First Order Integrated Rate Law Calculator

The First Order Integrated Rate Law Calculator helps you determine the concentration of a reactant at any time during a first-order chemical reaction. This tool is essential for chemistry students, researchers, and professionals working with reaction kinetics.

What is First Order Rate Law?

First order rate laws describe reactions where the rate of reaction is directly proportional to the concentration of one reactant. This type of reaction follows the integrated rate law equation:

First Order Rate Law Formula

Rate = k × [A]

Where:

Rate = Reaction rate (mol/L·s)
k = Rate constant (s⁻¹)
[A] = Concentration of reactant A (mol/L)

First order reactions are common in chemical kinetics and have several characteristic features:

Half-life is constant and independent of initial concentration
Graph of ln[A] vs time is linear
Rate depends only on the concentration of one reactant

Integrated Rate Law Formula

The integrated form of the first order rate law allows calculation of reactant concentration at any time:

Integrated Rate Law Equation

ln([A]ₜ / [A]₀) = -kt

Where:

[A]ₜ = Concentration at time t
[A]₀ = Initial concentration
k = Rate constant
t = Time

This equation can be rearranged to solve for [A]ₜ:

Solved Form

[A]ₜ = [A]₀ × e^-kt

The integrated rate law is particularly useful for predicting reaction progress over time and determining half-life.

How to Use This Calculator

Enter the initial concentration of your reactant in mol/L
Input the rate constant (k) in s⁻¹
Specify the time elapsed in seconds
Click "Calculate" to see the remaining concentration
View the result and chart showing concentration over time

Assumptions

This calculator assumes:

First order reaction conditions
Constant temperature
No side reactions
No catalyst changes

Example Calculation

Let's calculate the remaining concentration of a reactant after 100 seconds with:

Initial concentration [A]₀ = 0.5 mol/L
Rate constant k = 0.02 s⁻¹
Time t = 100 s

Using the formula:

[A]ₜ = 0.5 × e^{-(0.02 × 100)} = 0.5 × e^-2 ≈ 0.5 × 0.135 ≈ 0.0675 mol/L

After 100 seconds, approximately 0.0675 mol/L of the reactant remains.

Interpreting Results

The calculator provides several key outputs:

Current concentration at specified time
Percentage of reactant remaining
Visual chart showing concentration over time
Half-life calculation (t₁/₂ = ln(2)/k)

These results help you understand:

How quickly the reaction is proceeding
When significant amounts of reactant remain
Potential reaction completion time

Frequently Asked Questions

What is the difference between rate law and integrated rate law?

The rate law describes how the reaction rate depends on concentrations, while the integrated rate law shows how concentrations change over time. The integrated form allows you to calculate actual concentrations at specific times.

How do I determine if a reaction is first order?

A reaction is first order if a plot of ln[A] vs time is linear. You can verify this by checking if the rate depends only on one reactant's concentration.

What units should I use for the rate constant?

The rate constant (k) should be in units of s⁻¹ (seconds to the power of -1) for time in seconds. Adjust units accordingly if using different time units.

Can this calculator handle reactions with multiple reactants?

No, this calculator is specifically for first order reactions with a single reactant. For more complex reactions, you would need a different approach or calculator.