Reaction Rates

Overview

Reaction rate measures how quickly reactants are consumed or products are formed. Understanding reaction rates is essential for controlling chemical processes in industry and everyday life.

Defining Reaction Rate

For a general reaction:

$$a\text{A} + b\text{B} \rightarrow c\text{C} + d\text{D}$$

Rate of Reaction

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

Negative signs for reactants (concentration decreases) Positive signs for products (concentration increases)

Units

$$\text{Rate} = \text{mol/(L·s)} \text{ or M/s}$$

Average vs Instantaneous Rate

Average Rate

Change in concentration over a time interval:

$$\text{Average Rate} = -\frac{\Delta[\text{A}]}{\Delta t}$$

Instantaneous Rate

Rate at a specific moment (slope of tangent):

$$\text{Instantaneous Rate} = -\frac{d[\text{A}]}{dt}$$

Example Calculation

For: $2\text{N}_2\text{O}_5(g) \rightarrow 4\text{NO}_2(g) + \text{O}_2(g)$

If [N₂O₅] decreases by 0.050 M in 100 s:

$$\text{Rate of disappearance of N}_2\text{O}_5 = \frac{0.050 \text{ M}}{100 \text{ s}} = 5.0 \times 10^{-4} \text{ M/s}$$ $$\text{Rate of reaction} = \frac{1}{2} \times 5.0 \times 10^{-4} = 2.5 \times 10^{-4} \text{ M/s}$$ $$\text{Rate of appearance of NO}_2 = 4 \times 2.5 \times 10^{-4} = 1.0 \times 10^{-3} \text{ M/s}$$ $$\text{Rate of appearance of O}_2 = 1 \times 2.5 \times 10^{-4} = 2.5 \times 10^{-4} \text{ M/s}$$

Factors Affecting Reaction Rate

1. Concentration

Higher concentration → More collisions → Faster rate

2. Temperature

Higher temperature → Faster molecules → More energetic collisions → Faster rate

Rule of Thumb: Rate roughly doubles for every 10°C increase

3. Surface Area

Greater surface area → More exposed particles → Faster rate (For reactions involving solids)

4. Catalysts

Provide alternative pathway with lower activation energy Increase rate without being consumed

5. Nature of Reactants

• Ionic reactions in solution: Very fast
• Covalent bond breaking/forming: Slower
• More bonds to break: Slower

Collision Theory

Requirements for Reaction

1. Collision: Reactant particles must collide
2. Proper Orientation: Molecules must be aligned correctly
3. Sufficient Energy: Collision energy ≥ activation energy ($E_a$)

Effective Collision

Reaction occurs when collision has:

• Enough energy ($E_a$)
• Correct geometry

Measuring Reaction Rates

Common Methods

Method	Measures
Gas volume	Volume of gas produced
Mass change	Mass of solid/gas lost
Color change	Absorbance (spectrophotometry)
Conductivity	Ion concentration
pH change	H⁺ concentration
Pressure change	Gas pressure

Example: Magnesium + Acid

$$\text{Mg}(s) + 2\text{HCl}(aq) \rightarrow \text{MgCl}_2(aq) + \text{H}_2(g)$$

• Measure volume of H₂ produced over time
• Measure mass decrease over time

Initial Rate Method

Measure rate at the very beginning of reaction:

• Concentrations are known precisely
• Reverse reaction is negligible
• No product interference

Procedure

1. Mix reactants at known concentrations
2. Measure concentration change immediately
3. Repeat with different initial concentrations
4. Compare initial rates to determine rate law

Half-Life ($t_{1/2}$)

Time for concentration to decrease to half its initial value.

For First-Order Reactions

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

• Independent of initial concentration
• Constant for a given reaction

For Second-Order Reactions

$$t_{1/2} = \frac{1}{k[\text{A}]_0}$$

• Depends on initial concentration

For Zero-Order Reactions

$$t_{1/2} = \frac{[\text{A}]_0}{2k}$$

• Depends on initial concentration

Summary

Factor	Effect on Rate
↑ Concentration	↑ Rate
↑ Temperature	↑ Rate
↑ Surface Area	↑ Rate
Catalyst	↑ Rate
↑ Activation Energy	↓ Rate