficient 1.

Order of a Reaction

**Definition:** The **order of a reaction** is the sum of exponents of concentration terms in the rate law expression.

$$\text{Overall Order} = x + y$$

**Types Based on Order:**

**Zero Order Reaction:**

Rate independent of reactant concentration

Rate = k (constant)

Example: Photochemical reactions, heterogeneous catalysis on catalyst surface saturated with reactant

**First Order Reaction:**

Order = 1

Rate ∝ [A]

Rate = k[A]

**Second Order Reaction:**

Order = 2

Rate ∝ [A]² or [A][B]

**Fractional Order:**

Order can be ½, 3/2, etc. (common in complex mechanisms)

Example: **Rate = k[A]^(1/2)[B]^(3/2)** has order = 1/2 + 3/2 = 2

**Units of Rate Constant:**

Rate constant units depend on overall order:

| Order | Rate Equation Form | Units of k |

|-------|-------------------|------------|

| Zero | Rate = k | mol L⁻¹ s⁻¹ |

| First | Rate = k[A] | s⁻¹ |

| Second | Rate = k[A]² or k[A][B] | L mol⁻¹ s⁻¹ |

| Third | Rate = k[A]³ etc. | L² mol⁻² s⁻¹ |

**Derivation of units:** Since Rate (mol L⁻¹ s⁻¹) = k × [concentration terms]^n

$$k = \frac{\text{Rate}}{[\text{concentration}]^n} = \frac{\text{mol L}^{-1}\text{s}^{-1}}{(\text{mol L}^{-1})^n} = \text{mol}^{1-n}\text{L}^{n-1}\text{s}^{-1}$$

Molecularity and Order Distinction

**Molecularity:**

Number of molecules/atoms/ions that participate in an elementary reaction

Determined from reaction mechanism

Always positive integer

Cannot exceed 3 (termolecular reactions extremely rare)

Examples: **NH₄NO₂ → N₂ + 2H₂O** (unimolecular), **2HI → H₂ + I₂** (bimolecular), **2NO + O₂ → 2NO₂** (termolecular)

**Order:**

Sum of exponents in rate law

Determined experimentally

Can be 0, fractional, or integer

Related to overall reaction mechanism, not individual elementary steps

**Key Difference:** Molecularity applies to elementary steps; order applies to overall reaction. For complex reactions, they differ. Overall order = order of rate-determining step + contributions from pre-equilibrium steps.

Elementary and Complex Reactions

**Elementary Reaction:**

Reaction occurring in single step

Mechanism and rate law match with stoichiometric equation

No intermediates involved

**Complex (Multi-step) Reaction:**

Reaction occurring in 2+ elementary steps

Rate law is complex function of concentrations

Involves intermediates (substances formed and consumed within reaction)

Balanced equation is sum of all steps with intermediates canceling

**Example:** Hydrogen peroxide decomposition (catalyzed by I⁻):

**Overall:** 2H₂O₂ → 2H₂O + O₂

**Mechanism (2 elementary steps):**

1. H₂O₂ + I⁻ → H₂O + IO⁻ (slow step = rate-determining)

2. H₂O₂ + IO⁻ → H₂O + I⁻ + O₂ (fast step)

**Intermediate:** IO⁻ (formed in step 1, consumed in step 2)

**Observed Rate Law:** Rate = k[H₂O₂][I⁻]

This can be derived from mechanism:

Rate = rate of slowest step = k₁[H₂O₂][I⁻]

**Rate-Determining Step (RDS):**

The slowest elementary step controlling overall reaction rate

Determines rate law of overall reaction

Like relay race: team's speed = slowest runner's speed

**Determining Rate Law from Mechanism:**

If rate law involves species in fast pre-equilibrium steps, use equilibrium constant to eliminate intermediate:

Example: Reaction with fast pre-equilibrium:

1. A + B ⇌ C (fast, equilibrium)

2. C + D → E (slow)

For step 1: K = [C]/([A][B]), so [C] = K[A][B]

Rate = k₂[C][D] = k₂K[A][B][D] = k_overall[A][B][D]

Observed rate law includes all reactants affecting slow step via equilibrium.

Integrated Rate Laws

**For Zero Order Reaction:**

Rate = k (independent of concentration)

Integrated form (by integration):

$$[A] = [A]_0 - kt$$

Half-life: $$t_{1/2} = \frac{[A]_0}{2k}$$

Linear plot: [A] vs t gives straight line with slope = -k

**For First Order Reaction:**

Rate = k[A]

Integrated form (differential equation solution):

$$\ln[A] = \ln[A]_0 - kt$$

or

$$\ln\frac{[A]_0}{[A]} = kt$$

**Half-life formula (key result):**

$$t_{1/2} = \frac{0.693}{k} = \frac{\ln 2}{k}$$

Important: **First order half-life is independent of initial concentration** — each half-life period reduces concentration by half, regardless of starting amount.

Linear plot: ln[A] vs t gives straight line with slope = -k

**Determining Reaction Order Experimentally:**

1. **Graphical Method:** Plot concentration data in different forms:

If [A] vs t is linear → zero order

If ln[A] vs t is linear → first order

If 1/[A] vs t is linear → second order

2. **Half-life Method:** If each successive half-life doubles (for zero order) or remains constant (for first order), determine order

3. **Initial Rate Method:** Vary one reactant concentration while keeping others constant; compare how rate changes

Temperature Dependence: Arrhenius Equation

**Arrhenius Equation:**

$$k = Ae^{-E_a/RT}$$

or logarithmic form:

$$\ln k = \ln A - \frac{E_a}{RT}$$

where:

**k** = rate constant

**A** = pre-exponential factor (frequency factor, related to collision frequency)

**E_a** = activation energy (minimum energy required for reaction)

**R** = gas constant = 8.314 J mol⁻¹ K⁻¹

**T** = absolute temperature (K)

**Important Points:**

Activation energy determines temperature sensitivity: higher E_a → larger change in k with temperature

Pre-exponential factor A relates to entropy and orientation: A = frequency of collisions × steric factor × orientation factor

Increasing temperature increases k because more molecules have energy ≥ E_a

**Comparing Two Temperatures:**

$$\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$

Or rearranged:

$$\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{T_2 - T_1}{T_1T_2}\right)$$

**Determining E_a:**

From two rate constants at different temperatures or from slope of ln k vs 1/T plot (slope = -E_a/R)

Collision Theory of Reaction Rates

**Basic Postulates:**

1. **Collision Requirement:** For reaction to occur, reactant molecules must collide

2. **Energy Requirement:** Colliding molecules must possess minimum energy (activation energy E_a)

3. **Orientation Requirement:** Molecules must be properly oriented at moment of collision for bonds to form/break

**Reaction Rate Expression from Collision Theory:**

$$k = Az e^{-E_a/RT}$$

where:

**A** = collision frequency (number of collisions per unit volume per unit time)

**z** = collision frequency factor

**e^(-E_a/RT)** = fraction of molecules with energy ≥ E_a

**p** (steric factor) = accounts for proper orientation; included in A

Smaller steric factor p means fewer successful collisions despite proper energy.

**Example:** For 2HI → H₂ + I₂, collision theory predicts rate ∝ [HI]². Experimental rate law = k[HI]² confirms bimolecular mechanism. Activation energy from Arrhenius equation gives insight into reaction pathway.

**Energy Diagram:**

Reactants start at one energy level

Must surmount energy barrier (E_a) to reach transition state

Products may be at higher or lower energy depending on ΔG

For exothermic: products lower energy; endothermic: products higher energy

**Catalyst Effect:**

Catalyst lowers E_a by providing alternative reaction pathway

Increases fraction of molecules with E ≥ E_a(new)

Increases rate constant k without changing thermodynamic feasibility (ΔG)

Not consumed in reaction; recovered unchanged

Effect of Concentration on Rate

**Observation:** As reactants are consumed, rate decreases because:

Concentration of reactants decreases

For reactions with order > 0, lower [A] means lower rate

Zero order reactions maintain constant rate until reactant depleted

**Example:** Hydrolysis of butyl chloride C₄H₉Cl + H₂O → C₄H₉OH + HCl

Initial data at higher concentrations: rate ≈ 1.90 × 10⁻⁴ mol L⁻¹ s⁻¹

Later data at lower concentrations: rate ≈ 0.4 × 10⁻⁴ mol L⁻¹ s⁻¹

This continuous decrease is expected for any order > 0.

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Summary Table: Key Formulas and Relationships

| Concept | Formula | Key Points |

|---------|---------|-----------|

| Average Rate | r_av = -Δ[R]/Δt | Over finite interval |

| Instantaneous Rate | r_inst = d[R]/dt | At specific moment; graphical tangent |

| Rate Law | Rate = k[A]^x[B]^y | Must be determined experimentally |

| Order | n = x + y | Sum of exponents; can be fractional |

| Zero Order Half-life | t₁/₂ = [A]₀/2k | Depends on initial concentration |

| First Order Half-life | t₁/₂ = 0.693/k | Independent of initial concentration |

| Arrhenius Equation | k = Ae^(-E_a/RT) | Temperature dependence |

| Two-Temperature Arrhenius | ln(k₂/k₁) = (E_a/R)(1/T₁ - 1/T₂) | Calculating E_a |

| Stoichiometric Rate | Rate = -(1/a)Δ[A]/Δt = (1/c)Δ[C]/Δt | Coefficients normalize rates |