🎯⭐ INTERACTIVE LESSON

Integrated Rate Laws and Half-Life

Learn step-by-step with interactive practice!

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Integrated Rate Laws and Half-Life - Complete Interactive Lesson

Part 1: Zero-Order Reactions

📈 Zero-Order Integrated Rate Law

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

Integrated rate laws connect concentration to time directly. While the differential rate law tells us how rate depends on concentration, the integrated form lets us calculate concentrations at any future time, determine half-lives, and identify reaction order from graphical data.

We begin with the simplest case: zero-order reactions.

📏 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!

$y = mx + b$

Variable	Corresponds To
$y$	$[A]$
$m$ (slope)	$-k$
$x$

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

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:

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.

🔧 How to Identify Zero-Order from Data

Method: Test Different Plots

Given concentration-time data, make three plots:

Plot	Straight line if...
$[A]$ vs $t$	Zero-order
$\ln[A]$ vs $t$	First-order

Exit Quiz — Zero-Order Integrated Rate Law ✅

Part 2: First-Order Reactions

📉 First-Order Integrated Rate Law

Part 2 of 7 — Exponential Decay

First-order reactions are the most common type in chemistry. Radioactive decay, many decomposition reactions, and most biochemical processes follow first-order kinetics. The math involves logarithms, and the behavior is exponential decay.

📌 Derivation

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

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

Part 3: Second-Order Reactions

📊 Second-Order Integrated Rate Law

Part 3 of 7 — Inverse Concentration and Time

Second-order reactions complete our trio of integrated rate laws. The mathematics involves the reciprocal of concentration, and the behavior is distinctly different from both zero and first-order kinetics.

⏱️ Derivation (for Rate = k[A]²)

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

Part 4: Half-Life

📊 Graphical Analysis

Part 4 of 7 — Identifying Reaction Order from Plots

On the AP exam, you may be given data or graphs and asked to determine the reaction order. The key technique: make three plots and see which one is linear. This part gives you systematic practice with this approach.

🎯 The Three-Plot Strategy

Given concentration-vs-time data, create:

Plot	If Linear →	Slope =	y-intercept =
$[A]$ vs $t$	Zero-order	$-k$

Part 5: Graphical Analysis of Order

⏱️ Half-Life Problems

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

Half-life is one of the most-tested topics on the AP Chemistry exam. This part provides intensive practice with half-life calculations for all three orders, plus applications to radioactive decay (which is always first-order).

📋 Half-Life Formulas Summary

Order	Half-Life Formula	Dependence on $[A]_0$
Zero	$t_{1 / 2} = \frac{[A]_{0}}{2 k}$

Part 6: Problem-Solving Workshop

🔧 Problem-Solving Workshop

Part 6 of 7 — Mixed Order Identification and Calculations

This workshop combines all three integrated rate laws in problems that require you to first identify the order, then perform calculations. These mirror the multi-step problems found on the AP exam.

🛠️ Problem-Solving Flowchart

Step 1: Identify the Order

Method A — Graphical: Plot [A], ln[A], and 1/[A] vs t. The linear one wins.

Method B — Successive half-lives:

Half-lives equal → first-order
Half-lives decreasing → zero-order
Half-lives increasing (doubling) → second-order

Method C — Initial rates: Compare experiments (covered in earlier parts).

Step 2: Find k

Use the slope of the appropriate linear plot:

Zero: slope of [A] vs t = $-k$
First: slope of ln[A] vs t = $-k$
Second: slope of 1/[A] vs t =

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

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

This final part challenges you with comprehensive, exam-level problems that combine order identification, integrated rate law calculations, half-life analysis, and graphical interpretation.

📋 Complete Integrated Rate Laws Summary

	Zero-Order	First-Order	Second-Order
Differential	Rate = $k$	Rate = $k[A]$	Rate = $k[A]^2$

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

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

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

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

Half-life	[A] at start	[A] at end	Duration
1st	$[A]_0$	$[A]_0/2$	$[A]_0/(2k)$
2nd	$[A]_0/2$	$[A]_0/4$	$[A$
3rd	$[A]_0/4$	$[A]_0/8$	$[A$

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

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)

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

$t$ (s)	$[A]$ (M)	$\ln[A]$	$1/[A]$ (M⁻¹)
0	0.500	−0.693	2.00
10	0.450	−0.799	2.22
20	0.400	−0.916	2.50
30	0.350	−1.050	2.86

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

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

Separating variables and integrating:

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

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

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

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

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

Variable	Corresponds To
$y$	$\ln[A]$
$m$ (slope)	$-k$
$x$	$t$
$b$ (y-intercept)	$\ln[A]_0$

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

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

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

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

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

The Most Important Feature

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

Often you need to find how much reactant remains at time $t$ without knowing $[A]_0$ explicitly:

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

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

Quick Calculations with Half-Lives

Time	Fraction remaining	Percent remaining
$0$	1	100%
$t_{1/2}$	1/2	50%
$2t_{1/2}$

Exit Quiz — First-Order Integrated Rate Law ✅

Separating variables:

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

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

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

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

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

Variable	Corresponds To
$y$	$1/[A]$
$m$ (slope)	$+k$
$x$	$t$
$b$ (y-intercept)	$1/[A]_0$

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 Feature

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

Half-life	Starting [A]	Duration
1st	$[A]_0$	$1/(k[A]_0)$

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⁻¹s⁻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

Feature	Zero-Order	First-Order	Second-Order
Rate law	$k$	$k[A]$	$k[A]^2$
Integrated law	$[A] = -kt + [A]_0$	$\ln[A] = -kt + \ln[A]_0$
Linear plot	$[A]$ vs $t$	$\ln[A]$ vs $t$	$1/[A]$ vs
Slope	$-k$	$-k$	$+k$
Half-life	$[A]_0/(2k)$	$0.693/k$	$1/(k[A]_0)$
Successive $t_{1/2}$	Shorter	Constant	Longer
Units of $k$	M/s	s⁻¹	M⁻¹s⁻¹

Identify the Order 🔍

Exit Quiz — Second-Order Integrated Rate Law ✅

How to Check Linearity

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

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

$t$ (s)	$[A]$ (M)	$\ln[A]$	$1/[A]$ (M⁻¹)
0	1.000	0.000	1.000
10	0.607	−0.500	1.648
20	0.368	−1.000	2.718
30	0.223	−1.500	4.484
40	0.135	−2.000	7.407

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.

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

Graphical Analysis Practice 🎯

Given this data:

$t$ (s)	$[B]$ (M)
0	0.400
100	0.300
200	0.200
300	0.100

Identify the Order 🧮

$t$ (min)	$[C]$ (M)	$\ln[C]$	$1/[C]$ (M⁻¹)
0	0.500	−0.693	2.00
5	0.333	−1.099	3.00
10	0.250	−1.386	4.00
15	0.200	−1.609	5.00

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 ✅

t_{1/2} = \frac{[ A ] _{0}}{2 k}

Pattern of Successive Half-Lives

Order	1st t₁/₂	2nd t₁/₂	3rd t₁/₂	Pattern
Zero	$T$	$T/2$	$T/4$	Each is half the previous
First	$T$	$T$	$T$	All equal
Second	$T$	$2T$	$4T$	Each is double the previous

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⁻¹s⁻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:

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

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

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

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

Carbon-14 Dating

$^{14}$ C has $t_{1/2} = 5{,}730$ years
Living organisms maintain constant $^{14}$ C/C ratio through intake

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 ✅

Step 3: Solve the Problem

Use the appropriate integrated rate law to find concentration at any time, time to reach a certain concentration, or half-life.

Problem 1: Order Identification 🧮

$t$ (min)	$[A]$ (M)
0	0.800
10	0.400
20	0.200
30	0.100

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 🎯

$t$ (s)	$[B]$ (M)	$1/[B]$
0	0.500	2.00
50	0.333	3.00
100	0.250	4.00
150	0.200	5.00

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⁻¹s⁻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 ✅

AP Problem 1 🎯

The decomposition of SO₂Cl₂ 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:

$t$ (min)	$[A]$ (M)
0	1.00
20	0.50
40	0.33
60	0.25

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⁻¹min⁻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 ✅

When an organism dies,

^{14}

C decays without replacement

Measuring the remaining

^{14}

C fraction tells us when it died

Integrated Rate Laws and Half-Life

Key Feature

Successive Half-Lives

Data Test

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

Key Feature

📌 First-Order Half-Life

The Most Important Feature

Connection to Radioactive Decay

📌 Useful Ratio Form

Quick Calculations with Half-Lives

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

Key Feature

📌 Second-Order Half-Life

Key Feature

Successive Half-Lives

⚖️ Comparing All Three Orders

How to Check Linearity

AP Exam Shortcut

🧪 Worked Example: Identifying Order

Pattern of Successive Half-Lives

📌 Radioactive Decay: Always First-Order

Carbon-14 Dating

Example

Step 3: Solve the Problem

Integrated Rate Laws and Half-Life

Integrated Rate Laws and Half-Life - Complete Interactive Lesson

Part 1: Zero-Order Reactions

📈 Zero-Order Integrated Rate Law

📏 Derivation of the Zero-Order Integrated Rate Law

This is the equation of a straight line!

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

📌 Zero-Order Half-Life

🔧 How to Identify Zero-Order from Data

Method: Test Different Plots

Part 2: First-Order Reactions

📉 First-Order Integrated Rate Law

📌 Derivation

Part 3: Second-Order Reactions

📊 Second-Order Integrated Rate Law

⏱️ Derivation (for Rate = k[A]²)

Part 4: Half-Life

📊 Graphical Analysis

🎯 The Three-Plot Strategy

Part 5: Graphical Analysis of Order

⏱️ Half-Life Problems

📋 Half-Life Formulas Summary

Part 6: Problem-Solving Workshop

🔧 Problem-Solving Workshop

🛠️ Problem-Solving Flowchart

Step 1: Identify the Order

Step 2: Find k

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

📋 Complete Integrated Rate Laws Summary

Key Feature

Successive Half-Lives

Data Test

Linear Form: ln⁡[A]\ln[A]ln[A] vs ttt

Key Feature

📌 First-Order Half-Life

The Most Important Feature

Connection to Radioactive Decay

📌 Useful Ratio Form

Quick Calculations with Half-Lives

Linear Form: 1/[A]1/[A]1/[A] vs ttt

Key Feature

📌 Second-Order Half-Life

Key Feature

Successive Half-Lives

⚖️ Comparing All Three Orders

How to Check Linearity

AP Exam Shortcut

🧪 Worked Example: Identifying Order

Pattern of Successive Half-Lives

📌 Radioactive Decay: Always First-Order

Carbon-14 Dating

Example

Step 3: Solve the Problem

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

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

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