How to Calculate Rate Constant for Second Order Reaction
Understanding chemical kinetics is crucial for predicting how fast a chemical reaction proceeds. For a second-order reaction, the rate of reaction depends either on the square of the concentration of one reactant or on the product of the concentrations of two different reactants.
This calculator specifically solves for the rate constant (k) using the integrated rate law for a second-order reaction involving a single reactant type (or two reactants with equal initial concentrations).
The Second Order Integrated Rate Law Formula
The differential rate law for a second-order reaction is expressed as:
By integrating this equation from time 0 to time t, we derive the linear form of the integrated rate law:
Where:
- [A]ₜ = Concentration of reactant at time t (M)
- [A]₀ = Initial concentration of reactant (M)
- k = Rate constant (M⁻¹s⁻¹)
- t = Time elapsed
Rearranging to Solve for k
To use this calculator manually, you can rearrange the formula to isolate the rate constant k:
Units of the Rate Constant
Unlike first-order reactions where k has units of inverse time (s⁻¹), the units for a second-order rate constant depend on concentration. The standard unit is:
- L · mol⁻¹ · s⁻¹ (Liters per mole per second)
- Often written as M⁻¹s⁻¹
Note: If you input time in minutes, the unit becomes M⁻¹min⁻¹.
Example Calculation
Let's say nitrogen dioxide (NO₂) decomposes in a second-order reaction:
- Initial Concentration [A]₀: 0.0075 M
- Final Concentration [A]ₜ: 0.0025 M after 120 seconds.
Step 1: Calculate the inverse of concentrations.
1 / 0.0025 = 400
1 / 0.0075 = 133.33
Step 2: Subtract the inverse initial from the inverse final.
400 – 133.33 = 266.67
Step 3: Divide by time (120 s).
k = 266.67 / 120 ≈ 2.22 M⁻¹s⁻¹
Why is this important?
Calculating the rate constant allows chemists to determine the half-life of a reaction, which for second-order reactions changes as the concentration decreases. The formula for the half-life of a second-order reaction is t₁/₂ = 1 / (k[A]₀).