Chemical Kinetics

Chemical Kinetics studies the rates of chemical reactions and the factors affecting them.

Overview

graph TD
    A[Chemical Kinetics] --> B[Rate of Reaction]
    A --> C[Rate Laws]
    A --> D[Factors Affecting Rate]
    C --> C1[Order & Molecularity]
    C --> C2[Integrated Rate Laws]
    D --> D1[Temperature]
    D --> D2[Catalyst]
    D --> D3[Concentration]

Rate of Reaction

Average Rate

$$r_{avg} = -\frac{\Delta[R]}{\Delta t} = +\frac{\Delta[P]}{\Delta t}$$

Instantaneous Rate

$$r = -\frac{d[R]}{dt} = +\frac{d[P]}{dt}$$

For General Reaction

For $aA + bB \rightarrow cC + dD$:

$$\boxed{r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt}}$$

Rate Law

$$\boxed{r = k[A]^m[B]^n}$$

where:

• k = rate constant
• m, n = orders with respect to A, B
• (m + n) = overall order

Order vs Molecularity

Property	Order	Molecularity
Definition	Sum of powers in rate law	Number of molecules in elementary step
Value	Can be 0, fraction, integer	Always positive integer (1, 2, 3)
Determination	Experimental	From reaction mechanism
For complex reactions	May differ from stoichiometry	Not applicable

Units of Rate Constant

$$\boxed{k = \frac{r}{[A]^n} \Rightarrow \text{units} = (\text{conc})^{1-n}(\text{time})^{-1}}$$

Order	Units of k
0	mol L⁻¹ s⁻¹
1	s⁻¹
2	L mol⁻¹ s⁻¹
3	L² mol⁻² s⁻¹
n	L^(n-1) mol^(1-n) s⁻¹

Integrated Rate Laws

Zero Order

$$[A] = [A]_0 - kt$$ $$\boxed{t_{1/2} = \frac{[A]_0}{2k}}$$

Characteristics:

• Rate independent of concentration
• [A] vs t is linear

First Order

$$[A] = [A]_0 e^{-kt}$$ $$\boxed{k = \frac{2.303}{t}\log\frac{[A]_0}{[A]}}$$ $$\boxed{t_{1/2} = \frac{0.693}{k} = \frac{\ln 2}{k}}$$

Characteristics:

• t₁/₂ independent of initial concentration
• ln[A] vs t is linear

Second Order

For $A \rightarrow$ products:

$$\frac{1}{[A]} = \frac{1}{[A]_0} + kt$$ $$\boxed{t_{1/2} = \frac{1}{k[A]_0}}$$

Characteristics:

• t₁/₂ inversely proportional to initial concentration
• 1/[A] vs t is linear

Summary Table

Order	Differential	Integrated	Half-life	Linear Plot
0	$-\frac{d[A]}{dt} = k$	$[A] = [A]_0 - kt$	$\frac{[A]_0}{2k}$	[A] vs t
1	$-\frac{d[A]}{dt} = k[A]$	$\ln[A] = \ln[A]_0 - kt$	$\frac{0.693}{k}$	ln[A] vs t
2	$-\frac{d[A]}{dt} = k[A]^2$	$\frac{1}{[A]} = \frac{1}{[A]_0} + kt$	$\frac{1}{k[A]_0}$	1/[A] vs t

Methods to Determine Order

1. Initial rates method: Compare initial rates at different concentrations
2. Graphical method: Plot data and check linearity
3. Half-life method: Check dependence on concentration
4. Ostwald’s isolation method: Keep all but one reactant in excess

Effect of Temperature

Arrhenius Equation

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

Logarithmic form:

$$\ln k = \ln A - \frac{E_a}{RT}$$ $$\log k = \log A - \frac{E_a}{2.303RT}$$

Two-Temperature Form

$$\boxed{\log\frac{k_2}{k_1} = \frac{E_a}{2.303R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)}$$

Activation Energy

graph LR
    A[Reactants] -->|"Ea (forward)"| B[Activated Complex]
    B -->|"Ea (backward)"| C[Products]

$$E_a(forward) - E_a(backward) = \Delta H$$

Collision Theory

Rate:

$$r = Z \times f \times p$$

where:

• Z = collision frequency
• f = fraction with sufficient energy = $e^{-E_a/RT}$
• p = steric factor (probability of proper orientation)

Effective Collisions

For a reaction to occur:

1. Molecules must collide
2. With sufficient energy (≥ E_a)
3. With proper orientation

Factors Affecting Rate

1. Concentration: Higher concentration → more collisions → faster rate
2. Temperature: Higher T → more molecules with E ≥ E_a → faster rate
3. Catalyst: Lowers E_a → faster rate
4. Surface area: More surface → more contact → faster rate (for heterogeneous)
5. Nature of reactants: Ionic reactions are generally faster

Pseudo-First Order Reactions

When one reactant is in large excess, its concentration remains nearly constant.

Example: Hydrolysis of ester

$$CH_3COOC_2H_5 + H_2O \rightarrow CH_3COOH + C_2H_5OH$$

Rate = k[ester][H₂O] ≈ k’[ester] (since [H₂O] ≈ constant)

Practice Problems

1. The half-life of a first-order reaction is 20 minutes. Calculate the rate constant.

2. For a reaction, k = 2.5 × 10⁻³ s⁻¹ at 300 K and 7.5 × 10⁻³ s⁻¹ at 320 K. Calculate E_a.

3. The rate of reaction doubles when temperature increases from 27°C to 37°C. Calculate E_a.

4. For reaction A → B, [A] decreases from 0.1 M to 0.025 M in 40 min. Find order and k.

Further Reading

• Equilibrium
• Thermodynamics