Integrated Rate Laws and Half-Life

📈 Zero-Order Integrated Rate Law

Part 1 of 7 — When Rate Doesn't Depend on Concentration

Topics in This Part

🔑 Key Concept: Mastering this material will strengthen your foundation for both the AP Chemistry exam and more advanced chemistry topics.

What You'll Master in Part 1

• Understanding the core concepts covered in Part 1
• Applying these ideas to solve practice problems
• Building toward AP exam readiness for this topic

📏 Derivation of the Zero-Order Integrated Rate Law

For a zero-order reaction: $\text{Rate} = k$

$-\frac{d[A]}{dt} = k$

Integrating from $[A]_0$ to $[A]$ and from $0$ to $t$:

$\int_{[A]_0}^{[A]} d[A] = -\int_0^t k \, dt$

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

$\boxed{[A] = -kt + [A]_0}$

This is the equation of a straight line!

$y = mx + b$

Plot: $[A]$ vs $t$ → straight line for zero-order

🔑 Key Concept: If $[A]$ vs $t$ is a straight line, the reaction is zero-order. The slope gives $-k$.

• Slope = $-k$
• y-intercept = $[A]_0$
• $[A]$ decreases linearly with time

Zero-Order Graphical Analysis 🎯

📌 Zero-Order Half-Life

The half-life ($t_{1/2}$) is the time for the concentration to drop to half its initial value.

Set $[A] = [A]_0/2$ in the integrated rate law:

$\frac{[A]_0}{2} = -kt_{1/2} + [A]_0$

$kt_{1/2} = [A]_0 - \frac{[A]_0}{2} = \frac{[A]_0}{2}$

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

🔑 Key Concept: The zero-order half-life depends on $[A]_0$:

• Higher initial concentration → longer half-life
• Each successive half-life is shorter than the previous one
• The reaction reaches [A] = 0 in a finite time: $t_{\text{complete}} = [A]_0/k$

Successive Half-Lives

Each successive half-life is exactly half the duration of the previous one.

⚠️ Warning: Zero-order is the only order where the reaction reaches $[A] = 0$ in finite time. Don't assume all reactions behave this way!

Zero-Order Half-Life Concepts 🔍

Zero-Order Calculations 🧮

A zero-order reaction has $k = 5.0 \times 10^{-3}$ M/s and $[A]_0 = 0.60$ M.

1) What is [A] after 40 s? (in M, 3 significant figures)

2) What is the half-life? (in seconds, whole number)

3) How long until the reaction is complete ([A] = 0)? (in seconds, whole number)

🔧 How to Identify Zero-Order from Data

Method: Test Different Plots

Given concentration-time data, make three plots:

For zero-order, the $[A]$ vs $t$ plot will be linear with slope $= -k$.

Data Test

Check constant differences: Δ[A] = −0.050 M per 10 s → constant → zero-order ✓

$k = 0.050/10 = 0.005$ M/s

Exit Quiz — Zero-Order Integrated Rate Law ✅

📉 First-Order Integrated Rate Law

Part 2 of 7 — Exponential Decay

Topics in This Part

🔑 Key Concept: Mastering this material will strengthen your foundation for both the AP Chemistry exam and more advanced chemistry topics.

What You'll Master in Part 2

• Understanding the core concepts covered in Part 2
• Applying these ideas to solve practice problems
• Building toward AP exam readiness for this topic

📌 Derivation

For a first-order reaction: $\text{Rate} = k[A]$

$-\frac{d[A]}{dt} = k[A]$

Separating variables and integrating:

$\int_{[A]_0}^{[A]} \frac{d[A]}{[A]} = -\int_0^t k \, dt$

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

$\boxed{\ln[A] = -kt + \ln[A]_0}$

Or equivalently:

$\boxed{[A] = [A]_0 e^{-kt}}$

Linear Form: $\ln[A]$ vs $t$

🔑 Key Concept: A plot of $\ln[A]$ vs $t$ is linear for a first-order reaction. The slope equals $-k$.

First-Order Graphical Analysis 🎯

📌 First-Order Half-Life

Set $[A] = [A]_0/2$:

$\ln\frac{[A]_0/2}{[A]_0} = -kt_{1/2}$

$\ln\frac{1}{2} = -kt_{1/2}$

$-0.693 = -kt_{1/2}$

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

🔑 Key Concept: The half-life of a first-order reaction is independent of initial concentration.

This means:

• Every half-life has the same duration
• After 1 half-life: 50% remains
• After 2 half-lives: 25% remains
• After 3 half-lives: 12.5% remains
• After $n$ half-lives: $(1/2)^n$ remains

Connection to Radioactive Decay

All radioactive decay follows first-order kinetics:

$N = N_0 e^{-\lambda t}, \quad t_{1/2} = \frac{0.693}{\lambda}$

where $\lambda$ is the decay constant (equivalent to $k$).

First-Order Calculations 🧮

A first-order reaction has $k = 0.0100$ $s^{-1}$ and $[A]_0 = 2.00$ M.

1) What is the half-life? (in seconds, 3 significant figures)

2) What is [A] after 100 s? (in M, 3 significant figures)

3) How long until only 10% of A remains? (in seconds, 3 significant figures)

First-Order Properties 🔍

📌 Useful Ratio Form

💡 Tip: The ratio form lets you calculate the fraction remaining without knowing $[A]_0$ explicitly — very useful on the AP exam!

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

$\frac{[A]_t}{[A]_0} = e^{-kt}$

Quick Calculations with Half-Lives

Exit Quiz — First-Order Integrated Rate Law ✅

📊 Second-Order Integrated Rate Law

Part 3 of 7 — Inverse Concentration and Time

Topics in This Part

🔑 Key Concept: Mastering this material will strengthen your foundation for both the AP Chemistry exam and more advanced chemistry topics.

What You'll Master in Part 3

• Understanding the core concepts covered in Part 3
• Applying these ideas to solve practice problems
• Building toward AP exam readiness for this topic

⏱️ Derivation (for Rate = k[A]${}^{2}$)

$-\frac{d[A]}{dt} = k[A]^2$

Separating variables:

$\frac{d[A]}{[A]^2} = -k \, dt$

Integrating:

$\int_{[A]_0}^{[A]} [A]^{-2} \, d[A] = -\int_0^t k \, dt$

$-\frac{1}{[A]} + \frac{1}{[A]_0} = -kt$

$\boxed{\frac{1}{[A]} = kt + \frac{1}{[A]_0}}$

Linear Form: $1/[A]$ vs $t$

🔑 Key Concept: A plot of $1/[A]$ vs $t$ is linear for a second-order reaction. The slope equals $+k$ (positive!).

Second-Order Graphical Analysis 🎯

📌 Second-Order Half-Life

Set $[A] = [A]_0/2$:

$\frac{1}{[A]_0/2} = kt_{1/2} + \frac{1}{[A]_0}$

$\frac{2}{[A]_0} - \frac{1}{[A]_0} = kt_{1/2}$

$\frac{1}{[A]_0} = kt_{1/2}$

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

🔑 Key Concept: The half-life of a second-order reaction is inversely proportional to $[A]_0$:

• Higher $[A]_0$ → shorter half-life
• Each successive half-life is longer (doubles each time!)

Successive Half-Lives

Each successive half-life is twice the previous one. This is a telltale sign of second-order kinetics.

Second-Order Calculations 🧮

A second-order reaction has $k = 0.50$ $M^{-1}s^{-1}$ and $[A]_0 = 0.80$ M.

1) What is the half-life? (in seconds, to 3 significant figures)

2) What is [A] after 5.0 s? (in M, to 3 significant figures)

3) What is the second half-life (starting from [A] = 0.40 M)? (in seconds, to 3 significant figures)

⚖️ Comparing All Three Orders

💡 Tip: Only second-order has a positive slope in its linear plot. Zero and first-order both have slope $= -k$.

Identify the Order 🔍

Exit Quiz — Second-Order Integrated Rate Law ✅

📊 Graphical Analysis

Part 4 of 7 — Identifying Reaction Order from Plots

Topics in This Part

🔑 Key Concept: Mastering this material will strengthen your foundation for both the AP Chemistry exam and more advanced chemistry topics.

What You'll Master in Part 4

• Understanding the core concepts covered in Part 4
• Applying these ideas to solve practice problems
• Building toward AP exam readiness for this topic

🎯 The Three-Plot Strategy

Given concentration-vs-time data, create:

How to Check Linearity

1. Visual inspection: Does it look like a straight line?
2. Constant slope: Calculate $\Delta y / \Delta x$ between successive points — is it constant?
3. $R^2$ value: In a calculator, the best fit gives $R^2$ closest to 1.

⚠️ Warning: A plot of $[A]$ vs $t$ is always curved for first and second-order — don't mistake a gentle curve for a straight line!

💡 Tip — AP Exam Shortcut: If you are given just the raw data, calculate the transformed values and check which set has constant spacing:

• If $\Delta[A]$ is constant per $\Delta t$ → zero-order
• If $\Delta(\ln[A])$ is constant per $\Delta t$ → first-order
• If $\Delta(1/[A])$ is constant per $\Delta t$ → second-order

🧪 Worked Example: Identifying Order

Problem: Given the following data, determine the reaction order and find $k$.

Solution:

Test [A] vs t: Differences in [A]: −0.393, −0.239, −0.145, −0.088 → NOT constant ✗

Test ln[A] vs t: Differences in ln[A]: −0.500, −0.500, −0.500, −0.500 → CONSTANT ✓

Test 1/[A] vs t: Differences: +0.648, +1.070, +1.766, +2.923 → NOT constant ✗

Conclusion: The reaction is first-order.

$\boxed{k = -\text{slope} = -(-0.500/10) = 0.0500 \text{ s}^{-1}}$

Graphical Analysis Practice 🎯

Given this data:

Identify the Order 🧮

1) What is the order? (enter 0, 1, or 2)

2) What is k? (number only, to 3 significant figures)

3) What is [C] at t = 25 min? (in M, to 3 significant figures)

Slope Interpretation 🔍

AP-Style Problem 🎯

A student collects concentration-time data and plots all three standard graphs. She finds:

• [A] vs t: curved
• ln[A] vs t: curved
• 1/[A] vs t: straight line with slope 0.45

Exit Quiz — Graphical Analysis ✅

⏱️ Half-Life Problems

Part 5 of 7 — Calculations for Each Order and Radioactive Decay

Topics in This Part

🔑 Key Concept: Mastering this material will strengthen your foundation for both the AP Chemistry exam and more advanced chemistry topics.

What You'll Master in Part 5

• Understanding the core concepts covered in Part 5
• Applying these ideas to solve practice problems
• Building toward AP exam readiness for this topic

📋 Half-Life Formulas Summary

⚠️ Warning: For zero and second-order, "$n$ half-lives" does NOT mean $n \times t_{1/2}$ total time, because each successive half-life has a different duration!

🔑 Key Concept: The pattern of successive half-lives uniquely identifies reaction order:

Zero-Order Half-Life Problems 🧮

A zero-order reaction has $k = 0.0020$ M/s.

1) If [A]${}_{0}$ = 0.100 M, what is the half-life? (in seconds)

2) If [A]${}_{0}$ = 0.200 M, what is the half-life? (in seconds)

3) For [A]${}_{0}$ = 0.100 M, how long until 75% has reacted (only 25% remains)? Note: this is NOT simply 2 half-lives for zero-order! Use the integrated rate law. (in seconds, to 3 significant figures)

First-Order Half-Life Problems 🧮

1) A first-order reaction has $k = 0.0462$ $s^{-1}$. What is the half-life? (in seconds, to 3 significant figures)

2) If 93.75% of a first-order reactant has decomposed, how many half-lives have passed? (integer)

3) Iodine-131 has a half-life of 8.02 days. What fraction remains after 24.06 days? Express as a fraction with denominator 8 (e.g., enter 3/8).

Second-Order Half-Life Problems 🧮

A second-order reaction has $k = 0.40$ $M^{-1}s^{-1}$ and $[A]_0 = 0.50$ M.

1) What is the first half-life? (in seconds, to 3 significant figures)

2) What is the second half-life? (in seconds, to 3 significant figures)

3) What is [A] after 15 s? (in M, to 3 significant figures)

📌 Radioactive Decay: Always First-Order

All radioactive decay processes follow first-order kinetics:

$\boxed{N = N_0 e^{-\lambda t}}$

$\ln\frac{N}{N_0} = -\lambda t$

$\boxed{t_{1/2} = \frac{0.693}{\lambda}}$

where $\lambda$ = decay constant (same role as $k$), $N$ = number of atoms remaining.

🔑 Key Concept: All radioactive decay is first-order — the probability of any one nucleus decaying is constant per unit time, regardless of how many nuclei remain.

Carbon-14 Dating

• $^{14}$C has $t_{1/2} = 5{,}730$ years
• Living organisms maintain constant $^{14}$C/$^{12}$C ratio through intake
• When an organism dies, $^{14}$C decays without replacement
• Measuring the remaining $^{14}$C fraction tells us when it died

Example

Problem: A fossil has 25% of original $^{14}$C. How old is it?

Solution:

$\frac{N}{N_0} = 0.25 = \left(\frac{1}{2}\right)^n \Rightarrow n = 2 \text{ half-lives}$

$\text{Age} = 2 \times 5{,}730 = 11{,}460 \text{ years}$

Radioactive Decay Quiz 🎯

Half-Life Review 🔍

Exit Quiz — Half-Life Problems ✅

🔧 Problem-Solving Workshop

Part 6 of 7 — Mixed Order Identification and Calculations

Practice Makes Perfect

This workshop features multi-step problems that mirror the AP Chemistry exam format. Each problem requires you to combine concepts from previous parts and show your work clearly.

🔑 Why this matters: The AP Chemistry exam rewards students who can apply concepts to unfamiliar problems — structured practice is the best preparation.

What You'll Master in Part 6

• Working through complete multi-step problems from start to finish
• Building problem-solving strategies you can apply on the AP exam
• Identifying which concepts to apply and in what order

🛠️ Problem-Solving Flowchart

Step 1: Identify the Order

Step 2: Find $k$ from the Linear Plot

Step 3: Solve the Problem

Use the appropriate integrated rate law to find:

• Concentration at any time
• Time to reach a certain concentration
• Half-life

⚠️ Always check the units of $k$ to confirm the order you identified. Wrong units = wrong order!

Problem 1: Order Identification 🧮

1) What is the order of the reaction? (enter 0, 1, or 2)

2) What is k? (to 3 significant figures)

3) What is [A] at t = 50 min? (in M, to 3 significant figures)

Problem 2: Data Table Analysis 🎯

Problem 3: Working Backwards 🧮

A first-order reaction has a half-life of 25.0 minutes. The initial concentration is 1.20 M.

1) What is k? (in $min^{-1}$, to 3 significant figures)

2) What is [A] after 75.0 minutes? (in M, to 3 significant figures)

3) How long until [A] = 0.10 M? (in minutes, to 3 significant figures)

Problem 4: Conceptual Matching 🔍

Challenge Problem 🧮

A certain reaction is second-order with $k = 0.10$ $M^{-1}s^{-1}$ and $[A]_0 = 2.0$ M.

1) What is the first half-life? (in seconds, to 3 significant figures)

2) How long total until 87.5% of A has reacted? (in seconds, to 3 significant figures)

3) What is [A] at t = 35 s? (in M, to 3 significant figures)

Exit Quiz — Problem-Solving Workshop ✅

🎓 Synthesis & AP Review

Part 7 of 7 — AP-Style Integrated Rate Law Problems

Bringing It All Together

This comprehensive review connects every concept from Parts 1–6 with AP-style problems. The questions are designed to mirror what you'll see on the actual exam — multi-step, multi-concept, and requiring clear written explanations.

🔑 Why this matters: AP Chemistry exam questions rarely test one concept in isolation — success requires connecting ideas across topics.

What You'll Master in Part 7

• Solving AP-style questions that integrate multiple concepts from this unit
• Writing clear, concise explanations using proper chemistry terminology
• Identifying and avoiding common AP exam traps and mistakes

📋 Complete Integrated Rate Laws Summary

🔑 Key Concept: Memorize this table — it is the foundation for all integrated rate law problems on the AP exam.

⚠️ Warning: On the AP exam, always write the correct units of $k$ with your answer. M/s (zero), $s^{-1}$ (first), $M^{-1}s^{-1}$ (second) — getting units wrong can cost points!

AP Problem 1 🎯

The decomposition of $SO_{2}Cl_{2}$ is first-order:

$\text{SO}_2\text{Cl}_2(g) \rightarrow \text{SO}_2(g) + \text{Cl}_2(g)$

At 320°C, $k = 2.20 \times 10^{-5}$ $s^{-1}$.

AP Problem 2: Order Determination 🧮

The decomposition of a compound was studied. Data:

Verify: $1/[A]$ values are 1.00, 2.00, 3.00, 4.00 (constant Δ of 1.00 per 20 min).

1) What is the order? (enter 0, 1, or 2)

2) What is k? (in $M^{-1}min^{-1}$, to 3 significant figures)

3) What is [A] at t = 100 min? (in M, to 3 significant figures)

AP Problem 3: Carbon Dating 🎯

A wooden artifact is found to have 12.5% of the $^{14}$C content of living wood. The half-life of $^{14}$C is 5,730 years.

Synthesis: Match the Scenario 🔍

AP Problem 4: Comprehensive 🧮

A first-order reaction has $k = 3.46 \times 10^{-2}$ $s^{-1}$.

1) What is the half-life? (in seconds, to 3 significant figures)

2) How long until 90% has reacted? (in seconds, to 3 significant figures)

3) If [A]${}_{0}$ = 0.500 M, what is [A] after 30.0 s? (in M, to 3 significant figures)

Final Exit Quiz — Integrated Rate Laws ✅