Reaction Order Formula
Determine how concentration affects reaction rate through experimental methods
Rate Law & Order
Rate = k[A]m[B]n
m = order with respect to A
n = order with respect to B
m + n = overall reaction order
Orders are determined EXPERIMENTALLY, not from stoichiometry!
Method 1: Method of Initial Rates
Procedure
- Measure initial rate for different initial concentrations
- Compare trials where only ONE reactant concentration changes
- Calculate ratio: Rate₂/Rate₁ = ([A]₂/[A]₁)m
- Solve for m (order with respect to A)
- Repeat for other reactants
Order Determination
| If [A] doubles and... | Then order m = | Math |
|---|---|---|
| Rate stays same | 0 (zero order) | 20 = 1 |
| Rate doubles | 1 (first order) | 21 = 2 |
| Rate quadruples | 2 (second order) | 22 = 4 |
| Rate increases 8× | 3 (third order) | 23 = 8 |
Method 2: Graphical Analysis
Zero Order
Linear plot:
[A] vs t
Slope = -k
First Order
Linear plot:
ln[A] vs t
Slope = -k
Second Order
Linear plot:
1/[A] vs t
Slope = k
Worked Examples
Example 1: Method of Initial Rates
Reaction: 2NO(g) + O₂(g) → 2NO₂(g)
| Trial | [NO]₀ (M) | [O₂]₀ (M) | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 0.010 | 0.010 | 2.5×10⁻⁵ |
| 2 | 0.020 | 0.010 | 1.0×10⁻⁴ |
| 3 | 0.010 | 0.020 | 5.0×10⁻⁵ |
Solution:
Find order with respect to NO (compare trials 1 & 2):
Rate₂/Rate₁ = (1.0×10⁻⁴)/(2.5×10⁻⁵) = 4
[NO]₂/[NO]₁ = 0.020/0.010 = 2
4 = 2m → m = 2
Order with respect to NO = 2
Find order with respect to O₂ (compare trials 1 & 3):
Rate₃/Rate₁ = (5.0×10⁻⁵)/(2.5×10⁻⁵) = 2
[O₂]₃/[O₂]₁ = 0.020/0.010 = 2
2 = 2n → n = 1
Order with respect to O₂ = 1
Overall:
Rate = k[NO]²[O₂]
Overall order = 2 + 1 = 3 (third order)
Example 2: Calculate Rate Constant
Problem: Using data from Example 1, calculate k.
Solution:
Rate = k[NO]²[O₂]
Using Trial 1:
2.5×10⁻⁵ = k(0.010)²(0.010)
2.5×10⁻⁵ = k(1.0×10⁻⁶)
k = 2.5×10⁻⁵ / 1.0×10⁻⁶
k = 25 M⁻²s⁻¹
Units: M⁻²s⁻¹ for third order (overall order 3)
Example 3: Fractional Orders
Problem: When [A] triples, rate increases by factor of 5.2. Find order.
Solution:
Rate₂/Rate₁ = ([A]₂/[A]₁)m
5.2 = 3m
log(5.2) = m × log(3)
0.716 = m × 0.477
m = 0.716 / 0.477
m = 1.5 (order = 3/2)
Fractional orders are possible! Often indicates complex mechanisms.
Example 4: Half-Life Method
Problem: For decomposition of A, t1/2 is constant = 15.0 min. What is the order?
Solution:
Compare half-life behavior:
- Zero order: t1/2 ∝ [A]₀ (decreases with [A])
- First order: t1/2 = constant ✓
- Second order: t1/2 ∝ 1/[A]₀ (increases as [A] decreases)
First order reaction
k = 0.693/t1/2 = 0.693/15.0 = 0.0462 min⁻¹
Common Mistakes
Using Stoichiometric Coefficients
Order ≠ coefficient! For 2NO + O₂ → products, rate ≠ k[NO]²[O₂]. Must determine experimentally!
Comparing Wrong Trials
Only compare trials where ONE concentration changes and others stay constant!
Ignoring Units of k
Units depend on overall order: 0th (M/s), 1st (s⁻¹), 2nd (M⁻¹s⁻¹), 3rd (M⁻²s⁻¹)
Negative Orders Possible
If rate decreases when [A] increases, order can be negative! Example: inhibition.