Rate Law and Rate Constant- Chemistry Guide
What Is Rate Law in Chemistry?
Rate law describes how the reaction rate depends on the concentration of reactants. It's the mathematical relationship between concentration and speed.
Most students stumble here because they try to memorize formulas instead of understanding the logic. Don't do that. The logic is simple: changing reactant concentration changes how fast the reaction goes. Rate law quantifies exactly how much.
The Rate Law Equation
For a general reaction:
aA + bB → products
The rate law looks like this:
rate = k[A]m[B]n
Here's what each piece means:
- k = rate constant
- [A] and [B] = molar concentrations of reactants
- m and n = reaction orders (determined experimentally, not from stoichiometry)
The exponents m and n are NOT necessarily the stoichiometric coefficients. This trips up almost everyone. You must find them experimentally.
Understanding the Rate Constant (k)
The rate constant k is a proportionality constant that relates reactant concentrations to reaction rate. Key points:
- It's temperature-dependent — change the temp, change the k value
- Larger k = faster reaction
- It's determined experimentally, just like the reaction orders
- Units vary based on the overall reaction order
Reaction Order Explained
Reaction order tells you how sensitive the rate is to concentration changes. It's the exponent in the rate law equation.
Zero Order Reactions
Rate = k[A]0 = k
The rate is constant regardless of reactant concentration. Changing [A] has zero effect on rate. These are relatively rare — usually occur when a catalyst is saturated or the reaction happens on a surface.
First Order Reactions
Rate = k[A]1
Rate is directly proportional to concentration. Double the concentration, double the rate. Many radioactive decays and unimolecular decompositions are first order.
Second Order Reactions
Rate = k[A]2 or Rate = k[A][B]
Rate depends on concentration squared. Double [A], rate increases by a factor of 4. This is common in bimolecular reactions where two molecules must collide.
Order Can Be Fractional or Negative
Reaction order isn't limited to 0, 1, 2. It can be:
- Fractional (e.g., 0.5) — common in complex reaction mechanisms
- Negative (e.g., -1) — increasing that reactant actually slows the reaction
Rate Law Comparison Table
| Order | Rate Law | Integrated Law | Half-Life | Unit of k |
|---|---|---|---|---|
| Zero | k[A]⁰ | [A] = [A]₀ - kt | t½ = [A]₀/2k | M/s |
| First | k[A]¹ | ln[A] = ln[A]₀ - kt | t½ = ln(2)/k | s⁻¹ |
| Second | k[A]² | 1/[A] = 1/[A]₀ + kt | t½ = 1/k[A]₀ | M⁻¹s⁻¹ |
How to Determine Rate Law Experimentally
You can't just look at the balanced equation. You need data. Here's how:
The Initial Rates Method
Run multiple experiments with different initial concentrations. Measure the initial rate for each. Then compare.
Example: Suppose you're studying the reaction: 2NO + H₂ → N₂O + H₂O
You run three trials:
| Trial | [NO] | [H₂] | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 0.10 | 0.10 | 1.0 × 10⁻³ |
| 2 | 0.20 | 0.10 | 4.0 × 10⁻³ |
| 3 | 0.20 | 0.20 | 4.0 × 10⁻³ |
Step 1: Compare trials 1 and 2 (double [NO], keep [H₂] constant)
[NO] doubles, rate quadruples. NO is second order with respect to NO.
Step 2: Compare trials 2 and 3 (double [H₂], keep [NO] constant)
[H₂] doubles, rate stays the same. H₂ is zero order with respect to H₂.
Result: rate = k[NO]²[H₂]⁰ = k[NO]²
Getting Started: Solving Rate Law Problems
Here's your step-by-step approach:
- Identify what you know — list the given concentrations and rates
- Pick two experiments where only one concentration changed
- Divide the rate equations to find the order for each reactant
- Solve for k using one complete data set
- Write the complete rate law with all exponents and the k value
Don't try to skip steps. The "divide and compare" method works every time because it isolates one variable at a time.
Units of the Rate Constant
The units of k depend on the overall reaction order. They ensure the rate comes out in the right units (M/s).
- Zero order: M/s
- First order: s⁻¹
- Second order: M⁻¹s⁻¹
- Third order: M⁻²s⁻¹
General rule: units = M1-ns⁻¹ where n is the overall order.
Temperature and the Rate Constant
The Arrhenius equation shows how k changes with temperature:
k = Ae-Ea/RT
- A = frequency factor (collision frequency and orientation)
- Ea = activation energy
- R = gas constant (8.314 J/mol·K)
- T = temperature in Kelvin
Higher temperature = larger k = faster reaction. This is why heating speeds up chemical reactions. The exponential term dominates — small temperature increases cause large changes in k.
Common Mistakes to Avoid
- Stoichiometry isn't order — never assume m = a in the balanced equation
- Units matter — wrong units mean wrong answers on exams
- Order isn't always integer — watch for 0.5, 1.5, or negative values
- Half-life depends on initial concentration for second order — not constant like first order
Bottom Line
Rate law is experimental. The exponents come from data, not equations. The rate constant k varies with temperature. Master the initial rates method, and you can solve any rate law problem. Everything else is just applying that method to different numbers.