Derivation and Integrated Rate Law for Third Order Reactions
The Third Order Reaction in chemical kinetics describes a situation where the rate depends on the concentration of three reactant molecules or one reactant raised to the power of three. Understanding this rare kinetic order is crucial for JEE as it brings together mathematical derivation, graphical representation, units, and application in numericals. Most reactions of this order are theoretical, but knowing the principle strengthens your grasp of reaction mechanisms that occasionally appear in exam questions.
What is a Third Order Reaction?
A Third Order Reaction is defined by a rate law where the sum of the concentration exponents in the rate equation equals three. This means that the reaction rate depends either on the concentration of a single reactant cubed, the product of a first order and a second order reactant, or three distinct reactants each with exponent one. Such reactions are expressed as follows:
- Rate = k[A]3 (single reactant raised to three)
- Rate = k[A]2[B] (one squared, one first power)
- Rate = k[A][B][C] (three different reactants, each first order)
These configurations indicate the minimum number of simultaneous molecule collisions required—generally three. In reality, termolecular elementary reactions are extremely rare due to the low probability of three particles colliding at once with proper orientation and energy.
General Rate Law and Integrated Rate Equation
For a simple third order reaction of the type A → Products (all same concentration), the differential rate law is:
Rate = -d[A]/dt = k[A]3
To derive the integrated rate law, rearrange and integrate:
- Separate variables: d[A]/[A]3 = -k dt
- Integrate both sides: ∫ d[A]/[A]3 = -k ∫ dt
- This gives: -1/(2[A]2) = -kt + C
- On applying initial concentration [A]0 at t = 0:
1/(2[A]2) = kt + 1/(2[A]02)
Or, more compactly, for easier calculation in numericals:
1/[A]2 - 1/[A]02 = 2kt
Third Order Reaction: Units of Rate Constant
In a third order reaction, the unit of rate constant, k, can be found from the basic rate law:
- Rate = k [A]3
- SI unit of rate: mol L-1 s-1
- [A]3: (mol L-1)3 = mol3 L-3
Thus, k = Rate / [A]3 = (mol L-1 s-1) ÷ (mol3 L-3) = L2 mol-2 s-1.
Types and Examples of Third Order Reactions
Third order kinetics may follow different reactant concentration patterns. Common cases (relevant for JEE Main numericals) are:
- All Reactants Same:
- A + A + A → Products
- Rate = k[A]3
- Two Reactants Same, One Different:
- A + A + B → Products
- Rate = k[A]2[B]
- All Reactants Different:
- A + B + C → Products
- Rate = k[A][B][C]
A JEE-relevant real reaction is: 2 NO + Cl2 → 2 NOCl, with Rate = k[NO]2[Cl2], total order 2 + 1 = 3.
Compare with: 2 NO + O2 → 2 NO2, Rate = k[NO]2[O2]. Most textbook examples of third order kinetics are a combination of first and second order reactants.
Graphical Representation of Third Order Reactions
Graphing helps distinguish orders visually. For a third order reaction (A → Products, same initial concentrations):
- Time (t) vs. [A]: curve falls sharply, not exponential
- Plot of 1/[A]2 vs. t: gives a straight line, slope = 2k
This linearity aids in verifying the third order experimentally. Other orders (first, second) show distinct straight-line plots using appropriate transformations.
Half-Life of Third Order Reactions
The half-life (t1/2) is the time for [A]0 to drop to [A]0/2. For third order reactions:
t1/2 = 3 / (2 k [A]02)
Unlike first order reactions (where half-life is concentration independent), for third order, half-life is inversely proportional to the square of the initial concentration—so initial concentration increases lead to a dramatic drop in half-life.
Why Are Third Order Reactions Rare?
Third order reactions are rare due to the improbability of three particles colliding simultaneously with the required orientation and energy. Most observed “third order” reactions are actually a series of steps where one is rate determining, or a composite of lower order processes. In exams, third order examples are often derived from overall rate laws rather than actual elementary steps.
Comparison of Reaction Orders
| Order | Rate Law Form | Units of k | Half-Life (t1/2) |
|---|---|---|---|
| Zero | Rate = k | mol L-1 s-1 | [A]0 / 2k |
| First | Rate = k[A] | s-1 | 0.693 / k |
| Second | Rate = k[A]2 | L mol-1 s-1 | 1 / (k [A]0) |
| Third | Rate = k[A]3 | L2 mol-2 s-1 | 3 / (2 k [A]02) |
Understanding these differences is essential for JEE. Compare more on foundational reaction distinctions.
Numerical Example and Application
Example: If for a reaction 3A → Products, the initial concentration of A is 0.600 mol/L and after 50 seconds, the concentration drops to 0.300 mol/L, calculate the value of k.
Given:
- [A]0 = 0.600 mol/L
- [A] = 0.300 mol/L after t = 50 s
Use: 1/[A]2 - 1/[A]02 = 2kt
- 1/(0.300)2 - 1/(0.600)2 = 2k × 50
- 1/0.09 - 1/0.36 = 100k
- 11.11 - 2.78 = 100k
- 8.33 = 100k ⇒ k = 0.0833 L2 mol-2 s-1
Key Practice Points and JEE Tips
- Always check order by sum of exponents in the experimental rate law.
- Third order half-life sharply decreases as initial concentration increases.
- Look for rare third order exam numericals for speed and graph-linearity.
- Third order mechanisms are almost never elementary; actual steps can involve lower order kinetics.
For more on reaction orders, practice with chemical kinetics mock tests. Deep-dive into related topics like zero order reactions or first order kinetics, or integrated rate laws for further clarity. Vedantu regularly updates its JEE Main resources to align with exam trends, supporting student mastery at every stage.
FAQs on Third Order Reaction: Concept, Formula, and Examples
1. What is a third order reaction?
A third order reaction is a chemical reaction whose rate depends on the concentration of three reactants, or one reactant raised to the third power.
Key features of third order reactions include:
- The rate law is typically Rate = k[A]3 or k[A][B][C]
- The overall order of reaction is 3 (sum of powers in rate law)
- Such reactions are rare due to the low probability of three particles colliding simultaneously
2. What is the rate law for a third order reaction?
The rate law for a third order reaction states how the reaction rate depends on the concentration(s) of the reactant(s):
- If a single reactant: Rate = k[A]3
- If three different reactants: Rate = k[A][B][C]
3. Why are third order reactions rare?
Third order reactions are rare because simultaneous collisions between three reactant molecules are highly improbable.
Reasons include:
- The probability of three-particle collisions occurring at the same instant is extremely low
- Most observed third order reactions are actually the result of multi-step mechanisms (not elementary)
- Third order kinetics are uncommon in practical and biological systems
4. How do you determine the order of a reaction?
The order of a chemical reaction is found by summing the powers to which concentration terms are raised in the rate law.
To determine order:
- Experimentally measure the rate as you change the concentration of each reactant
- Fit data to the rate equation
- Total the exponents in the rate law (e.g., if Rate = k[A]2[B], the order is 2 + 1 = 3)
5. What is the integrated rate law for a third order reaction?
The integrated rate law for a third order reaction (with only one reactant) is:
1 / (2[A]2) = kt + 1 / (2[A]02)
Where:
- [A] = concentration at time t
- [A]0 = initial concentration
- k = rate constant
- t = time elapsed
6. How do you calculate the units of the rate constant for a third order reaction?
For a third order reaction, the units of the rate constant k are always mol-2 L2 s-1 in SI units.
Calculation:
- General formula: (mol L-1)1–n s-1
- For n = 3: (mol L-1)1–3 s-1 = (mol L-1)-2 s-1
- Which gives: L2 mol-2 s-1
7. Give an example of a third order reaction.
A classic example of a third order reaction is the reaction between nitrogen dioxide and carbon monoxide, but most true third order reactions are rare.
General Example:
- 3A → Products (rate law: Rate = k[A]3)
8. How does the half-life of a third order reaction depend on concentration?
The half-life (t1/2) of a third order reaction is inversely proportional to the square of the initial concentration.
Specifically:
- t1/2 = 3 / (2k[A]02)
- If initial concentration doubles, half-life becomes one-fourth
9. What is the graphical representation of a third order reaction?
The graph for a third order reaction (single reactant) is obtained by plotting 1 / [A]2 versus time (t), which yields a straight line.
Key points:
- Slope = 2k
- Y-intercept = 1 / [A]02
10. How does a third order reaction differ from first and second order reactions?
Third order reactions differ from first and second order reactions in their rate law, units, half-life dependence, and rarity.
Major differences:
- First order: Rate depends linearly on concentration (Rate = k[A]); half-life is constant
- Second order: Rate = k[A]2 or k[A][B]; half-life inversely proportional to initial concentration
- Third order: Rate = k[A]3 or similar; half-life inversely proportional to square of initial concentration; rare in nature
