Rate Law Practice- Problems and Solutions
Rate Law Practice: Problems and Solutions That Actually Make Sense
If you're staring at a rate law problem and feeling lost, you're not alone. Most textbooks bury the concepts under dense paragraphs and confusing notation. This guide cuts through that noise.
We'll work through real practice problems with step-by-step solutions. No filler. Just the math and logic you need to solve these problems on your own.
What Is a Rate Law Anyway?
A rate law expresses how the reaction rate depends on concentration. The general form is:
rate = k[A]m[B]n
Where:
- k = rate constant
- [A], [B] = concentrations of reactants
- m, n = reaction orders (determined experimentally)
The orders are not the stoichiometric coefficients. They're experimental values that tell you how sensitive the rate is to each reactant's concentration.
Zero, First, and Second Order Reactions
Understanding the three main reaction orders is essential. Here's what each one means:
Zero Order
Rate is independent of concentration. Doubling [A] does nothing to the rate.
Rate = k
The concentration decreases linearly over time. Use this equation:
[A]t = [A]0 - kt
First Order
Rate depends linearly on concentration. Double [A], rate doubles.
Rate = k[A]
Concentration decays exponentially. Use this equation:
ln[A]t = ln[A]0 - kt
Or the alternative form:
[A]t = [A]0 × e-kt
Second Order
Rate depends on concentration squared. Double [A], rate quadruples.
Rate = k[A]2
Concentration decreases hyperbolically. Use this equation:
1/[A]t = 1/[A]0 + kt
Practice Problem 1: Finding the Rate Law from Experimental Data
Problem: The reaction 2NO(g) + Cl2(g) → 2NOCl(g) was studied. Determine the rate law from this data:
| Experiment | [NO]0 (M) | [Cl2]0 (M) | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 0.10 | 0.10 | 0.018 |
| 2 | 0.20 | 0.10 | 0.072 |
| 3 | 0.10 | 0.20 | 0.036 |
Step 1: Compare experiments 1 and 2. [Cl2] stays constant while [NO] doubles.
When [NO] doubles (0.10 → 0.20), rate quadruples (0.018 → 0.072).
24 = 4. So the reaction is second order in NO.
Step 2: Compare experiments 1 and 3. [NO] stays constant while [Cl2] doubles.
When [Cl2] doubles (0.10 → 0.20), rate doubles (0.018 → 0.036).
21 = 2. So the reaction is first order in Cl2.
Step 3: Write the rate law and find k.
Rate = k[NO]2[Cl2]
Use experiment 1 to solve for k:
0.018 = k(0.10)2(0.10)
0.018 = k(0.001)
k = 18 M-2s-1
Practice Problem 2: Finding Half-Life
Problem: A first-order reaction has k = 0.045 s-1. Calculate the half-life.
For first-order reactions, there's a direct formula:
t1/2 = 0.693/k
Plug in:
t1/2 = 0.693 / 0.045
t1/2 = 15.4 seconds
That's it. First-order half-life is always independent of initial concentration. If you see a problem asking for half-life on a first-order reaction, use this formula.
Practice Problem 3: Concentration at a Given Time
Problem: A second-order reaction 2A → products has k = 0.015 M-1s-1. If [A]0 = 0.50 M, what is [A] after 120 seconds?
Use the second-order integrated rate law:
1/[A]t = 1/[A]0 + kt
Plug in the values:
1/[A]t = 1/0.50 + (0.015)(120)
1/[A]t = 2.0 + 1.8
1/[A]t = 3.8
[A]t = 0.263 M
Practice Problem 4: Determining Order from Concentration-Time Data
Problem: Use this data to determine the reaction order:
| Time (s) | [A] (M) |
|---|---|
| 0 | 1.00 |
| 50 | 0.75 |
| 100 | 0.56 |
| 150 | 0.42 |
Method: Test each integrated rate law to see which gives a straight line.
Test for zero order: Plot [A] vs. time. Check if concentration decreases by equal amounts in equal times.
Δ[A] from 0 to 50s: 0.25 M
Δ[A] from 50 to 100s: 0.19 M
Δ[A] from 100 to 150s: 0.14 M
Not linear. Not zero order.
Test for first order: Plot ln[A] vs. time.
ln(1.00) = 0
ln(0.75) = -0.287
ln(0.56) = -0.579
ln(0.42) = -0.868
The differences are roughly equal (-0.287, -0.292, -0.289). This is first order.
Rate Law Comparison Table
| Order | Rate Law | Integrated Law | Half-Life | Units of k |
|---|---|---|---|---|
| Zero | k | [A] = [A]0 - kt | [A]0/2k | M/s |
| First | k[A] | ln[A] = ln[A]0 - kt | 0.693/k | s-1 |
| Second | k[A]2 | 1/[A] = 1/[A]0 + kt | 1/(k[A]0) | M-1s-1 |
How to Solve Any Rate Law Problem
Follow this checklist:
- Identify what you're solving for. Rate law? Rate constant? Concentration at time t? Half-life?
- Determine the reaction order from experimental data or a graph.
- Pick the right equation from the table above.
- Plug in numbers and solve algebraically.
- Check units to verify your answer makes sense.
Common Mistakes to Avoid
- Don't assume stoichiometric coefficients equal reaction orders. They almost never do. You must determine them experimentally.
- Don't mix up the rate laws. Zero order uses [A]. First order uses ln[A]. Second order uses 1/[A].
- Don't forget to include all reactants in the rate law. The rate depends on everything that affects it.
- Don't skip the units on k. They tell you the reaction order. M-1s-1 means second order.
Quick Reference Formulas
Rate constant from initial rates:
Divide rate equations, cancel k, solve for orders. Then plug back in to find k.
Arrhenius equation (for temperature dependence):
k = Ae-Ea/RT
Or the two-point form:
ln(k2/k1) = -Ea/R (1/T2 - 1/T1)
You'll see this in later problems involving temperature changes.