💥 Collision Theory

Part 1 of 7 — Why Do Molecules Need to Collide?

For a chemical reaction to occur, reactant molecules must collide. But not just any collision will do — it must be an effective collision. Collision theory explains what makes a collision productive and connects molecular behavior to macroscopic reaction rates.

Requirements for an Effective Collision

For a collision to result in a reaction, two conditions must be met simultaneously:

Condition 1: Sufficient Energy

The colliding molecules must have kinetic energy at least equal to the activation energy ($E_a$):

$KE_{\text{collision}} \geq E_a$

If the collision energy is below $E_a$, the molecules simply bounce off each other without reacting.

Condition 2: Proper Orientation

Even with enough energy, the molecules must collide with the correct geometric orientation. The reactive parts of the molecules must be facing each other.

Example: NO + NO₃ → 2NO₂

• ✅ O of NO hits O of NO₃ → bonds can rearrange → reaction!
• ❌ N of NO hits N of NO₃ → wrong atoms in contact → no reaction

The Steric Factor

The fraction of collisions with correct orientation is called the steric factor ($p$), typically $0 < p < 1$. For simple atoms, $p \approx 1$; for complex molecules, $p$ can be very small.

Collision Theory Concepts 🎯

Maxwell-Boltzmann Distribution

At any temperature, molecules have a distribution of kinetic energies. The Maxwell-Boltzmann distribution shows:

• Most molecules have moderate energies
• A few have very low or very high energy
• The area under the curve beyond $E_a$ represents the fraction of molecules that can react

Effect of Temperature

When temperature increases:

1. The peak shifts to higher energy and becomes lower and broader
2. The fraction of molecules with $KE \geq E_a$ increases dramatically
3. This is why higher temperature → faster rate

The Boltzmann Factor

The fraction of molecules with energy $\geq E_a$ is approximately:

$f = e^{-E_a/(RT)}$

This exponential dependence explains why even small temperature changes can produce large rate changes.

Maxwell-Boltzmann Distribution 🔍

Collision Theory Rate Equation

Putting it all together, collision theory predicts:

$\text{Rate} = Z \cdot p \cdot e^{-E_a/(RT)}$

where:

• $Z$ = collision frequency (depends on concentration and temperature)
• $p$ = steric factor (orientation)
• $e^{-E_a/(RT)}$ = fraction with sufficient energy

Connection to the Arrhenius Equation

This leads directly to the Arrhenius equation:

$k = A \cdot e^{-E_a/(RT)}$

where $A = Z \cdot p$ is the frequency factor (also called the pre-exponential factor). We will derive this in detail in Part 3.

Collision Theory Calculations 🧮

1. At 300 K, the Boltzmann factor for a reaction with $E_a = 50$ kJ/mol is $e^{-E_a/(RT)}$. Calculate $E_a/(RT)$. ($R = 8.314$ J/(mol·K); round to 3 significant figures)

2. Calculate $e^{-20.0}$ to 2 significant figures. (Use scientific notation: e.g., 2.1e-9)

3. If temperature increases from 300 K to 310 K ($E_a = 50$ kJ/mol), calculate $E_a/(RT)$ at 310 K. (to 3 significant figures)

Exit Quiz — Collision Theory ✅

⛰️ Activation Energy & Energy Diagrams

Part 2 of 7 — The Energy Landscape of Reactions

Energy diagrams (also called reaction coordinate diagrams or potential energy diagrams) are one of the most important tools in chemistry. They show the energy changes that occur as reactants transform into products, and reveal the activation energy barrier that must be overcome.

Anatomy of an Energy Diagram

An energy diagram plots potential energy (y-axis) vs. reaction progress (x-axis):

Key Features

Mathematical Relationship

$E_a(\text{reverse}) = E_a(\text{forward}) - \Delta H$

Or equivalently:

$\Delta H = E_a(\text{forward}) - E_a(\text{reverse})$

Exothermic vs. Endothermic Diagrams

Exothermic ($\Delta H < 0$): Products LOWER than Reactants

• Products are more stable (lower energy)
• $E_a(\text{forward}) < E_a(\text{reverse})$
• Energy is released to surroundings
• Example: Combustion reactions

$\text{Reactants} \xrightarrow{E_a} \text{Transition State} \rightarrow \text{Products (lower energy)}$

Endothermic ($\Delta H > 0$): Products HIGHER than Reactants

• Products are less stable (higher energy)
• $E_a(\text{forward}) > E_a(\text{reverse})$
• Energy is absorbed from surroundings
• Example: Dissolving NH₄NO₃

$\text{Reactants} \xrightarrow{E_a} \text{Transition State} \rightarrow \text{Products (higher energy)}$

Important

$E_a$ is always positive — it is always an energy barrier that must be overcome, regardless of whether the reaction is exo- or endothermic.

Energy Diagram Quiz 🎯

The Transition State

The transition state (or activated complex) is the configuration of atoms at the energy maximum. It is:

• Not a real molecule — it cannot be isolated or observed directly
• Fleeting — exists for approximately $10^{-13}$ seconds
• Characterized by partial bonds — old bonds are partially broken, new bonds are partially formed
• Denoted with a double dagger: $\ddagger$ (e.g., $[ABC]^\ddagger$)

Example: SN2 Reaction

$\text{HO}^- + \text{CH}_3\text{Br} \rightarrow [\text{HO---CH}_3\text{---Br}]^\ddagger \rightarrow \text{CH}_3\text{OH} + \text{Br}^-$

In the transition state, both O−C and C−Br bonds are partial.

Transition State vs. Intermediate

Reading Energy Diagrams 🧮

An energy diagram shows:

• Reactants at 100 kJ
• Transition state at 250 kJ
• Products at 60 kJ

1. What is $E_a$ (forward)? (in kJ)

2. What is $\Delta H$? (in kJ, include sign)

3. What is $E_a$ (reverse)? (in kJ)

Energy Diagram Concepts 🔍

Exit Quiz — Energy Diagrams ✅

📐 The Arrhenius Equation

Part 3 of 7 — Connecting Rate Constants to Temperature

The Arrhenius equation is one of the most important equations in chemical kinetics. It quantitatively describes how the rate constant $k$ depends on temperature and activation energy.

The Arrhenius Equation

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

What Each Part Means

$A$ (frequency factor): Related to how often molecules collide with correct orientation

• $A = Z \cdot p$ (collision frequency × steric factor)
• Large $A$ → favorable collision geometry
• $A$ is approximately temperature-independent

$e^{-E_a/(RT)}$ (Boltzmann factor): Fraction of collisions with sufficient energy

• As $T \uparrow$: $E_a/(RT) \downarrow$, so $e^{-E_a/(RT)} \uparrow$, so $k \uparrow$
• As $E_a \uparrow$: $E_a/(RT) \uparrow$, so $e^{-E_a/(RT)} \downarrow$, so $k \downarrow$

Arrhenius Equation Concepts 🎯

Temperature Sensitivity and Ea

How Much Does k Change with Temperature?

The sensitivity of $k$ to temperature depends on $E_a$:

• Large $E_a$: $k$ is very sensitive to temperature changes (reaction speeds up dramatically)
• Small $E_a$: $k$ is less sensitive to temperature changes

Example Calculation

For $E_a = 100$ kJ/mol, comparing $k$ at 300 K and 310 K:

$\frac{k_{310}}{k_{300}} = \frac{e^{-E_a/(R \cdot 310)}}{e^{-E_a/(R \cdot 300)}} = e^{(E_a/R)(1/300 - 1/310)}$

$= e^{(100{,}000/8.314)(1/300 - 1/310)}$

$= e^{(12{,}027)(1.075 \times 10^{-4})} = e^{1.293} = 3.64$

A 10°C increase nearly quadruples the rate for this high-$E_a$ reaction!

For $E_a = 20$ kJ/mol: $\frac{k_{310}}{k_{300}} = e^{(20{,}000/8.314)(1.075 \times 10^{-4})} = e^{0.259} = 1.30$

Only a 30% increase — much less sensitive.

Arrhenius Calculations 🧮

1. Calculate $E_a/(RT)$ for $E_a = 75.0$ kJ/mol at $T = 500$ K. ($R = 8.314$ J/(mol·K); to 3 significant figures)

2. A reaction has $A = 1.0 \times 10^{13}$ s⁻¹ and $E_a = 100$ kJ/mol. Calculate $k$ at 300 K. (in s⁻¹, to 1 significant figure in scientific notation: e.g., 3e-5)

3. If $k = 0.010$ s⁻¹ at 300 K and $k = 0.040$ s⁻¹ at 310 K, by what factor does k increase? (to 3 significant figures)

The Frequency Factor A

The frequency factor $A$ represents the maximum possible rate constant — the value $k$ would have if every collision were effective ($E_a = 0$).

Typical Values

Key Point for AP

$A$ is assumed to be approximately independent of temperature (it has a very weak $T$ dependence that is negligible compared to the exponential). All the temperature dependence of $k$ comes from the $e^{-E_a/(RT)}$ term.

Arrhenius Equation Review 🔍

Exit Quiz — Arrhenius Equation ✅

📉 Linearized Arrhenius Equation

Part 4 of 7 — Finding Ea from Graphical Data

The Arrhenius equation in its exponential form is difficult to work with graphically. By taking the natural log, we convert it to a linear form that allows us to extract $E_a$ and $A$ from a plot.

Deriving the Linear Form

Starting from: $k = Ae^{-E_a/(RT)}$

Take the natural log of both sides:

$\ln k = \ln A + \ln(e^{-E_a/(RT)})$

$\boxed{\ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A}$

This is $y = mx + b$!

Key Result

A plot of $\ln k$ vs $1/T$ gives a straight line with:

• Slope $= -E_a/R$ → $E_a = -R \times \text{slope}$
• y-intercept $= \ln A$ → $A = e^{\text{intercept}}$

Linearized Arrhenius Quiz 🎯

Worked Example: Determining Ea from Data

The rate constant for a reaction was measured at several temperatures:

Finding the Slope

Using the first and last points: $\text{slope} = \frac{-3.91 - (-16.12)}{2.22 \times 10^{-3} - 3.33 \times 10^{-3}} = \frac{12.21}{-1.11 \times 10^{-3}} = -11{,}000 \; \text{K}$

Finding Ea

$E_a = -R \times \text{slope} = -(8.314)(-11{,}000) = 91{,}500 \; \text{J/mol} = 91.5 \; \text{kJ/mol}$

Arrhenius Plot Calculations 🧮

An Arrhenius plot of ln k vs 1/T has two data points:

• Point 1: $1/T = 3.00 \times 10^{-3}$ K⁻¹, $\ln k = -8.00$
• Point 2: $1/T = 2.50 \times 10^{-3}$ K⁻¹, $\ln k = -4.00$

1. What is the slope of the line? (in K, include sign)

2. What is $E_a$ in kJ/mol? (to 3 significant figures)

3. What is $\ln A$ (the y-intercept)? Use: $\ln A = \ln k + (E_a/R)(1/T)$, evaluated at point 1. (to 3 significant figures)

Practical Tips for AP

Converting Temperature

Always convert °C to K before using the Arrhenius equation:

$T(K) = T(°C) + 273.15$

Units of Ea

• In the Arrhenius equation, use $E_a$ in J/mol (not kJ/mol) when $R = 8.314$ J/(mol·K)
• Convert kJ to J by multiplying by 1000

Common Mistakes

1. ❌ Using temperature in °C instead of K
2. ❌ Mixing units: $E_a$ in kJ/mol with $R$ in J/(mol·K)
3. ❌ Forgetting the negative sign in the slope
4. ❌ Plotting $k$ vs $T$ instead of $\ln k$ vs $1/T$

Arrhenius Plot Analysis 🔍

Exit Quiz — Linearized Arrhenius ✅

🔄 Two-Point Arrhenius Equation

Part 5 of 7 — Finding Ea from Two Temperatures

Often you don't have enough data to make a full Arrhenius plot. Instead, you know $k$ at two different temperatures. The two-point form of the Arrhenius equation lets you find $E_a$ directly.

Deriving the Two-Point Form

Write the Arrhenius equation at two temperatures:

$\ln k_1 = -\frac{E_a}{R} \cdot \frac{1}{T_1} + \ln A$

$\ln k_2 = -\frac{E_a}{R} \cdot \frac{1}{T_2} + \ln A$

Subtract equation 1 from equation 2:

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

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

Using This Equation

To find $E_a$: $E_a = \frac{R \cdot \ln(k_2/k_1)}{1/T_1 - 1/T_2}$

To find $k$ at a new temperature: If you know $E_a$, $k_1$, and $T_1$, find $k_2$ at $T_2$: $\ln k_2 = \ln k_1 + \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

Worked Example

A reaction has $k_1 = 0.0120$ s⁻¹ at $T_1 = 400$ K and $k_2 = 0.150$ s⁻¹ at $T_2 = 500$ K.

Find $E_a$:

$\ln\frac{k_2}{k_1} = \ln\frac{0.150}{0.0120} = \ln(12.5) = 2.526$

$\frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{400} - \frac{1}{500} = 0.002500 - 0.002000 = 0.000500 \; \text{K}^{-1}$

$E_a = \frac{R \cdot \ln(k_2/k_1)}{1/T_1 - 1/T_2} = \frac{8.314 \times 2.526}{0.000500}$

$E_a = \frac{21.0}{0.000500} = 42{,}000 \; \text{J/mol} = 42.0 \; \text{kJ/mol}$

Practice: Finding Ea 🧮

A reaction has $k = 2.0 \times 10^{-3}$ s⁻¹ at 300 K and $k = 6.0 \times 10^{-2}$ s⁻¹ at 400 K.

1. Calculate $\ln(k_2/k_1)$. (to 3 significant figures)

2. Calculate $1/T_1 - 1/T_2$. (in K⁻¹, give as decimal: e.g., 0.000833)

3. Calculate $E_a$ in kJ/mol. (to 3 significant figures)

Practice: Predicting k at a New Temperature 🧮

A reaction has $E_a = 50.0$ kJ/mol and $k = 0.010$ s⁻¹ at 350 K.

1. Calculate $k$ at 400 K. First find $\ln(k_2/k_1)$. (to 3 significant figures)

2. Now find $k_2$. (in s⁻¹, to 3 significant figures)

3. By what factor did k increase from 350 K to 400 K? (to 3 significant figures)

Two-Point Arrhenius Concepts 🎯

Two-Point Arrhenius Review 🔍

Exit Quiz — Two-Point Arrhenius ✅

🧬 Catalysts

Part 6 of 7 — Lowering the Energy Barrier

Catalysts are substances that speed up reactions without being consumed. They are essential in industry, biology, and everyday life. This part explores how catalysts work at the molecular level and distinguishes between different types.

How Catalysts Work

A catalyst provides an alternative reaction pathway with a lower activation energy:

$E_a(\text{catalyzed}) < E_a(\text{uncatalyzed})$

On an Energy Diagram

The catalyzed pathway shows a lower peak (transition state) while the reactants and products remain at the same energy levels:

• $\Delta H$ is unchanged — the catalyst does not affect thermodynamics
• $E_a$ is reduced — more molecules have sufficient energy to react
• $k$ increases — from the Arrhenius equation: lower $E_a$ → larger $e^{-E_a/(RT)}$ → larger $k$

What Catalysts Do NOT Do

Types of Catalysts

1. Homogeneous Catalysts

Same phase as the reactants (typically all in solution).

2. Heterogeneous Catalysts

Different phase from reactants (typically a solid catalyst with gas or liquid reactants).

3. Biological Catalysts (Enzymes)

Proteins that catalyze specific biochemical reactions.

Catalyst Concepts Quiz 🎯

Heterogeneous Catalysis: The Four Steps

When a gaseous reactant reacts on a solid catalyst surface:

Step 1: Adsorption

Reactant molecules bind to the catalyst surface at active sites. Bonds in the reactant may be weakened.

Step 2: Migration / Diffusion

Adsorbed molecules move along the surface to find each other.

Step 3: Reaction

The weakened bonds allow the reaction to proceed with lower $E_a$. New bonds form.

Step 4: Desorption

Product molecules detach from the surface, freeing active sites for new reactant molecules.

Catalyst Poisoning

If a substance binds strongly to active sites and cannot be removed, the catalyst is poisoned:

• Lead poisons Pt catalytic converters (why leaded gas is banned)
• CO poisons iron catalysts in the Haber process
• Heavy metals poison enzymes

Catalyst Types and Properties 🔍

Catalyst Effect on Rate 🧮

An uncatalyzed reaction has $E_a = 120$ kJ/mol and $k = 1.0 \times 10^{-10}$ s⁻¹ at 300 K.

1. A catalyst lowers $E_a$ to 80 kJ/mol. What is the ratio $k_{\text{cat}}/k_{\text{uncat}}$ at 300 K? Use $\ln(k_{\text{cat}}/k_{\text{uncat}}) = (E_{a,\text{uncat}} - E_{a,\text{cat}})/(RT)$. Calculate this exponent first. (to 3 significant figures)

2. The catalyzed $k$ is approximately how many times larger? Express as a power of 10. (integer)

3. If the catalyzed half-life is $t_{1/2} = 0.693/k_{\text{cat}}$, and $k_{\text{cat}} \approx 1.0 \times 10^{-3}$ s⁻¹, what is the half-life? (in seconds, whole number)

Exit Quiz — Catalysts ✅

🎓 Synthesis & AP Review

Part 7 of 7 — Comprehensive Arrhenius and Catalyst Problems

This final part brings together collision theory, energy diagrams, the Arrhenius equation, and catalysts in AP exam-level problems.

Key Equations Summary

Arrhenius Equation

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

Linearized Form

$\ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A$

Two-Point Form

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

Energy Diagram Relationships

$\Delta H = E_a(\text{forward}) - E_a(\text{reverse})$

Constants

• $R = 8.314$ J/(mol·K)
• Remember: $E_a$ in J/mol when using $R$ in J/(mol·K)

AP Problem 1: Energy Diagram Analysis 🎯

A reaction energy diagram shows:

• Reactants: 50 kJ
• Transition state (uncatalyzed): 150 kJ
• Transition state (catalyzed): 100 kJ
• Products: 30 kJ

AP Problem 2: Arrhenius Calculation 🧮

The rate constant for the decomposition of N₂O₅ is $k = 3.46 \times 10^{-5}$ s⁻¹ at 298 K and $k = 4.87 \times 10^{-3}$ s⁻¹ at 338 K.

1. Calculate $E_a$ in kJ/mol. (to 3 significant figures)

2. Calculate the frequency factor $A$. (order of magnitude: enter the exponent, e.g., for 10¹³ enter 13)

3. What would $k$ be at 310 K? (in s⁻¹, to 1 significant figure in scientific notation, e.g., 2e-4)

AP Problem 3: Catalyst and Arrhenius 🎯

Comprehensive Review 🔍

Challenge: Complete Analysis 🧮

A catalyzed reaction has the following data:

1. Calculate Ea for the catalyzed reaction. (in kJ/mol, to 3 significant figures)

2. The uncatalyzed reaction has Ea = 100 kJ/mol. By how many kJ/mol does the catalyst lower Ea? (to 3 significant figures)

3. At 300 K, what is the ratio k(cat)/k(uncat)? (to 1 significant figure, scientific notation: e.g., 3e5)

Final Exit Quiz — Activation Energy & Arrhenius ✅