Integrated Rate Law Calculator

The Integrated Rate Law Calculator helps chemists and students determine the concentration of reactants over time in chemical reactions. This tool applies the integrated rate law formula to provide accurate results based on your input parameters.

What is Integrated Rate Law?

The integrated rate law describes how the concentration of reactants changes over time in a chemical reaction. It's derived from the differential rate law and provides a mathematical relationship between reactant concentration and time.

There are several forms of the integrated rate law depending on the reaction order:

Zero-order reactions: [A] = -kt + [A]₀
First-order reactions: ln[A] = -kt + ln[A]₀
Second-order reactions: 1/[A] = kt + 1/[A]₀

These equations are essential for predicting reaction progress and determining reaction rates.

Formula

The general form of the integrated rate law is:

[A] = [A]₀ - kt (for zero-order reactions)

ln[A] = -kt + ln[A]₀ (for first-order reactions)

1/[A] = kt + 1/[A]₀ (for second-order reactions)

Where:

[A] = concentration of reactant A at time t
[A]₀ = initial concentration of reactant A
k = rate constant
t = time

These equations allow chemists to model reaction progress and determine how quickly a reaction occurs.

How to Use the Calculator

Select the reaction order (zero, first, or second order)
Enter the initial concentration of the reactant ([A]₀)
Enter the rate constant (k)
Enter the time (t) at which you want to calculate the concentration
Click "Calculate" to see the result

The calculator will display the concentration of the reactant at the specified time and show a graph of the reaction progress over time.

Example Calculation

First-order reaction example

Given:

Initial concentration [A]₀ = 0.5 M
Rate constant k = 0.15 s⁻¹
Time t = 10 s

Using the first-order integrated rate law:

ln[A] = -kt + ln[A]₀

ln[A] = -0.15 × 10 + ln(0.5)

ln[A] = -1.5 + (-0.693)

ln[A] = -2.193

[A] = e⁻².¹⁹³ ≈ 0.11 M

This example shows how the concentration decreases over time in a first-order reaction.

Interpreting Results

The calculator provides several key pieces of information:

The concentration of the reactant at the specified time
A graph showing how the concentration changes over time
The half-life of the reaction (for first-order reactions)

Understanding these results helps chemists predict reaction progress, design experiments, and optimize reaction conditions.

FAQ

What is the difference between differential and integrated rate laws?

The differential rate law describes how the rate of reaction changes with reactant concentration, while the integrated rate law shows how the concentration of reactants changes over time.

Can I use this calculator for any reaction order?

Yes, the calculator supports zero-order, first-order, and second-order reactions. Simply select the appropriate reaction order before performing calculations.

What units should I use for the rate constant?

The units for the rate constant depend on the reaction order. For first-order reactions, the rate constant is typically in units of s⁻¹, while for second-order reactions it's in units of M⁻¹s⁻¹.