Determining Rate Laws
The MCAT will not ask you to derive a rate law from first principles. Instead, it will hand you a table of experimental data and ask you to figure out the rate law from the numbers. This is one of the most predictable question types on the exam, and the method is the same every time.
The Method of Initial Rates
Here is the step-by-step process:
Step 1: Find two trials where only ONE reactant’s concentration changes while everything else stays the same.
Step 2: Compare how the rate changed when that concentration changed.
Step 3: Use the relationship Δrate = (Δconcentration)^order to solve for the order.
Step 4: Repeat for each reactant.
Step 5: Plug the orders back into the rate law and use any trial to solve for k.
Worked Example
Consider the reaction A + B → C + D at 300 K:
| Trial | [A] (M) | [B] (M) | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 1.00 | 1.00 | 2.0 |
| 2 | 1.00 | 2.00 | 8.1 |
| 3 | 2.00 | 2.00 | 15.9 |
Finding the order with respect to B: Compare Trials 1 and 2. [A] is held constant at 1.00 M. [B] doubles from 1.00 to 2.00 M. The rate increases from 2.0 to 8.1 - approximately a factor of 4.
Since (2)^y = 4, y = 2. The reaction is second order with respect to B.
Finding the order with respect to A: Compare Trials 2 and 3. [B] is held constant at 2.00 M. [A] doubles from 1.00 to 2.00 M. The rate increases from 8.1 to 15.9 - approximately a factor of 2.
Since (2)^x = 2, x = 1. The reaction is first order with respect to A.
The rate law: rate = k[A]¹[B]² = k[A][B]²
Finding k: Plug in values from Trial 1:
2.0 = k(1.00)(1.00)²
k = 2.0 M⁻²·s⁻¹
Quick Reference: Common Patterns
| Concentration Change | Rate Change | Order |
|---|---|---|
| x2 | No change (x1) | 0 |
| x2 | x2 | 1 |
| x2 | x4 | 2 |
| x2 | x8 | 3 |
| x3 | x3 | 1 |
| x3 | x9 | 2 |
| x3 | x27 | 3 |
What If No Two Trials Hold Everything Else Constant?
Sometimes the MCAT will give you data where no pair of trials isolates a single variable perfectly. In that case, you may need to:
- Use the rate law algebraically. Write the rate law for two different trials and divide one equation by the other to cancel k.
- If you already know one order, substitute it in and solve for the remaining unknown.
This is less common, but it does appear in harder questions.